math.jsonl

{"id": "mat_0001", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is a direct proof?", "choices": null, "answer": "A direct proof assumes the premises and derives the conclusion step by step, so each line follows logically from earlier ones.", "steps": ["A direct proof assumes the premises and derives the conclusion step by step, so each line follows logically from earlier ones.", "So the answer stands: A direct proof assumes the premises and derives the conclusion step by step, so each line follows logically from earlier ones."], "text": "Answer: A direct proof assumes the premises and derives the conclusion step by step, so each line follows logically from earlier ones.\nSteps:\n1. A direct proof assumes the premises and derives the conclusion step by step, so each line follows logically from earlier ones.\n2. So the answer stands: A direct proof assumes the premises and derives the conclusion step by step, so each line follows logically from earlier ones."}
{"id": "mat_0002", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 2x² + 1x, compute f'(2) step by step.", "choices": null, "answer": "f'(2) = 9.", "steps": ["f'(x) = 4x + 1.", "Plug x = 2.", "4×2 + 1 = 9."], "text": "Answer: f'(2) = 9.\nSteps:\n1. f'(x) = 4x + 1.\n2. Plug x = 2.\n3. 4×2 + 1 = 9."}
{"id": "mat_0003", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej det [[2,1],[2,3]] krok za krokem.", "choices": null, "answer": "Determinant: 4.", "steps": ["ad − bc.", "2×3 − 1×2.", "= 4."], "text": "Answer: Determinant: 4.\nSteps:\n1. ad − bc.\n2. 2×3 − 1×2.\n3. = 4."}
{"id": "mat_0004", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is probability?", "choices": null, "answer": "Probability measures chance from 0 to 1, with axioms guaranteeing consistent reasoning about uncertainty.", "steps": ["Probability measures chance from 0 to 1, with axioms guaranteeing consistent reasoning about uncertainty.", "So the answer stands: Probability measures chance from 0 to 1, with axioms guaranteeing consistent reasoning about uncertainty."], "text": "Answer: Probability measures chance from 0 to 1, with axioms guaranteeing consistent reasoning about uncertainty.\nSteps:\n1. Probability measures chance from 0 to 1, with axioms guaranteeing consistent reasoning about uncertainty.\n2. So the answer stands: Probability measures chance from 0 to 1, with axioms guaranteeing consistent reasoning about uncertainty."}
{"id": "mat_0005", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is divisibility?", "choices": null, "answer": "a divides b if some integer k gives b = ak, the root relation of number theory.", "steps": ["a divides b if some integer k gives b = ak, the root relation of number theory.", "So the answer stands: a divides b if some integer k gives b = ak, the root relation of number theory."], "text": "Answer: a divides b if some integer k gives b = ak, the root relation of number theory.\nSteps:\n1. a divides b if some integer k gives b = ak, the root relation of number theory.\n2. So the answer stands: a divides b if some integer k gives b = ak, the root relation of number theory."}
{"id": "mat_0006", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(4,2).", "choices": {"A": "6", "B": "2", "C": "10", "D": "14"}, "answer": "C(4,2) is 6.", "steps": ["4×3/2.", "= 6.", "Select option A."], "text": "Answer: 6\nSteps:\n1. 4×3/2.\n2. = 6.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0007", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 6.", "choices": null, "answer": "Sum: 21.", "steps": ["Pair 1+6, 2+5, ...", "Use 6×7/2.", "= 21."], "text": "Answer: Sum: 21.\nSteps:\n1. Pair 1+6, 2+5, ...\n2. Use 6×7/2.\n3. = 21."}
{"id": "mat_0008", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 3.", "choices": null, "answer": "Integral: 27.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 3.", "3³ − 0 = 27."], "text": "Answer: Integral: 27.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 3.\n3. 3³ − 0 = 27."}
{"id": "mat_0009", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[2,1],[2,3]] step by step.", "choices": null, "answer": "Determinant: 4.", "steps": ["ad − bc.", "2×3 − 1×2.", "= 4."], "text": "Answer: Determinant: 4.\nSteps:\n1. ad − bc.\n2. 2×3 − 1×2.\n3. = 4."}
{"id": "mat_0010", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 6-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/6.", "steps": ["Favourable: 1.", "Possible: 6.", "P = 1/6."], "text": "Answer: P = 1/6.\nSteps:\n1. Favourable: 1.\n2. Possible: 6.\n3. P = 1/6."}
{"id": "mat_0011", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why study primes?", "choices": null, "answer": "Primes are the atoms of multiplication, since every integer factors uniquely into primes.", "steps": ["Primes are the atoms of multiplication, since every integer factors uniquely into primes.", "So the answer stands: Primes are the atoms of multiplication, since every integer factors uniquely into primes."], "text": "Answer: Primes are the atoms of multiplication, since every integer factors uniquely into primes.\nSteps:\n1. Primes are the atoms of multiplication, since every integer factors uniquely into primes.\n2. So the answer stands: Primes are the atoms of multiplication, since every integer factors uniquely into primes."}
{"id": "mat_0012", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(5,2).", "choices": null, "answer": "C(5,2) = 10.", "steps": ["Formula n(n−1)/2.", "5×4/2.", "= 10."], "text": "Answer: C(5,2) = 10.\nSteps:\n1. Formula n(n−1)/2.\n2. 5×4/2.\n3. = 10."}
{"id": "mat_0013", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 2 by computing its half.", "choices": null, "answer": "Half: 3, so 6 is even.", "steps": ["2² + 2 = 6.", "6 / 2 = 3.", "Integer, so even."], "text": "Answer: Half: 3, so 6 is even.\nSteps:\n1. 2² + 2 = 6.\n2. 6 / 2 = 3.\n3. Integer, so even."}
{"id": "mat_0014", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 3²)/(x − 3) as x → 3.", "choices": null, "answer": "Limit: 6.", "steps": ["Factor: (x−3)(x+3).", "Cancel (x−3).", "Plug 3: 3+3 = 6."], "text": "Answer: Limit: 6.\nSteps:\n1. Factor: (x−3)(x+3).\n2. Cancel (x−3).\n3. Plug 3: 3+3 = 6."}
{"id": "mat_0015", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[3,0],[0,4]].", "choices": null, "answer": "Trace: 7.", "steps": ["Sum diagonal.", "3 + 4.", "= 7."], "text": "Answer: Trace: 7.\nSteps:\n1. Sum diagonal.\n2. 3 + 4.\n3. = 7."}
{"id": "mat_0016", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..5}.", "choices": null, "answer": "E = 3.", "steps": ["Mean of 1..5.", "(5+1)/2.", "E = 3."], "text": "Answer: E = 3.\nSteps:\n1. Mean of 1..5.\n2. (5+1)/2.\n3. E = 3."}
{"id": "mat_0017", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Najdi největšího společného dělitele NSD(12, 18).", "choices": {"A": "6", "B": "3", "C": "9", "D": "12"}, "answer": "NSD je 6.", "steps": ["Vytkni 3 dvakrát.", "NSD = 6.", "Vyber možnost A."], "text": "Answer: 6\nSteps:\n1. Vytkni 3 dvakrát.\n2. NSD = 6.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0018", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(4,3).", "choices": null, "answer": "P(4,3) = 24.", "steps": ["n(n−1)(n−2).", "4×3×2.", "= 24."], "text": "Answer: P(4,3) = 24.\nSteps:\n1. n(n−1)(n−2).\n2. 4×3×2.\n3. = 24."}
{"id": "mat_0019", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 3 boolean variables. How many rows?", "choices": null, "answer": "Rows: 8.", "steps": ["2 choices each.", "2^3 = 8."], "text": "Answer: Rows: 8.\nSteps:\n1. 2 choices each.\n2. 2^3 = 8."}
{"id": "mat_0020", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is a limit?", "choices": null, "answer": "A limit is the value a function approaches as inputs near a point, even if the function is undefined there.", "steps": ["A limit is the value a function approaches as inputs near a point, even if the function is undefined there.", "So the answer stands: A limit is the value a function approaches as inputs near a point, even if the function is undefined there."], "text": "Answer: A limit is the value a function approaches as inputs near a point, even if the function is undefined there.\nSteps:\n1. A limit is the value a function approaches as inputs near a point, even if the function is undefined there.\n2. So the answer stands: A limit is the value a function approaches as inputs near a point, even if the function is undefined there."}
{"id": "mat_0021", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[1,2],[3,4]].", "choices": {"A": "-2", "B": "-5", "C": "1", "D": "4"}, "answer": "The determinant is -2.", "steps": ["1×4 − 2×3.", "Det = -2.", "Select option A."], "text": "Answer: -2\nSteps:\n1. 1×4 − 2×3.\n2. Det = -2.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0022", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why use sample spaces?", "choices": null, "answer": "Sample spaces list all outcomes, so probabilities attach to a complete well-defined set.", "steps": ["Sample spaces list all outcomes, so probabilities attach to a complete well-defined set.", "So the answer stands: Sample spaces list all outcomes, so probabilities attach to a complete well-defined set."], "text": "Answer: Sample spaces list all outcomes, so probabilities attach to a complete well-defined set.\nSteps:\n1. Sample spaces list all outcomes, so probabilities attach to a complete well-defined set.\n2. So the answer stands: Sample spaces list all outcomes, so probabilities attach to a complete well-defined set."}
{"id": "mat_0023", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(12, 18).", "choices": {"A": "6", "B": "3", "C": "9", "D": "12"}, "answer": "The gcd is 6.", "steps": ["Factor 3 out twice.", "gcd = 6.", "Select option A."], "text": "Answer: 6\nSteps:\n1. Factor 3 out twice.\n2. gcd = 6.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0024", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is combinatorics?", "choices": null, "answer": "Combinatorics counts arrangements and selections, turning listing questions into formulas.", "steps": ["Combinatorics counts arrangements and selections, turning listing questions into formulas.", "So the answer stands: Combinatorics counts arrangements and selections, turning listing questions into formulas."], "text": "Answer: Combinatorics counts arrangements and selections, turning listing questions into formulas.\nSteps:\n1. Combinatorics counts arrangements and selections, turning listing questions into formulas.\n2. So the answer stands: Combinatorics counts arrangements and selections, turning listing questions into formulas."}
{"id": "mat_0025", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 7.", "choices": null, "answer": "Sum: 28.", "steps": ["Pair 1+7, 2+6, ...", "Use 7×8/2.", "= 28."], "text": "Answer: Sum: 28.\nSteps:\n1. Pair 1+7, 2+6, ...\n2. Use 7×8/2.\n3. = 28."}
{"id": "mat_0026", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Pro f(x) = 1x² + 2x vypočítej f'(1).", "choices": {"A": "4", "B": "2", "C": "6", "D": "8"}, "answer": "Derivace je 4.", "steps": ["f'(x) = 2x + 2.", "V 1: 4.", "Vyber možnost A."], "text": "Answer: 4\nSteps:\n1. f'(x) = 2x + 2.\n2. V 1: 4.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0027", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[2,1],[2,5]].", "choices": {"A": "5", "B": "7", "C": "9", "D": "11"}, "answer": "The trace is 7.", "steps": ["2 + 5.", "Trace = 7.", "Select option B."], "text": "Answer: 7\nSteps:\n1. 2 + 5.\n2. Trace = 7.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0028", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is conditional probability?", "choices": null, "answer": "It updates chance given new evidence via P(A|B), the core of learning from data.", "steps": ["It updates chance given new evidence via P(A|B), the core of learning from data.", "So the answer stands: It updates chance given new evidence via P(A|B), the core of learning from data."], "text": "Answer: It updates chance given new evidence via P(A|B), the core of learning from data.\nSteps:\n1. It updates chance given new evidence via P(A|B), the core of learning from data.\n2. So the answer stands: It updates chance given new evidence via P(A|B), the core of learning from data."}
{"id": "mat_0029", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej NSD(8, 12) krok za krokem.", "choices": null, "answer": "NSD = 4.", "steps": ["Rozlož: 8=4×2, 12=6×2.", "Společné: 2×NSD(4,6).", "= 2×2 = 4."], "text": "Answer: NSD = 4.\nSteps:\n1. Rozlož: 8=4×2, 12=6×2.\n2. Společné: 2×NSD(4,6).\n3. = 2×2 = 4."}
{"id": "mat_0030", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why factorials?", "choices": null, "answer": "Factorials count orderings of distinct objects, since each slot has one fewer choice.", "steps": ["Factorials count orderings of distinct objects, since each slot has one fewer choice.", "So the answer stands: Factorials count orderings of distinct objects, since each slot has one fewer choice."], "text": "Answer: Factorials count orderings of distinct objects, since each slot has one fewer choice.\nSteps:\n1. Factorials count orderings of distinct objects, since each slot has one fewer choice.\n2. So the answer stands: Factorials count orderings of distinct objects, since each slot has one fewer choice."}
{"id": "mat_0031", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why use proof by contradiction?", "choices": null, "answer": "Contradiction assumes the claim is false and derives nonsense, so the original claim must be true.", "steps": ["Contradiction assumes the claim is false and derives nonsense, so the original claim must be true.", "So the answer stands: Contradiction assumes the claim is false and derives nonsense, so the original claim must be true."], "text": "Answer: Contradiction assumes the claim is false and derives nonsense, so the original claim must be true.\nSteps:\n1. Contradiction assumes the claim is false and derives nonsense, so the original claim must be true.\n2. So the answer stands: Contradiction assumes the claim is false and derives nonsense, so the original claim must be true."}
{"id": "mat_0032", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why study continuity?", "choices": null, "answer": "Continuous functions have no jumps, so small input changes give small output changes everywhere.", "steps": ["Continuous functions have no jumps, so small input changes give small output changes everywhere.", "So the answer stands: Continuous functions have no jumps, so small input changes give small output changes everywhere."], "text": "Answer: Continuous functions have no jumps, so small input changes give small output changes everywhere.\nSteps:\n1. Continuous functions have no jumps, so small input changes give small output changes everywhere.\n2. So the answer stands: Continuous functions have no jumps, so small input changes give small output changes everywhere."}
{"id": "mat_0033", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is a vector?", "choices": null, "answer": "A vector is a quantity with direction and size, added component-wise and scaled by numbers.", "steps": ["A vector is a quantity with direction and size, added component-wise and scaled by numbers.", "So the answer stands: A vector is a quantity with direction and size, added component-wise and scaled by numbers."], "text": "Answer: A vector is a quantity with direction and size, added component-wise and scaled by numbers.\nSteps:\n1. A vector is a quantity with direction and size, added component-wise and scaled by numbers.\n2. So the answer stands: A vector is a quantity with direction and size, added component-wise and scaled by numbers."}
{"id": "mat_0034", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 6-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/6", "B": "1/5", "C": "1/7", "D": "2/6"}, "answer": "The probability is 1/6.", "steps": ["1 face of 6.", "P = 1/6.", "Select option A."], "text": "Answer: 1/6\nSteps:\n1. 1 face of 6.\n2. P = 1/6.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0035", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is the fundamental theorem of arithmetic?", "choices": null, "answer": "Every integer above 1 factors uniquely into primes, grounding all of arithmetic.", "steps": ["Every integer above 1 factors uniquely into primes, grounding all of arithmetic.", "So the answer stands: Every integer above 1 factors uniquely into primes, grounding all of arithmetic."], "text": "Answer: Every integer above 1 factors uniquely into primes, grounding all of arithmetic.\nSteps:\n1. Every integer above 1 factors uniquely into primes, grounding all of arithmetic.\n2. So the answer stands: Every integer above 1 factors uniquely into primes, grounding all of arithmetic."}
{"id": "mat_0036", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(6,3).", "choices": null, "answer": "C(6,3) = 20.", "steps": ["n(n−1)(n−2)/6.", "6×5×4/6.", "= 20."], "text": "Answer: C(6,3) = 20.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 6×5×4/6.\n3. = 20."}
{"id": "mat_0037", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 2 variables have?", "choices": {"A": "4", "B": "2", "C": "6", "D": "8"}, "answer": "The table has 4 rows.", "steps": ["2^2 rows.", "Total: 4.", "Select option A."], "text": "Answer: 4\nSteps:\n1. 2^2 rows.\n2. Total: 4.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0038", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 3x² + 2x, compute f'(3) step by step.", "choices": null, "answer": "f'(3) = 20.", "steps": ["f'(x) = 6x + 2.", "Plug x = 3.", "6×3 + 2 = 20."], "text": "Answer: f'(3) = 20.\nSteps:\n1. f'(x) = 6x + 2.\n2. Plug x = 3.\n3. 6×3 + 2 = 20."}
{"id": "mat_0039", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why use matrices?", "choices": null, "answer": "Matrices encode linear maps and systems compactly, turning algebra into row operations.", "steps": ["Matrices encode linear maps and systems compactly, turning algebra into row operations.", "So the answer stands: Matrices encode linear maps and systems compactly, turning algebra into row operations."], "text": "Answer: Matrices encode linear maps and systems compactly, turning algebra into row operations.\nSteps:\n1. Matrices encode linear maps and systems compactly, turning algebra into row operations.\n2. So the answer stands: Matrices encode linear maps and systems compactly, turning algebra into row operations."}
{"id": "mat_0040", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 3 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 3/7.", "steps": ["Red: 3.", "Total: 7.", "P = 3/7."], "text": "Answer: P(red) = 3/7.\nSteps:\n1. Red: 3.\n2. Total: 7.\n3. P = 3/7."}
{"id": "mat_0041", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (100) mod 9.", "choices": {"A": "2", "B": "1", "C": "3", "D": "4"}, "answer": "The remainder is 1.", "steps": ["100 ÷ 9.", "Remainder 1.", "Select option B."], "text": "Answer: 1\nSteps:\n1. 100 ÷ 9.\n2. Remainder 1.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0042", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(6,2).", "choices": null, "answer": "C(6,2) = 15.", "steps": ["Formula n(n−1)/2.", "6×5/2.", "= 15."], "text": "Answer: C(6,2) = 15.\nSteps:\n1. Formula n(n−1)/2.\n2. 6×5/2.\n3. = 15."}
{"id": "mat_0043", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 5?", "choices": {"A": "10", "B": "15", "C": "20", "D": "25"}, "answer": "The sum is 15.", "steps": ["5×6/2.", "Sum = 15.", "Select option B."], "text": "Answer: 15\nSteps:\n1. 5×6/2.\n2. Sum = 15.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0044", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 4.", "choices": null, "answer": "Integral: 64.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 4.", "4³ − 0 = 64."], "text": "Answer: Integral: 64.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 4.\n3. 4³ − 0 = 64."}
{"id": "mat_0045", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej stopu matice [[3,0],[0,4]].", "choices": null, "answer": "Stopa: 7.", "steps": ["Sečti diagonálu.", "3 + 4.", "= 7."], "text": "Answer: Stopa: 7.\nSteps:\n1. Sečti diagonálu.\n2. 3 + 4.\n3. = 7."}
{"id": "mat_0046", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why Bayes' theorem?", "choices": null, "answer": "Bayes flips conditionals to infer causes from effects, powering diagnosis and spam filters.", "steps": ["Bayes flips conditionals to infer causes from effects, powering diagnosis and spam filters.", "So the answer stands: Bayes flips conditionals to infer causes from effects, powering diagnosis and spam filters."], "text": "Answer: Bayes flips conditionals to infer causes from effects, powering diagnosis and spam filters.\nSteps:\n1. Bayes flips conditionals to infer causes from effects, powering diagnosis and spam filters.\n2. So the answer stands: Bayes flips conditionals to infer causes from effects, powering diagnosis and spam filters."}
{"id": "mat_0047", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Co je dělitelnost?", "choices": null, "answer": "a dělí b, pokud existuje k s b = ak.", "steps": ["a dělí b, pokud existuje k s b = ak.", "Odpověď tedy platí: a dělí b, pokud existuje k s b = ak."], "text": "Answer: a dělí b, pokud existuje k s b = ak.\nSteps:\n1. a dělí b, pokud existuje k s b = ak.\n2. Odpověď tedy platí: a dělí b, pokud existuje k s b = ak."}
{"id": "mat_0048", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(3,2).", "choices": {"A": "3", "B": "6", "C": "9", "D": "12"}, "answer": "P(3,2) is 6.", "steps": ["3×2.", "= 6.", "Select option B."], "text": "Answer: 6\nSteps:\n1. 3×2.\n2. = 6.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0049", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Co je přímý důkaz?", "choices": null, "answer": "Přímý důkaz předpokládá předpoklady a odvozuje závěr krok za krokem.", "steps": ["Přímý důkaz předpokládá předpoklady a odvozuje závěr krok za krokem.", "Odpověď tedy platí: Přímý důkaz předpokládá předpoklady a odvozuje závěr krok za krokem."], "text": "Answer: Přímý důkaz předpokládá předpoklady a odvozuje závěr krok za krokem.\nSteps:\n1. Přímý důkaz předpokládá předpoklady a odvozuje závěr krok za krokem.\n2. Odpověď tedy platí: Přímý důkaz předpokládá předpoklady a odvozuje závěr krok za krokem."}
{"id": "mat_0050", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Pro f(x) = 2x² + 1x spočítej f'(2) krok za krokem.", "choices": null, "answer": "f'(2) = 9.", "steps": ["f'(x) = 4x + 1.", "Dosaď x = 2.", "4×2 + 1 = 9."], "text": "Answer: f'(2) = 9.\nSteps:\n1. f'(x) = 4x + 1.\n2. Dosaď x = 2.\n3. 4×2 + 1 = 9."}
{"id": "mat_0051", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is matrix multiplication?", "choices": null, "answer": "It composes linear maps via row-column dot products, and order matters since maps differ.", "steps": ["It composes linear maps via row-column dot products, and order matters since maps differ.", "So the answer stands: It composes linear maps via row-column dot products, and order matters since maps differ."], "text": "Answer: It composes linear maps via row-column dot products, and order matters since maps differ.\nSteps:\n1. It composes linear maps via row-column dot products, and order matters since maps differ.\n2. So the answer stands: It composes linear maps via row-column dot products, and order matters since maps differ."}
{"id": "mat_0052", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is independence?", "choices": null, "answer": "Independent events multiply probabilities, since one's outcome never affects the other's.", "steps": ["Independent events multiply probabilities, since one's outcome never affects the other's.", "So the answer stands: Independent events multiply probabilities, since one's outcome never affects the other's."], "text": "Answer: Independent events multiply probabilities, since one's outcome never affects the other's.\nSteps:\n1. Independent events multiply probabilities, since one's outcome never affects the other's.\n2. So the answer stands: Independent events multiply probabilities, since one's outcome never affects the other's."}
{"id": "mat_0053", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej (100) mod 9.", "choices": {"A": "2", "B": "1", "C": "3", "D": "4"}, "answer": "Zbytek je 1.", "steps": ["100 ÷ 9.", "Zbytek 1.", "Vyber možnost B."], "text": "Answer: 1\nSteps:\n1. 100 ÷ 9.\n2. Zbytek 1.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0054", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(5,3).", "choices": {"A": "5", "B": "15", "C": "10", "D": "20"}, "answer": "C(5,3) is 10.", "steps": ["5×4×3/6.", "= 10.", "Select option C."], "text": "Answer: 10\nSteps:\n1. 5×4×3/6.\n2. = 10.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0055", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is induction?", "choices": null, "answer": "Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers.", "steps": ["Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers.", "So the answer stands: Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers."], "text": "Answer: Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers.\nSteps:\n1. Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers.\n2. So the answer stands: Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers."}
{"id": "mat_0056", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 4²)/(x − 4) as x → 4.", "choices": null, "answer": "Limit: 8.", "steps": ["Factor: (x−4)(x+4).", "Cancel (x−4).", "Plug 4: 4+4 = 8."], "text": "Answer: Limit: 8.\nSteps:\n1. Factor: (x−4)(x+4).\n2. Cancel (x−4).\n3. Plug 4: 4+4 = 8."}
{"id": "mat_0057", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (1,2) · (3,4).", "choices": {"A": "7", "B": "15", "C": "11", "D": "19"}, "answer": "The dot product is 11.", "steps": ["1×3 + 2×4.", "Product = 11.", "Select option C."], "text": "Answer: 11\nSteps:\n1. 1×3 + 2×4.\n2. Product = 11.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0058", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 7-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/7.", "steps": ["Favourable: 1.", "Possible: 7.", "P = 1/7."], "text": "Answer: P = 1/7.\nSteps:\n1. Favourable: 1.\n2. Possible: 7.\n3. P = 1/7."}
{"id": "mat_0059", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(8, 12) step by step.", "choices": null, "answer": "gcd = 4.", "steps": ["Factor: 8=4×2, 12=6×2.", "Common: 2×gcd(4,6).", "= 2×2 = 4."], "text": "Answer: gcd = 4.\nSteps:\n1. Factor: 8=4×2, 12=6×2.\n2. Common: 2×gcd(4,6).\n3. = 2×2 = 4."}
{"id": "mat_0060", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(5,3).", "choices": null, "answer": "P(5,3) = 60.", "steps": ["n(n−1)(n−2).", "5×4×3.", "= 60."], "text": "Answer: P(5,3) = 60.\nSteps:\n1. n(n−1)(n−2).\n2. 5×4×3.\n3. = 60."}
{"id": "mat_0061", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 3 by computing its half.", "choices": null, "answer": "Half: 6, so 12 is even.", "steps": ["3² + 3 = 12.", "12 / 2 = 6.", "Integer, so even."], "text": "Answer: Half: 6, so 12 is even.\nSteps:\n1. 3² + 3 = 12.\n2. 12 / 2 = 6.\n3. Integer, so even."}
{"id": "mat_0062", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 1x² + 2x, find f'(1).", "choices": {"A": "4", "B": "2", "C": "6", "D": "8"}, "answer": "The derivative is 4.", "steps": ["f'(x) = 2x + 2.", "At 1: 4.", "Select option A."], "text": "Answer: 4\nSteps:\n1. f'(x) = 2x + 2.\n2. At 1: 4.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0063", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what detects invertibility?", "choices": {"A": "Big entries", "B": "Round shape", "C": "Long name", "D": "Nonzero determinant"}, "answer": "The answer is Nonzero determinant.", "steps": ["Zero means collapse.", "So: Nonzero determinant.", "Select option D."], "text": "Answer: Nonzero determinant\nSteps:\n1. Zero means collapse.\n2. So: Nonzero determinant.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0064", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why expected value?", "choices": null, "answer": "Expectation averages outcomes by probability, predicting long-run results of gambles.", "steps": ["Expectation averages outcomes by probability, predicting long-run results of gambles.", "So the answer stands: Expectation averages outcomes by probability, predicting long-run results of gambles."], "text": "Answer: Expectation averages outcomes by probability, predicting long-run results of gambles.\nSteps:\n1. Expectation averages outcomes by probability, predicting long-run results of gambles.\n2. So the answer stands: Expectation averages outcomes by probability, predicting long-run results of gambles."}
{"id": "mat_0065", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why infinitely many primes?", "choices": null, "answer": "Euclid multiplies known primes and adds one, forcing a new prime factor each time.", "steps": ["Euclid multiplies known primes and adds one, forcing a new prime factor each time.", "So the answer stands: Euclid multiplies known primes and adds one, forcing a new prime factor each time."], "text": "Answer: Euclid multiplies known primes and adds one, forcing a new prime factor each time.\nSteps:\n1. Euclid multiplies known primes and adds one, forcing a new prime factor each time.\n2. So the answer stands: Euclid multiplies known primes and adds one, forcing a new prime factor each time."}
{"id": "mat_0066", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(7,3).", "choices": null, "answer": "C(7,3) = 35.", "steps": ["n(n−1)(n−2)/6.", "7×6×5/6.", "= 35."], "text": "Answer: C(7,3) = 35.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 7×6×5/6.\n3. = 35."}
{"id": "mat_0067", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why prove the base case?", "choices": null, "answer": "The base case anchors induction, because without a true start the chain of implications proves nothing.", "steps": ["The base case anchors induction, because without a true start the chain of implications proves nothing.", "So the answer stands: The base case anchors induction, because without a true start the chain of implications proves nothing."], "text": "Answer: The base case anchors induction, because without a true start the chain of implications proves nothing.\nSteps:\n1. The base case anchors induction, because without a true start the chain of implications proves nothing.\n2. So the answer stands: The base case anchors induction, because without a true start the chain of implications proves nothing."}
{"id": "mat_0068", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 2.", "choices": {"A": "6", "B": "8", "C": "10", "D": "12"}, "answer": "The integral is 8.", "steps": ["Antiderivative x³.", "2³ = 8.", "Select option B."], "text": "Answer: 8\nSteps:\n1. Antiderivative x³.\n2. 2³ = 8.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0069", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why care about determinant?", "choices": null, "answer": "The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension.", "steps": ["The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension.", "So the answer stands: The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension."], "text": "Answer: The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension.\nSteps:\n1. The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension.\n2. So the answer stands: The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension."}
{"id": "mat_0070", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..6}.", "choices": null, "answer": "E = 3.5.", "steps": ["Mean of 1..6.", "(6+1)/2.", "E = 3.5."], "text": "Answer: E = 3.5.\nSteps:\n1. Mean of 1..6.\n2. (6+1)/2.\n3. E = 3.5."}
{"id": "mat_0071", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (50) mod 7.", "choices": null, "answer": "Remainder: 1.", "steps": ["Divide 50 by 7.", "7×7 = 49.", "50 − 49 = 1."], "text": "Answer: Remainder: 1.\nSteps:\n1. Divide 50 by 7.\n2. 7×7 = 49.\n3. 50 − 49 = 1."}
{"id": "mat_0072", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(7,2).", "choices": null, "answer": "C(7,2) = 21.", "steps": ["Formula n(n−1)/2.", "7×6/2.", "= 21."], "text": "Answer: C(7,2) = 21.\nSteps:\n1. Formula n(n−1)/2.\n2. 7×6/2.\n3. = 21."}
{"id": "mat_0073", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Kolik řádků má pravdivostní tabulka s 2 proměnnými?", "choices": {"A": "4", "B": "2", "C": "6", "D": "8"}, "answer": "Tabulka má 4 řádků.", "steps": ["2^2 řádků.", "Celkem: 4.", "Vyber možnost A."], "text": "Answer: 4\nSteps:\n1. 2^2 řádků.\n2. Celkem: 4.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0074", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is a derivative?", "choices": null, "answer": "A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point.", "steps": ["A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point.", "So the answer stands: A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point."], "text": "Answer: A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point.\nSteps:\n1. A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point.\n2. So the answer stands: A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point."}
{"id": "mat_0075", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (2,1) · (3,5) step by step.", "choices": null, "answer": "Dot product: 11.", "steps": ["Multiply components.", "2×3 + 1×5.", "= 11."], "text": "Answer: Dot product: 11.\nSteps:\n1. Multiply components.\n2. 2×3 + 1×5.\n3. = 11."}
{"id": "mat_0076", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 4 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 4/8.", "steps": ["Red: 4.", "Total: 8.", "P = 4/8."], "text": "Answer: P(red) = 4/8.\nSteps:\n1. Red: 4.\n2. Total: 8.\n3. P = 4/8."}
{"id": "mat_0077", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(4, 6).", "choices": {"A": "10", "B": "14", "C": "12", "D": "16"}, "answer": "The lcm is 12.", "steps": ["4×6/gcd.", "lcm = 12.", "Select option C."], "text": "Answer: 12\nSteps:\n1. 4×6/gcd.\n2. lcm = 12.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0078", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(6,3).", "choices": null, "answer": "P(6,3) = 120.", "steps": ["n(n−1)(n−2).", "6×5×4.", "= 120."], "text": "Answer: P(6,3) = 120.\nSteps:\n1. n(n−1)(n−2).\n2. 6×5×4.\n3. = 120."}
{"id": "mat_0079", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Dokaž vzorcem: najdi součet 1 + 2 + ... + 6.", "choices": null, "answer": "Součet: 21.", "steps": ["Páruj 1+6, 2+5, ...", "Použij 6×7/2.", "= 21."], "text": "Answer: Součet: 21.\nSteps:\n1. Páruj 1+6, 2+5, ...\n2. Použij 6×7/2.\n3. = 21."}
{"id": "mat_0080", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej integrál 3x² od 0 do 2.", "choices": {"A": "6", "B": "8", "C": "10", "D": "12"}, "answer": "Integrál je 8.", "steps": ["Primitivní x³.", "2³ = 8.", "Vyber možnost B."], "text": "Answer: 8\nSteps:\n1. Primitivní x³.\n2. 2³ = 8.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0081", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[3,1],[2,4]] step by step.", "choices": null, "answer": "Determinant: 10.", "steps": ["ad − bc.", "3×4 − 1×2.", "= 10."], "text": "Answer: Determinant: 10.\nSteps:\n1. ad − bc.\n2. 3×4 − 1×2.\n3. = 10."}
{"id": "mat_0082", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is variance?", "choices": null, "answer": "Variance averages squared deviations, measuring how spread out a random variable is.", "steps": ["Variance averages squared deviations, measuring how spread out a random variable is.", "So the answer stands: Variance averages squared deviations, measuring how spread out a random variable is."], "text": "Answer: Variance averages squared deviations, measuring how spread out a random variable is.\nSteps:\n1. Variance averages squared deviations, measuring how spread out a random variable is.\n2. So the answer stands: Variance averages squared deviations, measuring how spread out a random variable is."}
{"id": "mat_0083", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is the gcd?", "choices": null, "answer": "The greatest common divisor is the largest integer dividing both numbers at once.", "steps": ["The greatest common divisor is the largest integer dividing both numbers at once.", "So the answer stands: The greatest common divisor is the largest integer dividing both numbers at once."], "text": "Answer: The greatest common divisor is the largest integer dividing both numbers at once.\nSteps:\n1. The greatest common divisor is the largest integer dividing both numbers at once.\n2. So the answer stands: The greatest common divisor is the largest integer dividing both numbers at once."}
{"id": "mat_0084", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(8,3).", "choices": null, "answer": "C(8,3) = 56.", "steps": ["n(n−1)(n−2)/6.", "8×7×6/6.", "= 56."], "text": "Answer: C(8,3) = 56.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 8×7×6/6.\n3. = 56."}
{"id": "mat_0085", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 3.", "choices": {"A": "10", "B": "14", "C": "12", "D": "16"}, "answer": "The value is 12.", "steps": ["3² + 3.", "Value = 12.", "Select option C."], "text": "Answer: 12\nSteps:\n1. 3² + 3.\n2. Value = 12.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0086", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 2²)/(x − 2) as x → 2.", "choices": {"A": "2", "B": "6", "C": "4", "D": "8"}, "answer": "The limit is 4.", "steps": ["Factor (x−2)(x+2).", "Cancel, get 2×2 = 4.", "Select option C."], "text": "Answer: 4\nSteps:\n1. Factor (x−2)(x+2).\n2. Cancel, get 2×2 = 4.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0087", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[4,0],[0,5]].", "choices": null, "answer": "Trace: 9.", "steps": ["Sum diagonal.", "4 + 5.", "= 9."], "text": "Answer: Trace: 9.\nSteps:\n1. Sum diagonal.\n2. 4 + 5.\n3. = 9."}
{"id": "mat_0088", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..4}.", "choices": {"A": "3.5", "B": "2.5", "C": "1.5", "D": "4.5"}, "answer": "The expected value is 2.5.", "steps": ["(4+1)/2.", "E = 2.5.", "Select option B."], "text": "Answer: 2.5\nSteps:\n1. (4+1)/2.\n2. E = 2.5.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0089", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what are multiplication atoms?", "choices": {"A": "Evens", "B": "Squares", "C": "Cubes", "D": "Primes"}, "answer": "The answer is Primes.", "steps": ["Unique factorization.", "So: Primes.", "Select option D."], "text": "Answer: Primes\nSteps:\n1. Unique factorization.\n2. So: Primes.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0090", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what counts orderings?", "choices": {"A": "Sum", "B": "Average", "C": "Guess", "D": "Factorial"}, "answer": "The answer is Factorial.", "steps": ["Choices shrink per slot.", "So: Factorial.", "Select option D."], "text": "Answer: Factorial\nSteps:\n1. Choices shrink per slot.\n2. So: Factorial.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0091", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is a counterexample?", "choices": null, "answer": "A counterexample is one case where a universal claim fails, and a single one disproves the whole claim.", "steps": ["A counterexample is one case where a universal claim fails, and a single one disproves the whole claim.", "So the answer stands: A counterexample is one case where a universal claim fails, and a single one disproves the whole claim."], "text": "Answer: A counterexample is one case where a universal claim fails, and a single one disproves the whole claim.\nSteps:\n1. A counterexample is one case where a universal claim fails, and a single one disproves the whole claim.\n2. So the answer stands: A counterexample is one case where a universal claim fails, and a single one disproves the whole claim."}
{"id": "mat_0092", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 4x² + 3x, compute f'(4) step by step.", "choices": null, "answer": "f'(4) = 35.", "steps": ["f'(x) = 8x + 3.", "Plug x = 4.", "8×4 + 3 = 35."], "text": "Answer: f'(4) = 35.\nSteps:\n1. f'(x) = 8x + 3.\n2. Plug x = 4.\n3. 8×4 + 3 = 35."}
{"id": "mat_0093", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is rank?", "choices": null, "answer": "Rank counts independent rows or columns, measuring the true dimension of the map's image.", "steps": ["Rank counts independent rows or columns, measuring the true dimension of the map's image.", "So the answer stands: Rank counts independent rows or columns, measuring the true dimension of the map's image."], "text": "Answer: Rank counts independent rows or columns, measuring the true dimension of the map's image.\nSteps:\n1. Rank counts independent rows or columns, measuring the true dimension of the map's image.\n2. So the answer stands: Rank counts independent rows or columns, measuring the true dimension of the map's image."}
{"id": "mat_0094", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 2 red and 5 blue marbles. What is P(red)?", "choices": {"A": "5/7", "B": "2/8", "C": "2/7", "D": "3/7"}, "answer": "P(red) is 2/7.", "steps": ["2 of 7.", "P = 2/7.", "Select option C."], "text": "Answer: 2/7\nSteps:\n1. 2 of 7.\n2. P = 2/7.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0095", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why Euclid's algorithm?", "choices": null, "answer": "Repeated remainders shrink the pair fast, computing gcds in logarithmic steps.", "steps": ["Repeated remainders shrink the pair fast, computing gcds in logarithmic steps.", "So the answer stands: Repeated remainders shrink the pair fast, computing gcds in logarithmic steps."], "text": "Answer: Repeated remainders shrink the pair fast, computing gcds in logarithmic steps.\nSteps:\n1. Repeated remainders shrink the pair fast, computing gcds in logarithmic steps.\n2. So the answer stands: Repeated remainders shrink the pair fast, computing gcds in logarithmic steps."}
{"id": "mat_0096", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is nCr?", "choices": null, "answer": "nCr counts subsets of size r from n objects, ignoring order via division by r!.", "steps": ["nCr counts subsets of size r from n objects, ignoring order via division by r!.", "So the answer stands: nCr counts subsets of size r from n objects, ignoring order via division by r!."], "text": "Answer: nCr counts subsets of size r from n objects, ignoring order via division by r!.\nSteps:\n1. nCr counts subsets of size r from n objects, ignoring order via division by r!.\n2. So the answer stands: nCr counts subsets of size r from n objects, ignoring order via division by r!."}
{"id": "mat_0097", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what anchors induction?", "choices": {"A": "Fancy notation", "B": "Long chains", "C": "Big numbers", "D": "Base case"}, "answer": "The answer is Base case.", "steps": ["Without base, chain proves nothing.", "So: Base case.", "Select option D."], "text": "Answer: Base case\nSteps:\n1. Without base, chain proves nothing.\n2. So: Base case.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0098", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 5.", "choices": null, "answer": "Integral: 125.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 5.", "5³ − 0 = 125."], "text": "Answer: Integral: 125.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 5.\n3. 5³ − 0 = 125."}
{"id": "mat_0099", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why row-reduce?", "choices": null, "answer": "Row reduction exposes pivots and solutions, solving linear systems mechanically every time.", "steps": ["Row reduction exposes pivots and solutions, solving linear systems mechanically every time.", "So the answer stands: Row reduction exposes pivots and solutions, solving linear systems mechanically every time."], "text": "Answer: Row reduction exposes pivots and solutions, solving linear systems mechanically every time.\nSteps:\n1. Row reduction exposes pivots and solutions, solving linear systems mechanically every time.\n2. So the answer stands: Row reduction exposes pivots and solutions, solving linear systems mechanically every time."}
{"id": "mat_0100", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 8-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/8.", "steps": ["Favourable: 1.", "Possible: 8.", "P = 1/8."], "text": "Answer: P = 1/8.\nSteps:\n1. Favourable: 1.\n2. Possible: 8.\n3. P = 1/8."}
{"id": "mat_0101", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej (50) mod 7.", "choices": null, "answer": "Zbytek: 1.", "steps": ["Děl 50 sedmi.", "7×7 = 49.", "50 − 49 = 1."], "text": "Answer: Zbytek: 1.\nSteps:\n1. Děl 50 sedmi.\n2. 7×7 = 49.\n3. 50 − 49 = 1."}
{"id": "mat_0102", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej kombinační číslo C(4,2).", "choices": {"A": "6", "B": "2", "C": "10", "D": "14"}, "answer": "C(4,2) je 6.", "steps": ["4×3/2.", "= 6.", "Vyber možnost A."], "text": "Answer: 6\nSteps:\n1. 4×3/2.\n2. = 6.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0103", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 4 boolean variables. How many rows?", "choices": null, "answer": "Rows: 16.", "steps": ["2 choices each.", "2^4 = 16."], "text": "Answer: Rows: 16.\nSteps:\n1. 2 choices each.\n2. 2^4 = 16."}
{"id": "mat_0104", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Co je limita?", "choices": null, "answer": "Limita je hodnota, k níž se funkce blíží, i když v bodě není definována.", "steps": ["Limita je hodnota, k níž se funkce blíží, i když v bodě není definována.", "Odpověď tedy platí: Limita je hodnota, k níž se funkce blíží, i když v bodě není definována."], "text": "Answer: Limita je hodnota, k níž se funkce blíží, i když v bodě není definována.\nSteps:\n1. Limita je hodnota, k níž se funkce blíží, i když v bodě není definována.\n2. Odpověď tedy platí: Limita je hodnota, k níž se funkce blíží, i když v bodě není definována."}
{"id": "mat_0105", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (3,2) · (3,5) step by step.", "choices": null, "answer": "Dot product: 19.", "steps": ["Multiply components.", "3×3 + 2×5.", "= 19."], "text": "Answer: Dot product: 19.\nSteps:\n1. Multiply components.\n2. 3×3 + 2×5.\n3. = 19."}
{"id": "mat_0106", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..7}.", "choices": null, "answer": "E = 4.", "steps": ["Mean of 1..7.", "(7+1)/2.", "E = 4."], "text": "Answer: E = 4.\nSteps:\n1. Mean of 1..7.\n2. (7+1)/2.\n3. E = 4."}
{"id": "mat_0107", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what merges congruences?", "choices": {"A": "Chinese remainder", "B": "Blind luck", "C": "Wild guess", "D": "Long nap"}, "answer": "The answer is Chinese remainder.", "steps": ["One solution modulo product.", "So: Chinese remainder.", "Select option A."], "text": "Answer: Chinese remainder\nSteps:\n1. One solution modulo product.\n2. So: Chinese remainder.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0108", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why Pascal's triangle?", "choices": null, "answer": "It builds nCr rows by addition, revealing symmetry and binomial patterns visually.", "steps": ["It builds nCr rows by addition, revealing symmetry and binomial patterns visually.", "So the answer stands: It builds nCr rows by addition, revealing symmetry and binomial patterns visually."], "text": "Answer: It builds nCr rows by addition, revealing symmetry and binomial patterns visually.\nSteps:\n1. It builds nCr rows by addition, revealing symmetry and binomial patterns visually.\n2. So the answer stands: It builds nCr rows by addition, revealing symmetry and binomial patterns visually."}
{"id": "mat_0109", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 8.", "choices": null, "answer": "Sum: 36.", "steps": ["Pair 1+8, 2+7, ...", "Use 8×9/2.", "= 36."], "text": "Answer: Sum: 36.\nSteps:\n1. Pair 1+8, 2+7, ...\n2. Use 8×9/2.\n3. = 36."}
{"id": "mat_0110", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 5²)/(x − 5) as x → 5.", "choices": null, "answer": "Limit: 10.", "steps": ["Factor: (x−5)(x+5).", "Cancel (x−5).", "Plug 5: 5+5 = 10."], "text": "Answer: Limit: 10.\nSteps:\n1. Factor: (x−5)(x+5).\n2. Cancel (x−5).\n3. Plug 5: 5+5 = 10."}
{"id": "mat_0111", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[4,1],[2,5]] step by step.", "choices": null, "answer": "Determinant: 18.", "steps": ["ad − bc.", "4×5 − 1×2.", "= 18."], "text": "Answer: Determinant: 18.\nSteps:\n1. ad − bc.\n2. 4×5 − 1×2.\n3. = 18."}
{"id": "mat_0112", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Co je pravděpodobnost?", "choices": null, "answer": "Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování.", "steps": ["Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování.", "Odpověď tedy platí: Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování."], "text": "Answer: Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování.\nSteps:\n1. Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování.\n2. Odpověď tedy platí: Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování."}
{"id": "mat_0113", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(18, 27).", "choices": {"A": "6", "B": "9", "C": "12", "D": "15"}, "answer": "The gcd is 9.", "steps": ["Factor 3 out twice.", "gcd = 9.", "Select option B."], "text": "Answer: 9\nSteps:\n1. Factor 3 out twice.\n2. gcd = 9.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0114", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(8,2).", "choices": null, "answer": "C(8,2) = 28.", "steps": ["Formula n(n−1)/2.", "8×7/2.", "= 28."], "text": "Answer: C(8,2) = 28.\nSteps:\n1. Formula n(n−1)/2.\n2. 8×7/2.\n3. = 28."}
{"id": "mat_0115", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what proves existence?", "choices": {"A": "Witness or counting", "B": "Long names", "C": "Big fonts", "D": "Hope alone"}, "answer": "The answer is Witness or counting.", "steps": ["Build or count it.", "So: Witness or counting.", "Select option A."], "text": "Answer: Witness or counting\nSteps:\n1. Build or count it.\n2. So: Witness or counting.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0116", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why differentiate?", "choices": null, "answer": "Differentiation finds maxima, minima, and speeds, turning change questions into algebra.", "steps": ["Differentiation finds maxima, minima, and speeds, turning change questions into algebra.", "So the answer stands: Differentiation finds maxima, minima, and speeds, turning change questions into algebra."], "text": "Answer: Differentiation finds maxima, minima, and speeds, turning change questions into algebra.\nSteps:\n1. Differentiation finds maxima, minima, and speeds, turning change questions into algebra.\n2. So the answer stands: Differentiation finds maxima, minima, and speeds, turning change questions into algebra."}
{"id": "mat_0117", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[5,0],[0,6]].", "choices": null, "answer": "Trace: 11.", "steps": ["Sum diagonal.", "5 + 6.", "= 11."], "text": "Answer: Trace: 11.\nSteps:\n1. Sum diagonal.\n2. 5 + 6.\n3. = 11."}
{"id": "mat_0118", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Hází se férovou 6-stěnnou kostkou. Jaká je pravděpodobnost jednoho konkrétního čísla?", "choices": {"A": "1/6", "B": "1/5", "C": "1/7", "D": "2/6"}, "answer": "Pravděpodobnost je 1/6.", "steps": ["1 stěna z 6.", "P = 1/6.", "Vyber možnost A."], "text": "Answer: 1/6\nSteps:\n1. 1 stěna z 6.\n2. P = 1/6.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0119", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej NSN(5, 8) krok za krokem.", "choices": null, "answer": "NSN = 40.", "steps": ["NSD(5,8) = 1.", "5×8/1.", "= 40."], "text": "Answer: NSN = 40.\nSteps:\n1. NSD(5,8) = 1.\n2. 5×8/1.\n3. = 40."}
{"id": "mat_0120", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what defines from the past?", "choices": {"A": "Recurrence", "B": "Prophecy", "C": "Rumour", "D": "Dream"}, "answer": "The answer is Recurrence.", "steps": ["Base plus step.", "So: Recurrence.", "Select option A."], "text": "Answer: Recurrence\nSteps:\n1. Base plus step.\n2. So: Recurrence.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0121", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 4 by computing its half.", "choices": null, "answer": "Half: 10, so 20 is even.", "steps": ["4² + 4 = 20.", "20 / 2 = 10.", "Integer, so even."], "text": "Answer: Half: 10, so 20 is even.\nSteps:\n1. 4² + 4 = 20.\n2. 20 / 2 = 10.\n3. Integer, so even."}
{"id": "mat_0122", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is the chain rule?", "choices": null, "answer": "The chain rule differentiates compositions by multiplying outer and inner derivatives together.", "steps": ["The chain rule differentiates compositions by multiplying outer and inner derivatives together.", "So the answer stands: The chain rule differentiates compositions by multiplying outer and inner derivatives together."], "text": "Answer: The chain rule differentiates compositions by multiplying outer and inner derivatives together.\nSteps:\n1. The chain rule differentiates compositions by multiplying outer and inner derivatives together.\n2. So the answer stands: The chain rule differentiates compositions by multiplying outer and inner derivatives together."}
{"id": "mat_0123", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is an inverse?", "choices": null, "answer": "An inverse matrix undoes the original map, existing exactly when the determinant is nonzero.", "steps": ["An inverse matrix undoes the original map, existing exactly when the determinant is nonzero.", "So the answer stands: An inverse matrix undoes the original map, existing exactly when the determinant is nonzero."], "text": "Answer: An inverse matrix undoes the original map, existing exactly when the determinant is nonzero.\nSteps:\n1. An inverse matrix undoes the original map, existing exactly when the determinant is nonzero.\n2. So the answer stands: An inverse matrix undoes the original map, existing exactly when the determinant is nonzero."}
{"id": "mat_0124", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why standard deviation?", "choices": null, "answer": "It returns spread to original units, making variability interpretable in context.", "steps": ["It returns spread to original units, making variability interpretable in context.", "So the answer stands: It returns spread to original units, making variability interpretable in context."], "text": "Answer: It returns spread to original units, making variability interpretable in context.\nSteps:\n1. It returns spread to original units, making variability interpretable in context.\n2. So the answer stands: It returns spread to original units, making variability interpretable in context."}
{"id": "mat_0125", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is the lcm?", "choices": null, "answer": "The least common multiple is the smallest positive integer both numbers divide.", "steps": ["The least common multiple is the smallest positive integer both numbers divide.", "So the answer stands: The least common multiple is the smallest positive integer both numbers divide."], "text": "Answer: The least common multiple is the smallest positive integer both numbers divide.\nSteps:\n1. The least common multiple is the smallest positive integer both numbers divide.\n2. So the answer stands: The least common multiple is the smallest positive integer both numbers divide."}
{"id": "mat_0126", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(7,3).", "choices": null, "answer": "P(7,3) = 210.", "steps": ["n(n−1)(n−2).", "7×6×5.", "= 210."], "text": "Answer: P(7,3) = 210.\nSteps:\n1. n(n−1)(n−2).\n2. 7×6×5.\n3. = 210."}
{"id": "mat_0127", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 5 boolean variables. How many rows?", "choices": null, "answer": "Rows: 32.", "steps": ["2 choices each.", "2^5 = 32."], "text": "Answer: Rows: 32.\nSteps:\n1. 2 choices each.\n2. 2^5 = 32."}
{"id": "mat_0128", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why learn the product rule?", "choices": null, "answer": "The product rule splits (fg)' into f'g + fg', handling products no single rule covers.", "steps": ["The product rule splits (fg)' into f'g + fg', handling products no single rule covers.", "So the answer stands: The product rule splits (fg)' into f'g + fg', handling products no single rule covers."], "text": "Answer: The product rule splits (fg)' into f'g + fg', handling products no single rule covers.\nSteps:\n1. The product rule splits (fg)' into f'g + fg', handling products no single rule covers.\n2. So the answer stands: The product rule splits (fg)' into f'g + fg', handling products no single rule covers."}
{"id": "mat_0129", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why study eigenvalues?", "choices": null, "answer": "Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour.", "steps": ["Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour.", "So the answer stands: Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour."], "text": "Answer: Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour.\nSteps:\n1. Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour.\n2. So the answer stands: Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour."}
{"id": "mat_0130", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 5 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 5/9.", "steps": ["Red: 5.", "Total: 9.", "P = 5/9."], "text": "Answer: P(red) = 5/9.\nSteps:\n1. Red: 5.\n2. Total: 9.\n3. P = 5/9."}
{"id": "mat_0131", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (107) mod 9.", "choices": {"A": "0", "B": "1", "C": "8", "D": "2"}, "answer": "The remainder is 8.", "steps": ["107 ÷ 9.", "Remainder 8.", "Select option C."], "text": "Answer: 8\nSteps:\n1. 107 ÷ 9.\n2. Remainder 8.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0132", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is the binomial theorem?", "choices": null, "answer": "It expands (a+b)^n into nCr-weighted terms, linking algebra to counting.", "steps": ["It expands (a+b)^n into nCr-weighted terms, linking algebra to counting.", "So the answer stands: It expands (a+b)^n into nCr-weighted terms, linking algebra to counting."], "text": "Answer: It expands (a+b)^n into nCr-weighted terms, linking algebra to counting.\nSteps:\n1. It expands (a+b)^n into nCr-weighted terms, linking algebra to counting.\n2. So the answer stands: It expands (a+b)^n into nCr-weighted terms, linking algebra to counting."}
{"id": "mat_0133", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 3 variables have?", "choices": {"A": "6", "B": "8", "C": "10", "D": "12"}, "answer": "The table has 8 rows.", "steps": ["2^3 rows.", "Total: 8.", "Select option B."], "text": "Answer: 8\nSteps:\n1. 2^3 rows.\n2. Total: 8.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0134", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Proč spojitost?", "choices": null, "answer": "Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu.", "steps": ["Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu.", "Odpověď tedy platí: Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu."], "text": "Answer: Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu.\nSteps:\n1. Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu.\n2. Odpověď tedy platí: Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu."}
{"id": "mat_0135", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (4,3) · (3,5) step by step.", "choices": null, "answer": "Dot product: 27.", "steps": ["Multiply components.", "4×3 + 3×5.", "= 27."], "text": "Answer: Dot product: 27.\nSteps:\n1. Multiply components.\n2. 4×3 + 3×5.\n3. = 27."}
{"id": "mat_0136", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 9-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/9.", "steps": ["Favourable: 1.", "Possible: 9.", "P = 1/9."], "text": "Answer: P = 1/9.\nSteps:\n1. Favourable: 1.\n2. Possible: 9.\n3. P = 1/9."}
{"id": "mat_0137", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(5, 6).", "choices": {"A": "28", "B": "32", "C": "34", "D": "30"}, "answer": "The lcm is 30.", "steps": ["5×6/gcd.", "lcm = 30.", "Select option D."], "text": "Answer: 30\nSteps:\n1. 5×6/gcd.\n2. lcm = 30.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0138", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet C(5,2).", "choices": null, "answer": "C(5,2) = 10.", "steps": ["Vzorec n(n−1)/2.", "5×4/2.", "= 10."], "text": "Answer: C(5,2) = 10.\nSteps:\n1. Vzorec n(n−1)/2.\n2. 5×4/2.\n3. = 10."}
{"id": "mat_0139", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 9.", "choices": null, "answer": "Sum: 45.", "steps": ["Pair 1+9, 2+8, ...", "Use 9×10/2.", "= 45."], "text": "Answer: Sum: 45.\nSteps:\n1. Pair 1+9, 2+8, ...\n2. Use 9×10/2.\n3. = 45."}
{"id": "mat_0140", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Co je derivace?", "choices": null, "answer": "Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě.", "steps": ["Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě.", "Odpověď tedy platí: Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě."], "text": "Answer: Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě.\nSteps:\n1. Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě.\n2. Odpověď tedy platí: Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě."}
{"id": "mat_0141", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Co je vektor?", "choices": null, "answer": "Vektor je veličina se směrem a velikostí, sčítá se po složkách.", "steps": ["Vektor je veličina se směrem a velikostí, sčítá se po složkách.", "Odpověď tedy platí: Vektor je veličina se směrem a velikostí, sčítá se po složkách."], "text": "Answer: Vektor je veličina se směrem a velikostí, sčítá se po složkách.\nSteps:\n1. Vektor je veličina se směrem a velikostí, sčítá se po složkách.\n2. Odpověď tedy platí: Vektor je veličina se směrem a velikostí, sčítá se po složkách."}
{"id": "mat_0142", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is a random variable?", "choices": null, "answer": "A random variable maps outcomes to numbers, letting chance be studied with functions.", "steps": ["A random variable maps outcomes to numbers, letting chance be studied with functions.", "So the answer stands: A random variable maps outcomes to numbers, letting chance be studied with functions."], "text": "Answer: A random variable maps outcomes to numbers, letting chance be studied with functions.\nSteps:\n1. A random variable maps outcomes to numbers, letting chance be studied with functions.\n2. So the answer stands: A random variable maps outcomes to numbers, letting chance be studied with functions."}
{"id": "mat_0143", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(5, 8) step by step.", "choices": null, "answer": "lcm = 40.", "steps": ["gcd(5,8) = 1.", "5×8/1.", "= 40."], "text": "Answer: lcm = 40.\nSteps:\n1. gcd(5,8) = 1.\n2. 5×8/1.\n3. = 40."}
{"id": "mat_0144", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why stars and bars?", "choices": null, "answer": "It counts distributions of identical items into boxes via separator placements.", "steps": ["It counts distributions of identical items into boxes via separator placements.", "So the answer stands: It counts distributions of identical items into boxes via separator placements."], "text": "Answer: It counts distributions of identical items into boxes via separator placements.\nSteps:\n1. It counts distributions of identical items into boxes via separator placements.\n2. So the answer stands: It counts distributions of identical items into boxes via separator placements."}
{"id": "mat_0145", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 6?", "choices": {"A": "15", "B": "27", "C": "21", "D": "33"}, "answer": "The sum is 21.", "steps": ["6×7/2.", "Sum = 21.", "Select option C."], "text": "Answer: 21\nSteps:\n1. 6×7/2.\n2. Sum = 21.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0146", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: what does f' measure?", "choices": {"A": "Total area", "B": "Long length", "C": "Big volume", "D": "Instant rate"}, "answer": "The answer is Instant rate.", "steps": ["Tangent slope read.", "So: Instant rate.", "Select option D."], "text": "Answer: Instant rate\nSteps:\n1. Tangent slope read.\n2. So: Instant rate.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0147", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[5,1],[2,6]] step by step.", "choices": null, "answer": "Determinant: 28.", "steps": ["ad − bc.", "5×6 − 1×2.", "= 28."], "text": "Answer: Determinant: 28.\nSteps:\n1. ad − bc.\n2. 5×6 − 1×2.\n3. = 28."}
{"id": "mat_0148", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what lists all outcomes?", "choices": {"A": "Wish list", "B": "Guess bag", "C": "Hope chest", "D": "Sample space"}, "answer": "The answer is Sample space.", "steps": ["Events are subsets.", "So: Sample space.", "Select option D."], "text": "Answer: Sample space\nSteps:\n1. Events are subsets.\n2. So: Sample space.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0149", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why Bézout's identity?", "choices": null, "answer": "It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving.", "steps": ["It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving.", "So the answer stands: It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving."], "text": "Answer: It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving.\nSteps:\n1. It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving.\n2. So the answer stands: It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving."}
{"id": "mat_0150", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(5,2).", "choices": {"A": "5", "B": "10", "C": "15", "D": "20"}, "answer": "C(5,2) is 10.", "steps": ["5×4/2.", "= 10.", "Select option B."], "text": "Answer: 10\nSteps:\n1. 5×4/2.\n2. = 10.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0151", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 4.", "choices": {"A": "18", "B": "22", "C": "24", "D": "20"}, "answer": "The value is 20.", "steps": ["4² + 4.", "Value = 20.", "Select option D."], "text": "Answer: 20\nSteps:\n1. 4² + 4.\n2. Value = 20.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0152", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: what approximates functions?", "choices": {"A": "Taylor series", "B": "Random dots", "C": "Wild guesses", "D": "Blank lines"}, "answer": "The answer is Taylor series.", "steps": ["Polynomials with error bounds.", "So: Taylor series.", "Select option A."], "text": "Answer: Taylor series\nSteps:\n1. Polynomials with error bounds.\n2. So: Taylor series.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0153", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is an eigenvector?", "choices": null, "answer": "An eigenvector keeps its direction under the map, scaled only by its eigenvalue.", "steps": ["An eigenvector keeps its direction under the map, scaled only by its eigenvalue.", "So the answer stands: An eigenvector keeps its direction under the map, scaled only by its eigenvalue."], "text": "Answer: An eigenvector keeps its direction under the map, scaled only by its eigenvalue.\nSteps:\n1. An eigenvector keeps its direction under the map, scaled only by its eigenvalue.\n2. So the answer stands: An eigenvector keeps its direction under the map, scaled only by its eigenvalue."}
{"id": "mat_0154", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what measures spread?", "choices": {"A": "Variance", "B": "Mean", "C": "Mode", "D": "Hope"}, "answer": "The answer is Variance.", "steps": ["Squared deviations averaged.", "So: Variance.", "Select option A."], "text": "Answer: Variance\nSteps:\n1. Squared deviations averaged.\n2. So: Variance.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0155", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(12, 18) step by step.", "choices": null, "answer": "gcd = 6.", "steps": ["Factor: 12=4×3, 18=6×3.", "Common: 3×gcd(4,6).", "= 3×2 = 6."], "text": "Answer: gcd = 6.\nSteps:\n1. Factor: 12=4×3, 18=6×3.\n2. Common: 3×gcd(4,6).\n3. = 3×2 = 6."}
{"id": "mat_0156", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(4,2).", "choices": {"A": "8", "B": "16", "C": "12", "D": "20"}, "answer": "P(4,2) is 12.", "steps": ["4×3.", "= 12.", "Select option C."], "text": "Answer: 12\nSteps:\n1. 4×3.\n2. = 12.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0157", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Proč důkaz sporem?", "choices": null, "answer": "Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí.", "steps": ["Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí.", "Odpověď tedy platí: Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí."], "text": "Answer: Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí.\nSteps:\n1. Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí.\n2. Odpověď tedy platí: Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí."}
{"id": "mat_0158", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 2x² + 4x, compute f'(2) step by step.", "choices": null, "answer": "f'(2) = 12.", "steps": ["f'(x) = 4x + 4.", "Plug x = 2.", "4×2 + 4 = 12."], "text": "Answer: f'(2) = 12.\nSteps:\n1. f'(x) = 4x + 4.\n2. Plug x = 2.\n3. 4×2 + 4 = 12."}
{"id": "mat_0159", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why diagonalize?", "choices": null, "answer": "Diagonal form powers matrices easily, since powers act separately on each eigen-direction.", "steps": ["Diagonal form powers matrices easily, since powers act separately on each eigen-direction.", "So the answer stands: Diagonal form powers matrices easily, since powers act separately on each eigen-direction."], "text": "Answer: Diagonal form powers matrices easily, since powers act separately on each eigen-direction.\nSteps:\n1. Diagonal form powers matrices easily, since powers act separately on each eigen-direction.\n2. So the answer stands: Diagonal form powers matrices easily, since powers act separately on each eigen-direction."}
{"id": "mat_0160", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Proč prostor výsledků?", "choices": null, "answer": "Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu.", "steps": ["Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu.", "Odpověď tedy platí: Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu."], "text": "Answer: Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu.\nSteps:\n1. Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu.\n2. Odpověď tedy platí: Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu."}
{"id": "mat_0161", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (61) mod 7.", "choices": null, "answer": "Remainder: 5.", "steps": ["Divide 61 by 7.", "7×8 = 56.", "61 − 56 = 5."], "text": "Answer: Remainder: 5.\nSteps:\n1. Divide 61 by 7.\n2. 7×8 = 56.\n3. 61 − 56 = 5."}
{"id": "mat_0162", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is inclusion-exclusion?", "choices": null, "answer": "It fixes double-counting by alternating union sums, exact for overlapping sets.", "steps": ["It fixes double-counting by alternating union sums, exact for overlapping sets.", "So the answer stands: It fixes double-counting by alternating union sums, exact for overlapping sets."], "text": "Answer: It fixes double-counting by alternating union sums, exact for overlapping sets.\nSteps:\n1. It fixes double-counting by alternating union sums, exact for overlapping sets.\n2. So the answer stands: It fixes double-counting by alternating union sums, exact for overlapping sets."}
{"id": "mat_0163", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Ukaž, že N² + N je sudé pro N = 2, výpočtem poloviny.", "choices": null, "answer": "Polovina: 3, číslo 6 je sudé.", "steps": ["2² + 2 = 6.", "6 / 2 = 3.", "Celé číslo, tedy sudé."], "text": "Answer: Polovina: 3, číslo 6 je sudé.\nSteps:\n1. 2² + 2 = 6.\n2. 6 / 2 = 3.\n3. Celé číslo, tedy sudé."}
{"id": "mat_0164", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 6.", "choices": null, "answer": "Integral: 216.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 6.", "6³ − 0 = 216."], "text": "Answer: Integral: 216.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 6.\n3. 6³ − 0 = 216."}
{"id": "mat_0165", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[6,0],[0,7]].", "choices": null, "answer": "Trace: 13.", "steps": ["Sum diagonal.", "6 + 7.", "= 13."], "text": "Answer: Trace: 13.\nSteps:\n1. Sum diagonal.\n2. 6 + 7.\n3. = 13."}
{"id": "mat_0166", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why distributions?", "choices": null, "answer": "Distributions list value probabilities compactly, summarizing all chance behaviour at once.", "steps": ["Distributions list value probabilities compactly, summarizing all chance behaviour at once.", "So the answer stands: Distributions list value probabilities compactly, summarizing all chance behaviour at once."], "text": "Answer: Distributions list value probabilities compactly, summarizing all chance behaviour at once.\nSteps:\n1. Distributions list value probabilities compactly, summarizing all chance behaviour at once.\n2. So the answer stands: Distributions list value probabilities compactly, summarizing all chance behaviour at once."}
{"id": "mat_0167", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(6, 8) step by step.", "choices": null, "answer": "lcm = 24.", "steps": ["gcd(6,8) = 2.", "6×8/2.", "= 24."], "text": "Answer: lcm = 24.\nSteps:\n1. gcd(6,8) = 2.\n2. 6×8/2.\n3. = 24."}
{"id": "mat_0168", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej permutační číslo P(3,2).", "choices": {"A": "3", "B": "6", "C": "9", "D": "12"}, "answer": "P(3,2) je 6.", "steps": ["3×2.", "= 6.", "Vyber možnost B."], "text": "Answer: 6\nSteps:\n1. 3×2.\n2. = 6.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0169", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why state quantifiers clearly?", "choices": null, "answer": "For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy.", "steps": ["For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy.", "So the answer stands: For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy."], "text": "Answer: For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy.\nSteps:\n1. For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy.\n2. So the answer stands: For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy."}
{"id": "mat_0170", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Proč derivovat?", "choices": null, "answer": "Derivování nachází maxima, minima a rychlosti.", "steps": ["Derivování nachází maxima, minima a rychlosti.", "Odpověď tedy platí: Derivování nachází maxima, minima a rychlosti."], "text": "Answer: Derivování nachází maxima, minima a rychlosti.\nSteps:\n1. Derivování nachází maxima, minima a rychlosti.\n2. Odpověď tedy platí: Derivování nachází maxima, minima a rychlosti."}
{"id": "mat_0171", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is a vector space?", "choices": null, "answer": "A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra.", "steps": ["A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra.", "So the answer stands: A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra."], "text": "Answer: A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra.\nSteps:\n1. A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra.\n2. So the answer stands: A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra."}
{"id": "mat_0172", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 7-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/6", "B": "1/7", "C": "1/8", "D": "2/7"}, "answer": "The probability is 1/7.", "steps": ["1 face of 7.", "P = 1/7.", "Select option B."], "text": "Answer: 1/7\nSteps:\n1. 1 face of 7.\n2. P = 1/7.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0173", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is modular arithmetic?", "choices": null, "answer": "It computes with remainders upon division, wrapping numbers around a modulus clock.", "steps": ["It computes with remainders upon division, wrapping numbers around a modulus clock.", "So the answer stands: It computes with remainders upon division, wrapping numbers around a modulus clock."], "text": "Answer: It computes with remainders upon division, wrapping numbers around a modulus clock.\nSteps:\n1. It computes with remainders upon division, wrapping numbers around a modulus clock.\n2. So the answer stands: It computes with remainders upon division, wrapping numbers around a modulus clock."}
{"id": "mat_0174", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why pigeonhole?", "choices": null, "answer": "More items than boxes forces sharing, proving existence without construction.", "steps": ["More items than boxes forces sharing, proving existence without construction.", "So the answer stands: More items than boxes forces sharing, proving existence without construction."], "text": "Answer: More items than boxes forces sharing, proving existence without construction.\nSteps:\n1. More items than boxes forces sharing, proving existence without construction.\n2. So the answer stands: More items than boxes forces sharing, proving existence without construction."}
{"id": "mat_0175", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is a constructive proof?", "choices": null, "answer": "A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example.", "steps": ["A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example.", "So the answer stands: A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example."], "text": "Answer: A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example.\nSteps:\n1. A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example.\n2. So the answer stands: A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example."}
{"id": "mat_0176", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 6²)/(x − 6) as x → 6.", "choices": null, "answer": "Limit: 12.", "steps": ["Factor: (x−6)(x+6).", "Cancel (x−6).", "Plug 6: 6+6 = 12."], "text": "Answer: Limit: 12.\nSteps:\n1. Factor: (x−6)(x+6).\n2. Cancel (x−6).\n3. Plug 6: 6+6 = 12."}
{"id": "mat_0177", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what measures angles?", "choices": {"A": "Dot product", "B": "Cross guess", "C": "Wild sum", "D": "Blind hope"}, "answer": "The answer is Dot product.", "steps": ["Zero means orthogonal.", "So: Dot product.", "Select option A."], "text": "Answer: Dot product\nSteps:\n1. Zero means orthogonal.\n2. So: Dot product.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0178", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is the binomial distribution?", "choices": null, "answer": "It counts successes in fixed trials with constant p, modelling repeated yes-no experiments.", "steps": ["It counts successes in fixed trials with constant p, modelling repeated yes-no experiments.", "So the answer stands: It counts successes in fixed trials with constant p, modelling repeated yes-no experiments."], "text": "Answer: It counts successes in fixed trials with constant p, modelling repeated yes-no experiments.\nSteps:\n1. It counts successes in fixed trials with constant p, modelling repeated yes-no experiments.\n2. So the answer stands: It counts successes in fixed trials with constant p, modelling repeated yes-no experiments."}
{"id": "mat_0179", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why congruences?", "choices": null, "answer": "Congruences preserve sums and products, letting big-number questions shrink to small cases.", "steps": ["Congruences preserve sums and products, letting big-number questions shrink to small cases.", "So the answer stands: Congruences preserve sums and products, letting big-number questions shrink to small cases."], "text": "Answer: Congruences preserve sums and products, letting big-number questions shrink to small cases.\nSteps:\n1. Congruences preserve sums and products, letting big-number questions shrink to small cases.\n2. So the answer stands: Congruences preserve sums and products, letting big-number questions shrink to small cases."}
{"id": "mat_0180", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(9,3).", "choices": null, "answer": "C(9,3) = 84.", "steps": ["n(n−1)(n−2)/6.", "9×8×7/6.", "= 84."], "text": "Answer: C(9,3) = 84.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 9×8×7/6.\n3. = 84."}
{"id": "mat_0181", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why use contrapositive?", "choices": null, "answer": "Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier.", "steps": ["Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier.", "So the answer stands: Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier."], "text": "Answer: Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier.\nSteps:\n1. Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier.\n2. So the answer stands: Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier."}
{"id": "mat_0182", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is an integral?", "choices": null, "answer": "A definite integral sums infinite thin slices, measuring area under a curve exactly.", "steps": ["A definite integral sums infinite thin slices, measuring area under a curve exactly.", "So the answer stands: A definite integral sums infinite thin slices, measuring area under a curve exactly."], "text": "Answer: A definite integral sums infinite thin slices, measuring area under a curve exactly.\nSteps:\n1. A definite integral sums infinite thin slices, measuring area under a curve exactly.\n2. So the answer stands: A definite integral sums infinite thin slices, measuring area under a curve exactly."}
{"id": "mat_0183", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[2,2],[3,5]].", "choices": {"A": "1", "B": "4", "C": "7", "D": "10"}, "answer": "The determinant is 4.", "steps": ["2×5 − 2×3.", "Det = 4.", "Select option B."], "text": "Answer: 4\nSteps:\n1. 2×5 − 2×3.\n2. Det = 4.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0184", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why the normal distribution?", "choices": null, "answer": "Averages of many variables turn normal, so it models errors and natural variation.", "steps": ["Averages of many variables turn normal, so it models errors and natural variation.", "So the answer stands: Averages of many variables turn normal, so it models errors and natural variation."], "text": "Answer: Averages of many variables turn normal, so it models errors and natural variation.\nSteps:\n1. Averages of many variables turn normal, so it models errors and natural variation.\n2. So the answer stands: Averages of many variables turn normal, so it models errors and natural variation."}
{"id": "mat_0185", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what forces new primes?", "choices": {"A": "Euclid's argument", "B": "Big computers", "C": "Long lists", "D": "Loud claims"}, "answer": "The answer is Euclid's argument.", "steps": ["Product plus one.", "So: Euclid's argument.", "Select option A."], "text": "Answer: Euclid's argument\nSteps:\n1. Product plus one.\n2. So: Euclid's argument.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0186", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Co je kombinatorika?", "choices": null, "answer": "Kombinatorika počítá uspořádání a výběry.", "steps": ["Kombinatorika počítá uspořádání a výběry.", "Odpověď tedy platí: Kombinatorika počítá uspořádání a výběry."], "text": "Answer: Kombinatorika počítá uspořádání a výběry.\nSteps:\n1. Kombinatorika počítá uspořádání a výběry.\n2. Odpověď tedy platí: Kombinatorika počítá uspořádání a výběry."}
{"id": "mat_0187", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what disproves a for-all claim?", "choices": {"A": "One counterexample", "B": "Two examples", "C": "Loud denial", "D": "Long proof"}, "answer": "The answer is One counterexample.", "steps": ["Single failure kills it.", "So: One counterexample.", "Select option A."], "text": "Answer: One counterexample\nSteps:\n1. Single failure kills it.\n2. So: One counterexample.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0188", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 3x² + 5x, compute f'(3) step by step.", "choices": null, "answer": "f'(3) = 23.", "steps": ["f'(x) = 6x + 5.", "Plug x = 3.", "6×3 + 5 = 23."], "text": "Answer: f'(3) = 23.\nSteps:\n1. f'(x) = 6x + 5.\n2. Plug x = 3.\n3. 6×3 + 5 = 23."}
{"id": "mat_0189", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[3,1],[2,6]].", "choices": {"A": "7", "B": "11", "C": "9", "D": "13"}, "answer": "The trace is 9.", "steps": ["3 + 6.", "Trace = 9.", "Select option C."], "text": "Answer: 9\nSteps:\n1. 3 + 6.\n2. Trace = 9.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0190", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Najdi střední hodnotu rovnoměrného tahu z {1..4}.", "choices": {"A": "3,5", "B": "2,5", "C": "1,5", "D": "4,5"}, "answer": "Střední hodnota je 2,5.", "steps": ["(4+1)/2.", "E = 2,5.", "Vyber možnost B."], "text": "Answer: 2,5\nSteps:\n1. (4+1)/2.\n2. E = 2,5.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0191", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what seeks integer solutions?", "choices": {"A": "Real guess", "B": "Diophantine equation", "C": "Complex hope", "D": "Vague wish"}, "answer": "The answer is Diophantine equation.", "steps": ["Gcd conditions apply.", "So: Diophantine equation.", "Select option B."], "text": "Answer: Diophantine equation\nSteps:\n1. Gcd conditions apply.\n2. So: Diophantine equation.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0192", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(9,2).", "choices": null, "answer": "C(9,2) = 36.", "steps": ["Formula n(n−1)/2.", "9×8/2.", "= 36."], "text": "Answer: C(9,2) = 36.\nSteps:\n1. Formula n(n−1)/2.\n2. 9×8/2.\n3. = 36."}
{"id": "mat_0193", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is a lemma?", "choices": null, "answer": "A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable.", "steps": ["A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable.", "So the answer stands: A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable."], "text": "Answer: A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable.\nSteps:\n1. A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable.\n2. So the answer stands: A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable."}
{"id": "mat_0194", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Co je řetízkové pravidlo?", "choices": null, "answer": "Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace.", "steps": ["Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace.", "Odpověď tedy platí: Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace."], "text": "Answer: Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace.\nSteps:\n1. Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace.\n2. Odpověď tedy platí: Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace."}
{"id": "mat_0195", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why check linear independence?", "choices": null, "answer": "Independence means no vector is redundant, so each adds a new dimension to the span.", "steps": ["Independence means no vector is redundant, so each adds a new dimension to the span.", "So the answer stands: Independence means no vector is redundant, so each adds a new dimension to the span."], "text": "Answer: Independence means no vector is redundant, so each adds a new dimension to the span.\nSteps:\n1. Independence means no vector is redundant, so each adds a new dimension to the span.\n2. So the answer stands: Independence means no vector is redundant, so each adds a new dimension to the span."}
{"id": "mat_0196", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Co je podmíněná pravděpodobnost?", "choices": null, "answer": "Aktualizuje šanci při nové evidenci přes P(A|B).", "steps": ["Aktualizuje šanci při nové evidenci přes P(A|B).", "Odpověď tedy platí: Aktualizuje šanci při nové evidenci přes P(A|B)."], "text": "Answer: Aktualizuje šanci při nové evidenci přes P(A|B).\nSteps:\n1. Aktualizuje šanci při nové evidenci přes P(A|B).\n2. Odpověď tedy platí: Aktualizuje šanci při nové evidenci přes P(A|B)."}
{"id": "mat_0197", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(16, 24) step by step.", "choices": null, "answer": "gcd = 8.", "steps": ["Factor: 16=4×4, 24=6×4.", "Common: 4×gcd(4,6).", "= 4×2 = 8."], "text": "Answer: gcd = 8.\nSteps:\n1. Factor: 16=4×4, 24=6×4.\n2. Common: 4×gcd(4,6).\n3. = 4×2 = 8."}
{"id": "mat_0198", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet P(4,3).", "choices": null, "answer": "P(4,3) = 24.", "steps": ["n(n−1)(n−2).", "4×3×2.", "= 24."], "text": "Answer: P(4,3) = 24.\nSteps:\n1. n(n−1)(n−2).\n2. 4×3×2.\n3. = 24."}
{"id": "mat_0199", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Důkaz prochází všechny případy 3 booleovských proměnných. Kolik řádků?", "choices": null, "answer": "Řádků: 8.", "steps": ["2 volby každá.", "2^3 = 8."], "text": "Answer: Řádků: 8.\nSteps:\n1. 2 volby každá.\n2. 2^3 = 8."}
{"id": "mat_0200", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 7.", "choices": null, "answer": "Integral: 343.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 7.", "7³ − 0 = 343."], "text": "Answer: Integral: 343.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 7.\n3. 7³ − 0 = 343."}
{"id": "mat_0201", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Proč matice?", "choices": null, "answer": "Matice kódují lineární zobrazení a soustavy kompaktně.", "steps": ["Matice kódují lineární zobrazení a soustavy kompaktně.", "Odpověď tedy platí: Matice kódují lineární zobrazení a soustavy kompaktně."], "text": "Answer: Matice kódují lineární zobrazení a soustavy kompaktně.\nSteps:\n1. Matice kódují lineární zobrazení a soustavy kompaktně.\n2. Odpověď tedy platí: Matice kódují lineární zobrazení a soustavy kompaktně."}
{"id": "mat_0202", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..8}.", "choices": null, "answer": "E = 4.5.", "steps": ["Mean of 1..8.", "(8+1)/2.", "E = 4.5."], "text": "Answer: E = 4.5.\nSteps:\n1. Mean of 1..8.\n2. (8+1)/2.\n3. E = 4.5."}
{"id": "mat_0203", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (72) mod 7.", "choices": null, "answer": "Remainder: 2.", "steps": ["Divide 72 by 7.", "7×10 = 70.", "72 − 70 = 2."], "text": "Answer: Remainder: 2.\nSteps:\n1. Divide 72 by 7.\n2. 7×10 = 70.\n3. 72 − 70 = 2."}
{"id": "mat_0204", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Proč faktoriály?", "choices": null, "answer": "Faktoriály počítají pořadí různých objektů.", "steps": ["Faktoriály počítají pořadí různých objektů.", "Odpověď tedy platí: Faktoriály počítají pořadí různých objektů."], "text": "Answer: Faktoriály počítají pořadí různých objektů.\nSteps:\n1. Faktoriály počítají pořadí různých objektů.\n2. Odpověď tedy platí: Faktoriály počítají pořadí různých objektů."}
{"id": "mat_0205", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Co je indukce?", "choices": null, "answer": "Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla.", "steps": ["Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla.", "Odpověď tedy platí: Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla."], "text": "Answer: Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla.\nSteps:\n1. Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla.\n2. Odpověď tedy platí: Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla."}
{"id": "mat_0206", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 7²)/(x − 7) as x → 7.", "choices": null, "answer": "Limit: 14.", "steps": ["Factor: (x−7)(x+7).", "Cancel (x−7).", "Plug 7: 7+7 = 14."], "text": "Answer: Limit: 14.\nSteps:\n1. Factor: (x−7)(x+7).\n2. Cancel (x−7).\n3. Plug 7: 7+7 = 14."}
{"id": "mat_0207", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (2,3) · (3,4).", "choices": {"A": "14", "B": "22", "C": "26", "D": "18"}, "answer": "The dot product is 18.", "steps": ["2×3 + 3×4.", "Product = 18.", "Select option D."], "text": "Answer: 18\nSteps:\n1. 2×3 + 3×4.\n2. Product = 18.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0208", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 6 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 6/10.", "steps": ["Red: 6.", "Total: 10.", "P = 6/10."], "text": "Answer: P(red) = 6/10.\nSteps:\n1. Red: 6.\n2. Total: 10.\n3. P = 6/10."}
{"id": "mat_0209", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(24, 36).", "choices": {"A": "9", "B": "15", "C": "12", "D": "18"}, "answer": "The gcd is 12.", "steps": ["Factor 3 out twice.", "gcd = 12.", "Select option C."], "text": "Answer: 12\nSteps:\n1. Factor 3 out twice.\n2. gcd = 12.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0210", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej kombinační číslo C(5,3).", "choices": {"A": "5", "B": "15", "C": "10", "D": "20"}, "answer": "C(5,3) je 10.", "steps": ["5×4×3/6.", "= 10.", "Vyber možnost C."], "text": "Answer: 10\nSteps:\n1. 5×4×3/6.\n2. = 10.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0211", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 5 by computing its half.", "choices": null, "answer": "Half: 15, so 30 is even.", "steps": ["5² + 5 = 30.", "30 / 2 = 15.", "Integer, so even."], "text": "Answer: Half: 15, so 30 is even.\nSteps:\n1. 5² + 5 = 30.\n2. 30 / 2 = 15.\n3. Integer, so even."}
{"id": "mat_0212", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 4x² + 6x, compute f'(4) step by step.", "choices": null, "answer": "f'(4) = 38.", "steps": ["f'(x) = 8x + 6.", "Plug x = 4.", "8×4 + 6 = 38."], "text": "Answer: f'(4) = 38.\nSteps:\n1. f'(x) = 8x + 6.\n2. Plug x = 4.\n3. 8×4 + 6 = 38."}
{"id": "mat_0213", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (5,4) · (3,5) step by step.", "choices": null, "answer": "Dot product: 35.", "steps": ["Multiply components.", "5×3 + 4×5.", "= 35."], "text": "Answer: Dot product: 35.\nSteps:\n1. Multiply components.\n2. 5×3 + 4×5.\n3. = 35."}
{"id": "mat_0214", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..5}.", "choices": {"A": "4", "B": "2", "C": "3", "D": "5"}, "answer": "The expected value is 3.", "steps": ["(5+1)/2.", "E = 3.", "Select option C."], "text": "Answer: 3\nSteps:\n1. (5+1)/2.\n2. E = 3.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0215", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(7, 8) step by step.", "choices": null, "answer": "lcm = 56.", "steps": ["gcd(7,8) = 1.", "7×8/1.", "= 56."], "text": "Answer: lcm = 56.\nSteps:\n1. gcd(7,8) = 1.\n2. 7×8/1.\n3. = 56."}
{"id": "mat_0216", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet C(6,3).", "choices": null, "answer": "C(6,3) = 20.", "steps": ["n(n−1)(n−2)/6.", "6×5×4/6.", "= 20."], "text": "Answer: C(6,3) = 20.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 6×5×4/6.\n3. = 20."}
{"id": "mat_0217", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why define terms first?", "choices": null, "answer": "Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity.", "steps": ["Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity.", "So the answer stands: Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity."], "text": "Answer: Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity.\nSteps:\n1. Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity.\n2. So the answer stands: Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity."}
{"id": "mat_0218", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Co je integrál?", "choices": null, "answer": "Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou.", "steps": ["Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou.", "Odpověď tedy platí: Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou."], "text": "Answer: Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou.\nSteps:\n1. Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou.\n2. Odpověď tedy platí: Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou."}
{"id": "mat_0219", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what counts image dimension?", "choices": {"A": "Rank", "B": "Size", "C": "Color", "D": "Age"}, "answer": "The answer is Rank.", "steps": ["Independent rows tallied.", "So: Rank.", "Select option A."], "text": "Answer: Rank\nSteps:\n1. Independent rows tallied.\n2. So: Rank.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0220", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 3 red and 7 blue marbles. What is P(red)?", "choices": {"A": "7/10", "B": "3/11", "C": "4/10", "D": "3/10"}, "answer": "P(red) is 3/10.", "steps": ["3 of 10.", "P = 3/10.", "Select option D."], "text": "Answer: 3/10\nSteps:\n1. 3 of 10.\n2. P = 3/10.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0221", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej NSD(12, 18) krok za krokem.", "choices": null, "answer": "NSD = 6.", "steps": ["Rozlož: 12=4×3, 18=6×3.", "Společné: 3×NSD(4,6).", "= 3×2 = 6."], "text": "Answer: NSD = 6.\nSteps:\n1. Rozlož: 12=4×3, 18=6×3.\n2. Společné: 3×NSD(4,6).\n3. = 3×2 = 6."}
{"id": "mat_0222", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(8,3).", "choices": null, "answer": "P(8,3) = 336.", "steps": ["n(n−1)(n−2).", "8×7×6.", "= 336."], "text": "Answer: P(8,3) = 336.\nSteps:\n1. n(n−1)(n−2).\n2. 8×7×6.\n3. = 336."}
{"id": "mat_0223", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 6 boolean variables. How many rows?", "choices": null, "answer": "Rows: 64.", "steps": ["2 choices each.", "2^6 = 64."], "text": "Answer: Rows: 64.\nSteps:\n1. 2 choices each.\n2. 2^6 = 64."}
{"id": "mat_0224", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 2x² + 3x, find f'(2).", "choices": {"A": "9", "B": "11", "C": "13", "D": "15"}, "answer": "The derivative is 11.", "steps": ["f'(x) = 4x + 3.", "At 2: 11.", "Select option B."], "text": "Answer: 11\nSteps:\n1. f'(x) = 4x + 3.\n2. At 2: 11.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0225", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej (2,1) · (3,5) krok za krokem.", "choices": null, "answer": "Skalární součin: 11.", "steps": ["Násob složky.", "2×3 + 1×5.", "= 11."], "text": "Answer: Skalární součin: 11.\nSteps:\n1. Násob složky.\n2. 2×3 + 1×5.\n3. = 11."}
{"id": "mat_0226", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what updates with evidence?", "choices": {"A": "Conditional probability", "B": "Blind guess", "C": "Stubborn prior", "D": "Wild hope"}, "answer": "The answer is Conditional probability.", "steps": ["Space shrinks, ratio taken.", "So: Conditional probability.", "Select option A."], "text": "Answer: Conditional probability\nSteps:\n1. Space shrinks, ratio taken.\n2. So: Conditional probability.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0227", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (114) mod 9.", "choices": {"A": "7", "B": "8", "C": "0", "D": "6"}, "answer": "The remainder is 6.", "steps": ["114 ÷ 9.", "Remainder 6.", "Select option D."], "text": "Answer: 6\nSteps:\n1. 114 ÷ 9.\n2. Remainder 6.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0228", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(6,3).", "choices": {"A": "14", "B": "26", "C": "32", "D": "20"}, "answer": "C(6,3) is 20.", "steps": ["6×5×4/6.", "= 20.", "Select option D."], "text": "Answer: 20\nSteps:\n1. 6×5×4/6.\n2. = 20.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0229", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Co je protipříklad?", "choices": null, "answer": "Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí.", "steps": ["Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí.", "Odpověď tedy platí: Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí."], "text": "Answer: Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí.\nSteps:\n1. Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí.\n2. Odpověď tedy platí: Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí."}
{"id": "mat_0230", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 3.", "choices": {"A": "24", "B": "30", "C": "27", "D": "33"}, "answer": "The integral is 27.", "steps": ["Antiderivative x³.", "3³ = 27.", "Select option C."], "text": "Answer: 27\nSteps:\n1. Antiderivative x³.\n2. 3³ = 27.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0231", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what finds closest point?", "choices": {"A": "Guessing", "B": "Projection", "C": "Hoping", "D": "Waiting"}, "answer": "The answer is Projection.", "steps": ["Error orthogonal.", "So: Projection.", "Select option B."], "text": "Answer: Projection\nSteps:\n1. Error orthogonal.\n2. So: Projection.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0232", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 10-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/10.", "steps": ["Favourable: 1.", "Possible: 10.", "P = 1/10."], "text": "Answer: P = 1/10.\nSteps:\n1. Favourable: 1.\n2. Possible: 10.\n3. P = 1/10."}
{"id": "mat_0233", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Proč prvočísla?", "choices": null, "answer": "Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá.", "steps": ["Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá.", "Odpověď tedy platí: Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá."], "text": "Answer: Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá.\nSteps:\n1. Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá.\n2. Odpověď tedy platí: Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá."}
{"id": "mat_0234", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what counts subsets?", "choices": {"A": "nCr", "B": "n+r", "C": "n*r", "D": "n+r!"}, "answer": "The answer is nCr.", "steps": ["Order divided out.", "So: nCr.", "Select option A."], "text": "Answer: nCr\nSteps:\n1. Order divided out.\n2. So: nCr.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0235", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what must cases cover?", "choices": {"A": "Easy ones", "B": "Every possibility", "C": "First two", "D": "Odd cases"}, "answer": "The answer is Every possibility.", "steps": ["Gaps break proofs.", "So: Every possibility.", "Select option B."], "text": "Answer: Every possibility\nSteps:\n1. Gaps break proofs.\n2. So: Every possibility.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0236", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 3²)/(x − 3) as x → 3.", "choices": {"A": "4", "B": "8", "C": "10", "D": "6"}, "answer": "The limit is 6.", "steps": ["Factor (x−3)(x+3).", "Cancel, get 2×3 = 6.", "Select option D."], "text": "Answer: 6\nSteps:\n1. Factor (x−3)(x+3).\n2. Cancel, get 2×3 = 6.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0237", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is a basis?", "choices": null, "answer": "A basis is an independent spanning set, giving every vector unique coordinates.", "steps": ["A basis is an independent spanning set, giving every vector unique coordinates.", "So the answer stands: A basis is an independent spanning set, giving every vector unique coordinates."], "text": "Answer: A basis is an independent spanning set, giving every vector unique coordinates.\nSteps:\n1. A basis is an independent spanning set, giving every vector unique coordinates.\n2. So the answer stands: A basis is an independent spanning set, giving every vector unique coordinates."}
{"id": "mat_0238", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..9}.", "choices": null, "answer": "E = 5.", "steps": ["Mean of 1..9.", "(9+1)/2.", "E = 5."], "text": "Answer: E = 5.\nSteps:\n1. Mean of 1..9.\n2. (9+1)/2.\n3. E = 5."}
{"id": "mat_0239", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej (61) mod 7.", "choices": null, "answer": "Zbytek: 5.", "steps": ["Děl 61 sedmi.", "7×8 = 56.", "61 − 56 = 5."], "text": "Answer: Zbytek: 5.\nSteps:\n1. Děl 61 sedmi.\n2. 7×8 = 56.\n3. 61 − 56 = 5."}
{"id": "mat_0240", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what pairs sets equally?", "choices": {"A": "Wild map", "B": "Bijection", "C": "Blind guess", "D": "Vague hope"}, "answer": "The answer is Bijection.", "steps": ["Inverse proves size.", "So: Bijection.", "Select option B."], "text": "Answer: Bijection\nSteps:\n1. Inverse proves size.\n2. So: Bijection.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0241", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 4 variables have?", "choices": {"A": "14", "B": "18", "C": "16", "D": "20"}, "answer": "The table has 16 rows.", "steps": ["2^4 rows.", "Total: 16.", "Select option C."], "text": "Answer: 16\nSteps:\n1. 2^4 rows.\n2. Total: 16.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0242", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Vyčísli integrál 3x² od 0 do 3.", "choices": null, "answer": "Integrál: 27.", "steps": ["Primitivní: x³.", "Vyčísli 0 až 3.", "3³ − 0 = 27."], "text": "Answer: Integrál: 27.\nSteps:\n1. Primitivní: x³.\n2. Vyčísli 0 až 3.\n3. 3³ − 0 = 27."}
{"id": "mat_0243", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why count dimension?", "choices": null, "answer": "Dimension is the basis size, fixing how many numbers describe every vector exactly.", "steps": ["Dimension is the basis size, fixing how many numbers describe every vector exactly.", "So the answer stands: Dimension is the basis size, fixing how many numbers describe every vector exactly."], "text": "Answer: Dimension is the basis size, fixing how many numbers describe every vector exactly.\nSteps:\n1. Dimension is the basis size, fixing how many numbers describe every vector exactly.\n2. So the answer stands: Dimension is the basis size, fixing how many numbers describe every vector exactly."}
{"id": "mat_0244", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is the law of large numbers?", "choices": null, "answer": "Sample averages converge to expectation as trials grow, justifying statistics from data.", "steps": ["Sample averages converge to expectation as trials grow, justifying statistics from data.", "So the answer stands: Sample averages converge to expectation as trials grow, justifying statistics from data."], "text": "Answer: Sample averages converge to expectation as trials grow, justifying statistics from data.\nSteps:\n1. Sample averages converge to expectation as trials grow, justifying statistics from data.\n2. So the answer stands: Sample averages converge to expectation as trials grow, justifying statistics from data."}
{"id": "mat_0245", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(20, 30) step by step.", "choices": null, "answer": "gcd = 10.", "steps": ["Factor: 20=4×5, 30=6×5.", "Common: 5×gcd(4,6).", "= 5×2 = 10."], "text": "Answer: gcd = 10.\nSteps:\n1. Factor: 20=4×5, 30=6×5.\n2. Common: 5×gcd(4,6).\n3. = 5×2 = 10."}
{"id": "mat_0246", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is a permutation?", "choices": null, "answer": "A permutation orders objects, and n distinct objects admit exactly n! orders.", "steps": ["A permutation orders objects, and n distinct objects admit exactly n! orders.", "So the answer stands: A permutation orders objects, and n distinct objects admit exactly n! orders."], "text": "Answer: A permutation orders objects, and n distinct objects admit exactly n! orders.\nSteps:\n1. A permutation orders objects, and n distinct objects admit exactly n! orders.\n2. So the answer stands: A permutation orders objects, and n distinct objects admit exactly n! orders."}
{"id": "mat_0247", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 10.", "choices": null, "answer": "Sum: 55.", "steps": ["Pair 1+10, 2+9, ...", "Use 10×11/2.", "= 55."], "text": "Answer: Sum: 55.\nSteps:\n1. Pair 1+10, 2+9, ...\n2. Use 10×11/2.\n3. = 55."}
{"id": "mat_0248", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Proč integrovat?", "choices": null, "answer": "Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy.", "steps": ["Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy.", "Odpověď tedy platí: Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy."], "text": "Answer: Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy.\nSteps:\n1. Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy.\n2. Odpověď tedy platí: Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy."}
{"id": "mat_0249", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[6,1],[2,7]] step by step.", "choices": null, "answer": "Determinant: 40.", "steps": ["ad − bc.", "6×7 − 1×2.", "= 40."], "text": "Answer: Determinant: 40.\nSteps:\n1. ad − bc.\n2. 6×7 − 1×2.\n3. = 40."}
{"id": "mat_0250", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what models error bells?", "choices": {"A": "Wild guess", "B": "Normal distribution", "C": "Flat line", "D": "Single dot"}, "answer": "The answer is Normal distribution.", "steps": ["Averages turn normal.", "So: Normal distribution.", "Select option B."], "text": "Answer: Normal distribution\nSteps:\n1. Averages turn normal.\n2. So: Normal distribution.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0251", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(6, 6).", "choices": {"A": "6", "B": "4", "C": "8", "D": "10"}, "answer": "The lcm is 6.", "steps": ["6×6/gcd.", "lcm = 6.", "Select option A."], "text": "Answer: 6\nSteps:\n1. 6×6/gcd.\n2. lcm = 6.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0252", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(10,3).", "choices": null, "answer": "C(10,3) = 120.", "steps": ["n(n−1)(n−2)/6.", "10×9×8/6.", "= 120."], "text": "Answer: C(10,3) = 120.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 10×9×8/6.\n3. = 120."}
{"id": "mat_0253", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 6 by computing its half.", "choices": null, "answer": "Half: 21, so 42 is even.", "steps": ["6² + 6 = 42.", "42 / 2 = 21.", "Integer, so even."], "text": "Answer: Half: 21, so 42 is even.\nSteps:\n1. 6² + 6 = 42.\n2. 42 / 2 = 21.\n3. Integer, so even."}
{"id": "mat_0254", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Vyčísli limitu (x² − 3²)/(x − 3) pro x → 3.", "choices": null, "answer": "Limita: 6.", "steps": ["Rozlož: (x−3)(x+3).", "Pokrať (x−3).", "Dosaď 3: 3+3 = 6."], "text": "Answer: Limita: 6.\nSteps:\n1. Rozlož: (x−3)(x+3).\n2. Pokrať (x−3).\n3. Dosaď 3: 3+3 = 6."}
{"id": "mat_0255", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[3,2],[3,6]].", "choices": {"A": "9", "B": "15", "C": "12", "D": "18"}, "answer": "The determinant is 12.", "steps": ["3×6 − 2×3.", "Det = 12.", "Select option C."], "text": "Answer: 12\nSteps:\n1. 3×6 − 2×3.\n2. Det = 12.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0256", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Proč Bayesova věta?", "choices": null, "answer": "Bayes obrací podmíněnosti a usuzuje příčiny z následků.", "steps": ["Bayes obrací podmíněnosti a usuzuje příčiny z následků.", "Odpověď tedy platí: Bayes obrací podmíněnosti a usuzuje příčiny z následků."], "text": "Answer: Bayes obrací podmíněnosti a usuzuje příčiny z následků.\nSteps:\n1. Bayes obrací podmíněnosti a usuzuje příčiny z následků.\n2. Odpověď tedy platí: Bayes obrací podmíněnosti a usuzuje příčiny z následků."}
{"id": "mat_0257", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Najdi nejmenší společný násobek NSN(4, 6).", "choices": {"A": "10", "B": "14", "C": "12", "D": "16"}, "answer": "NSN je 12.", "steps": ["4×6/NSD.", "NSN = 12.", "Vyber možnost C."], "text": "Answer: 12\nSteps:\n1. 4×6/NSD.\n2. NSN = 12.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0258", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(10,2).", "choices": null, "answer": "C(10,2) = 45.", "steps": ["Formula n(n−1)/2.", "10×9/2.", "= 45."], "text": "Answer: C(10,2) = 45.\nSteps:\n1. Formula n(n−1)/2.\n2. 10×9/2.\n3. = 45."}
{"id": "mat_0259", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Dokaž vzorcem: najdi součet 1 + 2 + ... + 7.", "choices": null, "answer": "Součet: 28.", "steps": ["Páruj 1+7, 2+6, ...", "Použij 7×8/2.", "= 28."], "text": "Answer: Součet: 28.\nSteps:\n1. Páruj 1+7, 2+6, ...\n2. Použij 7×8/2.\n3. = 28."}
{"id": "mat_0260", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why integrate?", "choices": null, "answer": "Integration recovers totals from rates, computing areas, volumes, and accumulated change.", "steps": ["Integration recovers totals from rates, computing areas, volumes, and accumulated change.", "So the answer stands: Integration recovers totals from rates, computing areas, volumes, and accumulated change."], "text": "Answer: Integration recovers totals from rates, computing areas, volumes, and accumulated change.\nSteps:\n1. Integration recovers totals from rates, computing areas, volumes, and accumulated change.\n2. So the answer stands: Integration recovers totals from rates, computing areas, volumes, and accumulated change."}
{"id": "mat_0261", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[7,0],[0,8]].", "choices": null, "answer": "Trace: 15.", "steps": ["Sum diagonal.", "7 + 8.", "= 15."], "text": "Answer: Trace: 15.\nSteps:\n1. Sum diagonal.\n2. 7 + 8.\n3. = 15."}
{"id": "mat_0262", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 8-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/7", "B": "1/9", "C": "1/8", "D": "2/8"}, "answer": "The probability is 1/8.", "steps": ["1 face of 8.", "P = 1/8.", "Select option C."], "text": "Answer: 1/8\nSteps:\n1. 1 face of 8.\n2. P = 1/8.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0263", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej NSN(6, 8) krok za krokem.", "choices": null, "answer": "NSN = 24.", "steps": ["NSD(6,8) = 2.", "6×8/2.", "= 24."], "text": "Answer: NSN = 24.\nSteps:\n1. NSD(6,8) = 2.\n2. 6×8/2.\n3. = 24."}
{"id": "mat_0264", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why derangements?", "choices": null, "answer": "Derangements count fixed-point-free orders, modelling secret Santas and mismatches.", "steps": ["Derangements count fixed-point-free orders, modelling secret Santas and mismatches.", "So the answer stands: Derangements count fixed-point-free orders, modelling secret Santas and mismatches."], "text": "Answer: Derangements count fixed-point-free orders, modelling secret Santas and mismatches.\nSteps:\n1. Derangements count fixed-point-free orders, modelling secret Santas and mismatches.\n2. So the answer stands: Derangements count fixed-point-free orders, modelling secret Santas and mismatches."}
{"id": "mat_0265", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 7?", "choices": {"A": "21", "B": "35", "C": "42", "D": "28"}, "answer": "The sum is 28.", "steps": ["7×8/2.", "Sum = 28.", "Select option D."], "text": "Answer: 28\nSteps:\n1. 7×8/2.\n2. Sum = 28.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0266", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 8.", "choices": null, "answer": "Integral: 512.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 8.", "8³ − 0 = 512."], "text": "Answer: Integral: 512.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 8.\n3. 8³ − 0 = 512."}
{"id": "mat_0267", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[4,1],[2,7]].", "choices": {"A": "9", "B": "13", "C": "15", "D": "11"}, "answer": "The trace is 11.", "steps": ["4 + 7.", "Trace = 11.", "Select option D."], "text": "Answer: 11\nSteps:\n1. 4 + 7.\n2. Trace = 11.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0268", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Hází se férovou 6-stěnnou kostkou. Ukaž P(jedno konkrétní číslo).", "choices": null, "answer": "P = 1/6.", "steps": ["Příznivé: 1.", "Možné: 6.", "P = 1/6."], "text": "Answer: P = 1/6.\nSteps:\n1. Příznivé: 1.\n2. Možné: 6.\n3. P = 1/6."}
{"id": "mat_0269", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Co je základní věta aritmetiky?", "choices": null, "answer": "Každé číslo nad 1 se jednoznačně rozkládá na prvočísla.", "steps": ["Každé číslo nad 1 se jednoznačně rozkládá na prvočísla.", "Odpověď tedy platí: Každé číslo nad 1 se jednoznačně rozkládá na prvočísla."], "text": "Answer: Každé číslo nad 1 se jednoznačně rozkládá na prvočísla.\nSteps:\n1. Každé číslo nad 1 se jednoznačně rozkládá na prvočísla.\n2. Odpověď tedy platí: Každé číslo nad 1 se jednoznačně rozkládá na prvočísla."}
{"id": "mat_0270", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(6,2).", "choices": {"A": "9", "B": "21", "C": "15", "D": "27"}, "answer": "C(6,2) is 15.", "steps": ["6×5/2.", "= 15.", "Select option C."], "text": "Answer: 15\nSteps:\n1. 6×5/2.\n2. = 15.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0271", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 5.", "choices": {"A": "30", "B": "28", "C": "32", "D": "34"}, "answer": "The value is 30.", "steps": ["5² + 5.", "Value = 30.", "Select option A."], "text": "Answer: 30\nSteps:\n1. 5² + 5.\n2. Value = 30.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0272", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is the fundamental theorem of calculus?", "choices": null, "answer": "It links derivatives and integrals, so definite integrals equal antiderivative differences.", "steps": ["It links derivatives and integrals, so definite integrals equal antiderivative differences.", "So the answer stands: It links derivatives and integrals, so definite integrals equal antiderivative differences."], "text": "Answer: It links derivatives and integrals, so definite integrals equal antiderivative differences.\nSteps:\n1. It links derivatives and integrals, so definite integrals equal antiderivative differences.\n2. So the answer stands: It links derivatives and integrals, so definite integrals equal antiderivative differences."}
{"id": "mat_0273", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (6,5) · (3,5) step by step.", "choices": null, "answer": "Dot product: 43.", "steps": ["Multiply components.", "6×3 + 5×5.", "= 43."], "text": "Answer: Dot product: 43.\nSteps:\n1. Multiply components.\n2. 6×3 + 5×5.\n3. = 43."}
{"id": "mat_0274", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Co je nezávislost?", "choices": null, "answer": "Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý.", "steps": ["Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý.", "Odpověď tedy platí: Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý."], "text": "Answer: Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý.\nSteps:\n1. Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý.\n2. Odpověď tedy platí: Nezávislé jevy násobí pravděpodobnosti, výsledek jednoho neovlivní druhý."}
{"id": "mat_0275", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (83) mod 7.", "choices": null, "answer": "Remainder: 6.", "steps": ["Divide 83 by 7.", "7×11 = 77.", "83 − 77 = 6."], "text": "Answer: Remainder: 6.\nSteps:\n1. Divide 83 by 7.\n2. 7×11 = 77.\n3. 83 − 77 = 6."}
{"id": "mat_0276", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(5,2).", "choices": {"A": "15", "B": "25", "C": "30", "D": "20"}, "answer": "P(5,2) is 20.", "steps": ["5×4.", "= 20.", "Select option D."], "text": "Answer: 20\nSteps:\n1. 5×4.\n2. = 20.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0277", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 7 boolean variables. How many rows?", "choices": null, "answer": "Rows: 128.", "steps": ["2 choices each.", "2^7 = 128."], "text": "Answer: Rows: 128.\nSteps:\n1. 2 choices each.\n2. 2^7 = 128."}
{"id": "mat_0278", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why substitute in integrals?", "choices": null, "answer": "Substitution rewrites messy integrals in simpler variables, undoing the chain rule.", "steps": ["Substitution rewrites messy integrals in simpler variables, undoing the chain rule.", "So the answer stands: Substitution rewrites messy integrals in simpler variables, undoing the chain rule."], "text": "Answer: Substitution rewrites messy integrals in simpler variables, undoing the chain rule.\nSteps:\n1. Substitution rewrites messy integrals in simpler variables, undoing the chain rule.\n2. So the answer stands: Substitution rewrites messy integrals in simpler variables, undoing the chain rule."}
{"id": "mat_0279", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[7,1],[2,8]] step by step.", "choices": null, "answer": "Determinant: 54.", "steps": ["ad − bc.", "7×8 − 1×2.", "= 54."], "text": "Answer: Determinant: 54.\nSteps:\n1. ad − bc.\n2. 7×8 − 1×2.\n3. = 54."}
{"id": "mat_0280", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 7 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 7/11.", "steps": ["Red: 7.", "Total: 11.", "P = 7/11."], "text": "Answer: P(red) = 7/11.\nSteps:\n1. Red: 7.\n2. Total: 11.\n3. P = 7/11."}
{"id": "mat_0281", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(8, 8) step by step.", "choices": null, "answer": "lcm = 8.", "steps": ["gcd(8,8) = 8.", "8×8/8.", "= 8."], "text": "Answer: lcm = 8.\nSteps:\n1. gcd(8,8) = 8.\n2. 8×8/8.\n3. = 8."}
{"id": "mat_0282", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is a combination?", "choices": null, "answer": "A combination selects without order, counted by nCr for r from n.", "steps": ["A combination selects without order, counted by nCr for r from n.", "So the answer stands: A combination selects without order, counted by nCr for r from n."], "text": "Answer: A combination selects without order, counted by nCr for r from n.\nSteps:\n1. A combination selects without order, counted by nCr for r from n.\n2. So the answer stands: A combination selects without order, counted by nCr for r from n."}
{"id": "mat_0283", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what does contradiction assume?", "choices": {"A": "Claim is true", "B": "Claim is false", "C": "Nothing at all", "D": "Two claims"}, "answer": "The answer is Claim is false.", "steps": ["Absurdity follows.", "So: Claim is false.", "Select option B."], "text": "Answer: Claim is false\nSteps:\n1. Absurdity follows.\n2. So: Claim is false.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0284", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is integration by parts?", "choices": null, "answer": "It trades one hard integral for an easier one via uv minus the integral of v du.", "steps": ["It trades one hard integral for an easier one via uv minus the integral of v du.", "So the answer stands: It trades one hard integral for an easier one via uv minus the integral of v du."], "text": "Answer: It trades one hard integral for an easier one via uv minus the integral of v du.\nSteps:\n1. It trades one hard integral for an easier one via uv minus the integral of v du.\n2. So the answer stands: It trades one hard integral for an easier one via uv minus the integral of v du."}
{"id": "mat_0285", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[8,0],[0,9]].", "choices": null, "answer": "Trace: 17.", "steps": ["Sum diagonal.", "8 + 9.", "= 17."], "text": "Answer: Trace: 17.\nSteps:\n1. Sum diagonal.\n2. 8 + 9.\n3. = 17."}
{"id": "mat_0286", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Spočítej E rovnoměrného tahu z {1..5}.", "choices": null, "answer": "E = 3.", "steps": ["Průměr 1..5.", "(5+1)/2.", "E = 3."], "text": "Answer: E = 3.\nSteps:\n1. Průměr 1..5.\n2. (5+1)/2.\n3. E = 3."}
{"id": "mat_0287", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is Fermat's little theorem?", "choices": null, "answer": "For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses.", "steps": ["For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses.", "So the answer stands: For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses."], "text": "Answer: For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses.\nSteps:\n1. For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses.\n2. So the answer stands: For prime p, a^(p-1) is 1 mod p, powering fast primality checks and inverses."}
{"id": "mat_0288", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(9,3).", "choices": null, "answer": "P(9,3) = 504.", "steps": ["n(n−1)(n−2).", "9×8×7.", "= 504."], "text": "Answer: P(9,3) = 504.\nSteps:\n1. n(n−1)(n−2).\n2. 9×8×7.\n3. = 504."}
{"id": "mat_0289", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what is a lemma?", "choices": {"A": "Long book", "B": "Final answer", "C": "Helper claim", "D": "New axiom"}, "answer": "The answer is Helper claim.", "steps": ["Main proof uses it.", "So: Helper claim.", "Select option C."], "text": "Answer: Helper claim\nSteps:\n1. Main proof uses it.\n2. So: Helper claim.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0290", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 8²)/(x − 8) as x → 8.", "choices": null, "answer": "Limit: 16.", "steps": ["Factor: (x−8)(x+8).", "Cancel (x−8).", "Plug 8: 8+8 = 16."], "text": "Answer: Limit: 16.\nSteps:\n1. Factor: (x−8)(x+8).\n2. Cancel (x−8).\n3. Plug 8: 8+8 = 16."}
{"id": "mat_0291", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (3,4) · (3,4).", "choices": {"A": "25", "B": "21", "C": "29", "D": "33"}, "answer": "The dot product is 25.", "steps": ["3×3 + 4×4.", "Product = 25.", "Select option A."], "text": "Answer: 25\nSteps:\n1. 3×3 + 4×4.\n2. Product = 25.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0292", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 11-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/11.", "steps": ["Favourable: 1.", "Possible: 11.", "P = 1/11."], "text": "Answer: P = 1/11.\nSteps:\n1. Favourable: 1.\n2. Possible: 11.\n3. P = 1/11."}
{"id": "mat_0293", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what shrinks gcd pairs?", "choices": {"A": "Guesses", "B": "Remainders", "C": "Hopes", "D": "Dreams"}, "answer": "The answer is Remainders.", "steps": ["Euclid runs fast.", "So: Remainders.", "Select option B."], "text": "Answer: Remainders\nSteps:\n1. Euclid runs fast.\n2. So: Remainders.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0294", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why generating functions?", "choices": null, "answer": "They encode sequences as series coefficients, turning recurrences into algebra.", "steps": ["They encode sequences as series coefficients, turning recurrences into algebra.", "So the answer stands: They encode sequences as series coefficients, turning recurrences into algebra."], "text": "Answer: They encode sequences as series coefficients, turning recurrences into algebra.\nSteps:\n1. They encode sequences as series coefficients, turning recurrences into algebra.\n2. So the answer stands: They encode sequences as series coefficients, turning recurrences into algebra."}
{"id": "mat_0295", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 11.", "choices": null, "answer": "Sum: 66.", "steps": ["Pair 1+11, 2+10, ...", "Use 11×12/2.", "= 66."], "text": "Answer: Sum: 66.\nSteps:\n1. Pair 1+11, 2+10, ...\n2. Use 11×12/2.\n3. = 66."}
{"id": "mat_0296", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: what resolves 0/0?", "choices": {"A": "L'Hôpital's rule", "B": "Guessing", "C": "Ignoring", "D": "Deleting"}, "answer": "The answer is L'Hôpital's rule.", "steps": ["Differentiate top, bottom.", "So: L'Hôpital's rule.", "Select option A."], "text": "Answer: L'Hôpital's rule\nSteps:\n1. Differentiate top, bottom.\n2. So: L'Hôpital's rule.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0297", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Co je násobení matic?", "choices": null, "answer": "Skládá zobrazení přes skalární součiny řádků a sloupců.", "steps": ["Skládá zobrazení přes skalární součiny řádků a sloupců.", "Odpověď tedy platí: Skládá zobrazení přes skalární součiny řádků a sloupců."], "text": "Answer: Skládá zobrazení přes skalární součiny řádků a sloupců.\nSteps:\n1. Skládá zobrazení přes skalární součiny řádků a sloupců.\n2. Odpověď tedy platí: Skládá zobrazení přes skalární součiny řádků a sloupců."}
{"id": "mat_0298", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Sáček má 3 červených a 4 modré kuličky. Ukaž P(červená).", "choices": null, "answer": "P(červená) = 3/7.", "steps": ["Červené: 3.", "Celkem: 7.", "P = 3/7."], "text": "Answer: P(červená) = 3/7.\nSteps:\n1. Červené: 3.\n2. Celkem: 7.\n3. P = 3/7."}
{"id": "mat_0299", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why Euler's theorem?", "choices": null, "answer": "It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs.", "steps": ["It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs.", "So the answer stands: It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs."], "text": "Answer: It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs.\nSteps:\n1. It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs.\n2. So the answer stands: It generalizes Fermat via totient, so a^phi(n) is 1 mod n for coprime pairs."}
{"id": "mat_0300", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(11,3).", "choices": null, "answer": "C(11,3) = 165.", "steps": ["n(n−1)(n−2)/6.", "11×10×9/6.", "= 165."], "text": "Answer: C(11,3) = 165.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 11×10×9/6.\n3. = 165."}
{"id": "mat_0301", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is an existence proof?", "choices": null, "answer": "An existence proof shows some object has a property, either by building one or by counting arguments.", "steps": ["An existence proof shows some object has a property, either by building one or by counting arguments.", "So the answer stands: An existence proof shows some object has a property, either by building one or by counting arguments."], "text": "Answer: An existence proof shows some object has a property, either by building one or by counting arguments.\nSteps:\n1. An existence proof shows some object has a property, either by building one or by counting arguments.\n2. So the answer stands: An existence proof shows some object has a property, either by building one or by counting arguments."}
{"id": "mat_0302", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why study series?", "choices": null, "answer": "Series sum infinite terms, and convergence tests tell whether the total is finite.", "steps": ["Series sum infinite terms, and convergence tests tell whether the total is finite.", "So the answer stands: Series sum infinite terms, and convergence tests tell whether the total is finite."], "text": "Answer: Series sum infinite terms, and convergence tests tell whether the total is finite.\nSteps:\n1. Series sum infinite terms, and convergence tests tell whether the total is finite.\n2. So the answer stands: Series sum infinite terms, and convergence tests tell whether the total is finite."}
{"id": "mat_0303", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is a subspace?", "choices": null, "answer": "A subspace is a subset closed under the operations, inheriting the vector-space structure.", "steps": ["A subspace is a subset closed under the operations, inheriting the vector-space structure.", "So the answer stands: A subspace is a subset closed under the operations, inheriting the vector-space structure."], "text": "Answer: A subspace is a subset closed under the operations, inheriting the vector-space structure.\nSteps:\n1. A subspace is a subset closed under the operations, inheriting the vector-space structure.\n2. So the answer stands: A subspace is a subset closed under the operations, inheriting the vector-space structure."}
{"id": "mat_0304", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why central limit theorem?", "choices": null, "answer": "It says standardized sums approach normal, explaining the bell curve everywhere.", "steps": ["It says standardized sums approach normal, explaining the bell curve everywhere.", "So the answer stands: It says standardized sums approach normal, explaining the bell curve everywhere."], "text": "Answer: It says standardized sums approach normal, explaining the bell curve everywhere.\nSteps:\n1. It says standardized sums approach normal, explaining the bell curve everywhere.\n2. So the answer stands: It says standardized sums approach normal, explaining the bell curve everywhere."}
{"id": "mat_0305", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(24, 36) step by step.", "choices": null, "answer": "gcd = 12.", "steps": ["Factor: 24=4×6, 36=6×6.", "Common: 6×gcd(4,6).", "= 6×2 = 12."], "text": "Answer: gcd = 12.\nSteps:\n1. Factor: 24=4×6, 36=6×6.\n2. Common: 6×gcd(4,6).\n3. = 6×2 = 12."}
{"id": "mat_0306", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(7,3).", "choices": {"A": "35", "B": "28", "C": "42", "D": "49"}, "answer": "C(7,3) is 35.", "steps": ["7×6×5/6.", "= 35.", "Select option A."], "text": "Answer: 35\nSteps:\n1. 7×6×5/6.\n2. = 35.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0307", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Proč uvádět kvantifikátory?", "choices": null, "answer": "Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii.", "steps": ["Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii.", "Odpověď tedy platí: Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii."], "text": "Answer: Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii.\nSteps:\n1. Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii.\n2. Odpověď tedy platí: Tvrzení pro všechna a existuje se dokazují různě, kvantifikátor volí strategii."}
{"id": "mat_0308", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 2x² + 7x, compute f'(2) step by step.", "choices": null, "answer": "f'(2) = 15.", "steps": ["f'(x) = 4x + 7.", "Plug x = 2.", "4×2 + 7 = 15."], "text": "Answer: f'(2) = 15.\nSteps:\n1. f'(x) = 4x + 7.\n2. Plug x = 2.\n3. 4×2 + 7 = 15."}
{"id": "mat_0309", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (7,6) · (3,5) step by step.", "choices": null, "answer": "Dot product: 51.", "steps": ["Multiply components.", "7×3 + 6×5.", "= 51."], "text": "Answer: Dot product: 51.\nSteps:\n1. Multiply components.\n2. 7×3 + 6×5.\n3. = 51."}
{"id": "mat_0310", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..6}.", "choices": {"A": "4.5", "B": "2.5", "C": "5.5", "D": "3.5"}, "answer": "The expected value is 3.5.", "steps": ["(6+1)/2.", "E = 3.5.", "Select option D."], "text": "Answer: 3.5\nSteps:\n1. (6+1)/2.\n2. E = 3.5.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0311", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Proč nekonečně prvočísel?", "choices": null, "answer": "Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel.", "steps": ["Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel.", "Odpověď tedy platí: Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel."], "text": "Answer: Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel.\nSteps:\n1. Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel.\n2. Odpověď tedy platí: Euklides vynásobí známá a přičte jedna, vynutí nový prvočinitel."}
{"id": "mat_0312", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co počítá pořadí?", "choices": {"A": "součet", "B": "průměr", "C": "hádání", "D": "faktoriál"}, "answer": "Odpověď je faktoriál.", "steps": ["Volby na pozici ubývají.", "Tedy: faktoriál.", "Vyber možnost D."], "text": "Answer: faktoriál\nSteps:\n1. Volby na pozici ubývají.\n2. Tedy: faktoriál.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0313", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 7 by computing its half.", "choices": null, "answer": "Half: 28, so 56 is even.", "steps": ["7² + 7 = 56.", "56 / 2 = 28.", "Integer, so even."], "text": "Answer: Half: 28, so 56 is even.\nSteps:\n1. 7² + 7 = 56.\n2. 56 / 2 = 28.\n3. Integer, so even."}
{"id": "mat_0314", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej limitu (x² − 2²)/(x − 2) pro x → 2.", "choices": {"A": "2", "B": "6", "C": "4", "D": "8"}, "answer": "Limita je 4.", "steps": ["Rozlož (x−2)(x+2).", "Pokrať, dostaneš 2×2 = 4.", "Vyber možnost C."], "text": "Answer: 4\nSteps:\n1. Rozlož (x−2)(x+2).\n2. Pokrať, dostaneš 2×2 = 4.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0315", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej determinant matice [[1,2],[3,4]].", "choices": {"A": "-2", "B": "-5", "C": "1", "D": "4"}, "answer": "Determinant je -2.", "steps": ["1×4 − 2×3.", "Det = -2.", "Vyber možnost A."], "text": "Answer: -2\nSteps:\n1. 1×4 − 2×3.\n2. Det = -2.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0316", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..10}.", "choices": null, "answer": "E = 5.5.", "steps": ["Mean of 1..10.", "(10+1)/2.", "E = 5.5."], "text": "Answer: E = 5.5.\nSteps:\n1. Mean of 1..10.\n2. (10+1)/2.\n3. E = 5.5."}
{"id": "mat_0317", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej NSD(16, 24) krok za krokem.", "choices": null, "answer": "NSD = 8.", "steps": ["Rozlož: 16=4×4, 24=6×4.", "Společné: 4×NSD(4,6).", "= 4×2 = 8."], "text": "Answer: NSD = 8.\nSteps:\n1. Rozlož: 16=4×4, 24=6×4.\n2. Společné: 4×NSD(4,6).\n3. = 4×2 = 8."}
{"id": "mat_0318", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet C(6,2).", "choices": null, "answer": "C(6,2) = 15.", "steps": ["Vzorec n(n−1)/2.", "6×5/2.", "= 15."], "text": "Answer: C(6,2) = 15.\nSteps:\n1. Vzorec n(n−1)/2.\n2. 6×5/2.\n3. = 15."}
{"id": "mat_0319", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Ukaž, že N² + N je sudé pro N = 3, výpočtem poloviny.", "choices": null, "answer": "Polovina: 6, číslo 12 je sudé.", "steps": ["3² + 3 = 12.", "12 / 2 = 6.", "Celé číslo, tedy sudé."], "text": "Answer: Polovina: 6, číslo 12 je sudé.\nSteps:\n1. 3² + 3 = 12.\n2. 12 / 2 = 6.\n3. Celé číslo, tedy sudé."}
{"id": "mat_0320", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: where can maxima hide?", "choices": {"A": "Nowhere else", "B": "Endpoints too", "C": "Only zero", "D": "Far away"}, "answer": "The answer is Endpoints too.", "steps": ["Check ends always.", "So: Endpoints too.", "Select option B."], "text": "Answer: Endpoints too\nSteps:\n1. Check ends always.\n2. So: Endpoints too.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0321", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[8,1],[2,9]] step by step.", "choices": null, "answer": "Determinant: 70.", "steps": ["ad − bc.", "8×9 − 1×2.", "= 70."], "text": "Answer: Determinant: 70.\nSteps:\n1. ad − bc.\n2. 8×9 − 1×2.\n3. = 70."}
{"id": "mat_0322", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 4 red and 9 blue marbles. What is P(red)?", "choices": {"A": "4/13", "B": "9/13", "C": "4/14", "D": "5/13"}, "answer": "P(red) is 4/13.", "steps": ["4 of 13.", "P = 4/13.", "Select option A."], "text": "Answer: 4/13\nSteps:\n1. 4 of 13.\n2. P = 4/13.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0323", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what lists primes fast?", "choices": {"A": "Random picks", "B": "Lucky dips", "C": "Eratosthenes sieve", "D": "Slow walks"}, "answer": "The answer is Eratosthenes sieve.", "steps": ["Multiples crossed out.", "So: Eratosthenes sieve.", "Select option C."], "text": "Answer: Eratosthenes sieve\nSteps:\n1. Multiples crossed out.\n2. So: Eratosthenes sieve.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0324", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is recurrence?", "choices": null, "answer": "A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors.", "steps": ["A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors.", "So the answer stands: A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors."], "text": "Answer: A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors.\nSteps:\n1. A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors.\n2. So the answer stands: A recurrence defines terms from earlier ones, like Fibonacci from its two predecessors."}
{"id": "mat_0325", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why prove uniqueness?", "choices": null, "answer": "Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal.", "steps": ["Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal.", "So the answer stands: Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal."], "text": "Answer: Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal.\nSteps:\n1. Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal.\n2. So the answer stands: Uniqueness shows at most one object fits, usually by assuming two and deriving they are equal."}
{"id": "mat_0326", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co měří f'?", "choices": {"A": "celkovou plochu", "B": "dlouhou délku", "C": "velký objem", "D": "okamžitou rychlost"}, "answer": "Odpověď je okamžitou rychlost.", "steps": ["Směrnice tečny čtena.", "Tedy: okamžitou rychlost.", "Vyber možnost D."], "text": "Answer: okamžitou rychlost\nSteps:\n1. Směrnice tečny čtena.\n2. Tedy: okamžitou rychlost.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0327", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[9,0],[0,10]].", "choices": null, "answer": "Trace: 19.", "steps": ["Sum diagonal.", "9 + 10.", "= 19."], "text": "Answer: Trace: 19.\nSteps:\n1. Sum diagonal.\n2. 9 + 10.\n3. = 19."}
{"id": "mat_0328", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Proč střední hodnota?", "choices": null, "answer": "Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování.", "steps": ["Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování.", "Odpověď tedy platí: Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování."], "text": "Answer: Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování.\nSteps:\n1. Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování.\n2. Odpověď tedy platí: Střední hodnota průměruje výsledky a předpovídá dlouhodobé chování."}
{"id": "mat_0329", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(30, 45).", "choices": {"A": "12", "B": "18", "C": "21", "D": "15"}, "answer": "The gcd is 15.", "steps": ["Factor 3 out twice.", "gcd = 15.", "Select option D."], "text": "Answer: 15\nSteps:\n1. Factor 3 out twice.\n2. gcd = 15.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0330", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why solve recurrences?", "choices": null, "answer": "Closed forms compute terms fast, skipping step-by-step iteration entirely.", "steps": ["Closed forms compute terms fast, skipping step-by-step iteration entirely.", "So the answer stands: Closed forms compute terms fast, skipping step-by-step iteration entirely."], "text": "Answer: Closed forms compute terms fast, skipping step-by-step iteration entirely.\nSteps:\n1. Closed forms compute terms fast, skipping step-by-step iteration entirely.\n2. So the answer stands: Closed forms compute terms fast, skipping step-by-step iteration entirely."}
{"id": "mat_0331", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 8 boolean variables. How many rows?", "choices": null, "answer": "Rows: 256.", "steps": ["2 choices each.", "2^8 = 256."], "text": "Answer: Rows: 256.\nSteps:\n1. 2 choices each.\n2. 2^8 = 256."}
{"id": "mat_0332", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is the ratio test?", "choices": null, "answer": "The ratio test compares successive terms, and a limit below one guarantees absolute convergence.", "steps": ["The ratio test compares successive terms, and a limit below one guarantees absolute convergence.", "So the answer stands: The ratio test compares successive terms, and a limit below one guarantees absolute convergence."], "text": "Answer: The ratio test compares successive terms, and a limit below one guarantees absolute convergence.\nSteps:\n1. The ratio test compares successive terms, and a limit below one guarantees absolute convergence.\n2. So the answer stands: The ratio test compares successive terms, and a limit below one guarantees absolute convergence."}
{"id": "mat_0333", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what undoes a matrix?", "choices": {"A": "Its transpose", "B": "Its inverse", "C": "Its trace", "D": "Its copy"}, "answer": "The answer is Its inverse.", "steps": ["Product is identity.", "So: Its inverse.", "Select option B."], "text": "Answer: Its inverse\nSteps:\n1. Product is identity.\n2. So: Its inverse.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0334", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is a confidence interval?", "choices": null, "answer": "It brackets the true parameter with stated confidence, quantifying statistical uncertainty.", "steps": ["It brackets the true parameter with stated confidence, quantifying statistical uncertainty.", "So the answer stands: It brackets the true parameter with stated confidence, quantifying statistical uncertainty."], "text": "Answer: It brackets the true parameter with stated confidence, quantifying statistical uncertainty.\nSteps:\n1. It brackets the true parameter with stated confidence, quantifying statistical uncertainty.\n2. So the answer stands: It brackets the true parameter with stated confidence, quantifying statistical uncertainty."}
{"id": "mat_0335", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (121) mod 9.", "choices": {"A": "4", "B": "5", "C": "6", "D": "7"}, "answer": "The remainder is 4.", "steps": ["121 ÷ 9.", "Remainder 4.", "Select option A."], "text": "Answer: 4\nSteps:\n1. 121 ÷ 9.\n2. Remainder 4.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0336", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(11,2).", "choices": null, "answer": "C(11,2) = 55.", "steps": ["Formula n(n−1)/2.", "11×10/2.", "= 55."], "text": "Answer: C(11,2) = 55.\nSteps:\n1. Formula n(n−1)/2.\n2. 11×10/2.\n3. = 55."}
{"id": "mat_0337", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 12.", "choices": null, "answer": "Sum: 78.", "steps": ["Pair 1+12, 2+11, ...", "Use 12×13/2.", "= 78."], "text": "Answer: Sum: 78.\nSteps:\n1. Pair 1+12, 2+11, ...\n2. Use 12×13/2.\n3. = 78."}
{"id": "mat_0338", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 9.", "choices": null, "answer": "Integral: 729.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 9.", "9³ − 0 = 729."], "text": "Answer: Integral: 729.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 9.\n3. 9³ − 0 = 729."}
{"id": "mat_0339", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what holds map-to-zero?", "choices": {"A": "Top row", "B": "Last column", "C": "Null space", "D": "Big entry"}, "answer": "The answer is Null space.", "steps": ["Collapse measured.", "So: Null space.", "Select option C."], "text": "Answer: Null space\nSteps:\n1. Collapse measured.\n2. So: Null space.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0340", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why hypothesis testing?", "choices": null, "answer": "Testing weighs evidence against a null claim, rejecting it only when data is extreme.", "steps": ["Testing weighs evidence against a null claim, rejecting it only when data is extreme.", "So the answer stands: Testing weighs evidence against a null claim, rejecting it only when data is extreme."], "text": "Answer: Testing weighs evidence against a null claim, rejecting it only when data is extreme.\nSteps:\n1. Testing weighs evidence against a null claim, rejecting it only when data is extreme.\n2. So the answer stands: Testing weighs evidence against a null claim, rejecting it only when data is extreme."}
{"id": "mat_0341", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co jsou atomy násobení?", "choices": {"A": "sudá čísla", "B": "čtverce", "C": "krychle", "D": "prvočísla"}, "answer": "Odpověď je prvočísla.", "steps": ["Jednoznačný rozklad.", "Tedy: prvočísla.", "Vyber možnost D."], "text": "Answer: prvočísla\nSteps:\n1. Jednoznačný rozklad.\n2. Tedy: prvočísla.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0342", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is the Fibonacci sequence?", "choices": null, "answer": "Each term sums the prior two from 1,1, growing exponentially with golden ratio ties.", "steps": ["Each term sums the prior two from 1,1, growing exponentially with golden ratio ties.", "So the answer stands: Each term sums the prior two from 1,1, growing exponentially with golden ratio ties."], "text": "Answer: Each term sums the prior two from 1,1, growing exponentially with golden ratio ties.\nSteps:\n1. Each term sums the prior two from 1,1, growing exponentially with golden ratio ties.\n2. So the answer stands: Each term sums the prior two from 1,1, growing exponentially with golden ratio ties."}
{"id": "mat_0343", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 8 by computing its half.", "choices": null, "answer": "Half: 36, so 72 is even.", "steps": ["8² + 8 = 72.", "72 / 2 = 36.", "Integer, so even."], "text": "Answer: Half: 36, so 72 is even.\nSteps:\n1. 8² + 8 = 72.\n2. 72 / 2 = 36.\n3. Integer, so even."}
{"id": "mat_0344", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 9²)/(x − 9) as x → 9.", "choices": null, "answer": "Limit: 18.", "steps": ["Factor: (x−9)(x+9).", "Cancel (x−9).", "Plug 9: 9+9 = 18."], "text": "Answer: Limit: 18.\nSteps:\n1. Factor: (x−9)(x+9).\n2. Cancel (x−9).\n3. Plug 9: 9+9 = 18."}
{"id": "mat_0345", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (8,7) · (3,5) step by step.", "choices": null, "answer": "Dot product: 59.", "steps": ["Multiply components.", "8×3 + 7×5.", "= 59."], "text": "Answer: Dot product: 59.\nSteps:\n1. Multiply components.\n2. 8×3 + 7×5.\n3. = 59."}
{"id": "mat_0346", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what flips conditionals?", "choices": {"A": "Coin toss", "B": "Bayes' theorem", "C": "Dice roll", "D": "Card draw"}, "answer": "The answer is Bayes' theorem.", "steps": ["Causes from effects.", "So: Bayes' theorem.", "Select option B."], "text": "Answer: Bayes' theorem\nSteps:\n1. Causes from effects.\n2. So: Bayes' theorem.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0347", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is the totient?", "choices": null, "answer": "phi(n) counts numbers below n coprime to it, measuring reduced residue systems.", "steps": ["phi(n) counts numbers below n coprime to it, measuring reduced residue systems.", "So the answer stands: phi(n) counts numbers below n coprime to it, measuring reduced residue systems."], "text": "Answer: phi(n) counts numbers below n coprime to it, measuring reduced residue systems.\nSteps:\n1. phi(n) counts numbers below n coprime to it, measuring reduced residue systems.\n2. So the answer stands: phi(n) counts numbers below n coprime to it, measuring reduced residue systems."}
{"id": "mat_0348", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what fixes double-counting?", "choices": {"A": "More counting", "B": "Inclusion-exclusion", "C": "Less thinking", "D": "Wild hope"}, "answer": "The answer is Inclusion-exclusion.", "steps": ["Alternate the sums.", "So: Inclusion-exclusion.", "Select option B."], "text": "Answer: Inclusion-exclusion\nSteps:\n1. Alternate the sums.\n2. So: Inclusion-exclusion.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0349", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 5 variables have?", "choices": {"A": "30", "B": "34", "C": "36", "D": "32"}, "answer": "The table has 32 rows.", "steps": ["2^5 rows.", "Total: 32.", "Select option D."], "text": "Answer: 32\nSteps:\n1. 2^5 rows.\n2. Total: 32.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0350", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 3x² + 8x, compute f'(3) step by step.", "choices": null, "answer": "f'(3) = 26.", "steps": ["f'(x) = 6x + 8.", "Plug x = 3.", "6×3 + 8 = 26."], "text": "Answer: f'(3) = 26.\nSteps:\n1. f'(x) = 6x + 8.\n2. Plug x = 3.\n3. 6×3 + 8 = 26."}
{"id": "mat_0351", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[9,1],[2,10]] step by step.", "choices": null, "answer": "Determinant: 88.", "steps": ["ad − bc.", "9×10 − 1×2.", "= 88."], "text": "Answer: Determinant: 88.\nSteps:\n1. ad − bc.\n2. 9×10 − 1×2.\n3. = 88."}
{"id": "mat_0352", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is a p-value?", "choices": null, "answer": "A p-value is the chance of data this extreme under the null, small values casting doubt.", "steps": ["A p-value is the chance of data this extreme under the null, small values casting doubt.", "So the answer stands: A p-value is the chance of data this extreme under the null, small values casting doubt."], "text": "Answer: A p-value is the chance of data this extreme under the null, small values casting doubt.\nSteps:\n1. A p-value is the chance of data this extreme under the null, small values casting doubt.\n2. So the answer stands: A p-value is the chance of data this extreme under the null, small values casting doubt."}
{"id": "mat_0353", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (94) mod 7.", "choices": null, "answer": "Remainder: 3.", "steps": ["Divide 94 by 7.", "7×13 = 91.", "94 − 91 = 3."], "text": "Answer: Remainder: 3.\nSteps:\n1. Divide 94 by 7.\n2. 7×13 = 91.\n3. 94 − 91 = 3."}
{"id": "mat_0354", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why Catalan numbers?", "choices": null, "answer": "They count trees, brackets, and paths, popping up across recursive structures.", "steps": ["They count trees, brackets, and paths, popping up across recursive structures.", "So the answer stands: They count trees, brackets, and paths, popping up across recursive structures."], "text": "Answer: They count trees, brackets, and paths, popping up across recursive structures.\nSteps:\n1. They count trees, brackets, and paths, popping up across recursive structures.\n2. So the answer stands: They count trees, brackets, and paths, popping up across recursive structures."}
{"id": "mat_0355", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is proof by cases?", "choices": null, "answer": "Proof by cases splits possibilities into an exhaustive list and proves the claim under each one.", "steps": ["Proof by cases splits possibilities into an exhaustive list and proves the claim under each one.", "So the answer stands: Proof by cases splits possibilities into an exhaustive list and proves the claim under each one."], "text": "Answer: Proof by cases splits possibilities into an exhaustive list and proves the claim under each one.\nSteps:\n1. Proof by cases splits possibilities into an exhaustive list and proves the claim under each one.\n2. So the answer stands: Proof by cases splits possibilities into an exhaustive list and proves the claim under each one."}
{"id": "mat_0356", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why learn Taylor series?", "choices": null, "answer": "Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds.", "steps": ["Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds.", "So the answer stands: Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds."], "text": "Answer: Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds.\nSteps:\n1. Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds.\n2. So the answer stands: Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds."}
{"id": "mat_0357", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Proč determinant?", "choices": null, "answer": "Determinant škáluje objemy a detekuje regularitu.", "steps": ["Determinant škáluje objemy a detekuje regularitu.", "Odpověď tedy platí: Determinant škáluje objemy a detekuje regularitu."], "text": "Answer: Determinant škáluje objemy a detekuje regularitu.\nSteps:\n1. Determinant škáluje objemy a detekuje regularitu.\n2. Odpověď tedy platí: Determinant škáluje objemy a detekuje regularitu."}
{"id": "mat_0358", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 8 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 8/12.", "steps": ["Red: 8.", "Total: 12.", "P = 8/12."], "text": "Answer: P(red) = 8/12.\nSteps:\n1. Red: 8.\n2. Total: 12.\n3. P = 8/12."}
{"id": "mat_0359", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why Chinese remainder?", "choices": null, "answer": "It merges congruences into one solution modulo the product, splitting hard mods apart.", "steps": ["It merges congruences into one solution modulo the product, splitting hard mods apart.", "So the answer stands: It merges congruences into one solution modulo the product, splitting hard mods apart."], "text": "Answer: It merges congruences into one solution modulo the product, splitting hard mods apart.\nSteps:\n1. It merges congruences into one solution modulo the product, splitting hard mods apart.\n2. So the answer stands: It merges congruences into one solution modulo the product, splitting hard mods apart."}
{"id": "mat_0360", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(10,3).", "choices": null, "answer": "P(10,3) = 720.", "steps": ["n(n−1)(n−2).", "10×9×8.", "= 720."], "text": "Answer: P(10,3) = 720.\nSteps:\n1. n(n−1)(n−2).\n2. 10×9×8.\n3. = 720."}
{"id": "mat_0361", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 8?", "choices": {"A": "36", "B": "28", "C": "44", "D": "52"}, "answer": "The sum is 36.", "steps": ["8×9/2.", "Sum = 36.", "Select option A."], "text": "Answer: 36\nSteps:\n1. 8×9/2.\n2. Sum = 36.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0362", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is L'Hôpital's rule?", "choices": null, "answer": "It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones.", "steps": ["It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones.", "So the answer stands: It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones."], "text": "Answer: It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones.\nSteps:\n1. It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones.\n2. So the answer stands: It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones."}
{"id": "mat_0363", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Co je hodnost?", "choices": null, "answer": "Hodnost počítá nezávislé řádky a měří dimenzi obrazu.", "steps": ["Hodnost počítá nezávislé řádky a měří dimenzi obrazu.", "Odpověď tedy platí: Hodnost počítá nezávislé řádky a měří dimenzi obrazu."], "text": "Answer: Hodnost počítá nezávislé řádky a měří dimenzi obrazu.\nSteps:\n1. Hodnost počítá nezávislé řádky a měří dimenzi obrazu.\n2. Odpověď tedy platí: Hodnost počítá nezávislé řádky a měří dimenzi obrazu."}
{"id": "mat_0364", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what settles averages?", "choices": {"A": "Small samples", "B": "Wild hopes", "C": "Large numbers law", "D": "Quick peeks"}, "answer": "The answer is Large numbers law.", "steps": ["Trials many, mean neared.", "So: Large numbers law.", "Select option C."], "text": "Answer: Large numbers law\nSteps:\n1. Trials many, mean neared.\n2. So: Large numbers law.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0365", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co slučuje kongruence?", "choices": {"A": "čínská zbytková věta", "B": "slepé štěstí", "C": "divoké hádání", "D": "dlouhý spánek"}, "answer": "Odpověď je čínská zbytková věta.", "steps": ["Jedno řešení modulo součin.", "Tedy: čínská zbytková věta.", "Vyber možnost A."], "text": "Answer: čínská zbytková věta\nSteps:\n1. Jedno řešení modulo součin.\n2. Tedy: čínská zbytková věta.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0366", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(12,3).", "choices": null, "answer": "C(12,3) = 220.", "steps": ["n(n−1)(n−2)/6.", "12×11×10/6.", "= 220."], "text": "Answer: C(12,3) = 220.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 12×11×10/6.\n3. = 220."}
{"id": "mat_0367", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 6.", "choices": {"A": "40", "B": "42", "C": "44", "D": "46"}, "answer": "The value is 42.", "steps": ["6² + 6.", "Value = 42.", "Select option B."], "text": "Answer: 42\nSteps:\n1. 6² + 6.\n2. Value = 42.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0368", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 3x² + 4x, find f'(3).", "choices": {"A": "20", "B": "24", "C": "22", "D": "26"}, "answer": "The derivative is 22.", "steps": ["f'(x) = 6x + 4.", "At 3: 22.", "Select option C."], "text": "Answer: 22\nSteps:\n1. f'(x) = 6x + 4.\n2. At 3: 22.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0369", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej det [[3,1],[2,4]] krok za krokem.", "choices": null, "answer": "Determinant: 10.", "steps": ["ad − bc.", "3×4 − 1×2.", "= 10."], "text": "Answer: Determinant: 10.\nSteps:\n1. ad − bc.\n2. 3×4 − 1×2.\n3. = 10."}
{"id": "mat_0370", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why avoid gambler's fallacy?", "choices": null, "answer": "Past independent outcomes never change future odds, so streaks carry no memory at all.", "steps": ["Past independent outcomes never change future odds, so streaks carry no memory at all.", "So the answer stands: Past independent outcomes never change future odds, so streaks carry no memory at all."], "text": "Answer: Past independent outcomes never change future odds, so streaks carry no memory at all.\nSteps:\n1. Past independent outcomes never change future odds, so streaks carry no memory at all.\n2. So the answer stands: Past independent outcomes never change future odds, so streaks carry no memory at all."}
{"id": "mat_0371", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is a Diophantine equation?", "choices": null, "answer": "It seeks integer solutions only, turning algebra into number puzzles with gcd conditions.", "steps": ["It seeks integer solutions only, turning algebra into number puzzles with gcd conditions.", "So the answer stands: It seeks integer solutions only, turning algebra into number puzzles with gcd conditions."], "text": "Answer: It seeks integer solutions only, turning algebra into number puzzles with gcd conditions.\nSteps:\n1. It seeks integer solutions only, turning algebra into number puzzles with gcd conditions.\n2. So the answer stands: It seeks integer solutions only, turning algebra into number puzzles with gcd conditions."}
{"id": "mat_0372", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is graph counting?", "choices": null, "answer": "It counts graphs, paths, and colorings, blending combinatorics with networks.", "steps": ["It counts graphs, paths, and colorings, blending combinatorics with networks.", "So the answer stands: It counts graphs, paths, and colorings, blending combinatorics with networks."], "text": "Answer: It counts graphs, paths, and colorings, blending combinatorics with networks.\nSteps:\n1. It counts graphs, paths, and colorings, blending combinatorics with networks.\n2. So the answer stands: It counts graphs, paths, and colorings, blending combinatorics with networks."}
{"id": "mat_0373", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why check all cases?", "choices": null, "answer": "Missing one case breaks the proof, so the split must cover every possibility with no gaps.", "steps": ["Missing one case breaks the proof, so the split must cover every possibility with no gaps.", "So the answer stands: Missing one case breaks the proof, so the split must cover every possibility with no gaps."], "text": "Answer: Missing one case breaks the proof, so the split must cover every possibility with no gaps.\nSteps:\n1. Missing one case breaks the proof, so the split must cover every possibility with no gaps.\n2. So the answer stands: Missing one case breaks the proof, so the split must cover every possibility with no gaps."}
{"id": "mat_0374", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 10.", "choices": null, "answer": "Integral: 1000.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 10.", "10³ − 0 = 1000."], "text": "Answer: Integral: 1000.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 10.\n3. 10³ − 0 = 1000."}
{"id": "mat_0375", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why use dot products?", "choices": null, "answer": "Dot products measure angles and lengths, turning geometry into algebra on coordinates.", "steps": ["Dot products measure angles and lengths, turning geometry into algebra on coordinates.", "So the answer stands: Dot products measure angles and lengths, turning geometry into algebra on coordinates."], "text": "Answer: Dot products measure angles and lengths, turning geometry into algebra on coordinates.\nSteps:\n1. Dot products measure angles and lengths, turning geometry into algebra on coordinates.\n2. So the answer stands: Dot products measure angles and lengths, turning geometry into algebra on coordinates."}
{"id": "mat_0376", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 9-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/8", "B": "1/10", "C": "2/9", "D": "1/9"}, "answer": "The probability is 1/9.", "steps": ["1 face of 9.", "P = 1/9.", "Select option D."], "text": "Answer: 1/9\nSteps:\n1. 1 face of 9.\n2. P = 1/9.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0377", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why solve linear congruences?", "choices": null, "answer": "They model cycles and schedules, solved by inverses when gcd permits.", "steps": ["They model cycles and schedules, solved by inverses when gcd permits.", "So the answer stands: They model cycles and schedules, solved by inverses when gcd permits."], "text": "Answer: They model cycles and schedules, solved by inverses when gcd permits.\nSteps:\n1. They model cycles and schedules, solved by inverses when gcd permits.\n2. So the answer stands: They model cycles and schedules, solved by inverses when gcd permits."}
{"id": "mat_0378", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(12,2).", "choices": null, "answer": "C(12,2) = 66.", "steps": ["Formula n(n−1)/2.", "12×11/2.", "= 66."], "text": "Answer: C(12,2) = 66.\nSteps:\n1. Formula n(n−1)/2.\n2. 12×11/2.\n3. = 66."}
{"id": "mat_0379", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 9 boolean variables. How many rows?", "choices": null, "answer": "Rows: 512.", "steps": ["2 choices each.", "2^9 = 512."], "text": "Answer: Rows: 512.\nSteps:\n1. 2 choices each.\n2. 2^9 = 512."}
{"id": "mat_0380", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 4.", "choices": {"A": "60", "B": "68", "C": "72", "D": "64"}, "answer": "The integral is 64.", "steps": ["Antiderivative x³.", "4³ = 64.", "Select option D."], "text": "Answer: 64\nSteps:\n1. Antiderivative x³.\n2. 4³ = 64.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0381", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[10,0],[0,11]].", "choices": null, "answer": "Trace: 21.", "steps": ["Sum diagonal.", "10 + 11.", "= 21."], "text": "Answer: Trace: 21.\nSteps:\n1. Sum diagonal.\n2. 10 + 11.\n3. = 21."}
{"id": "mat_0382", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..7}.", "choices": {"A": "4", "B": "5", "C": "3", "D": "6"}, "answer": "The expected value is 4.", "steps": ["(7+1)/2.", "E = 4.", "Select option A."], "text": "Answer: 4\nSteps:\n1. (7+1)/2.\n2. E = 4.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0383", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is a primitive root?", "choices": null, "answer": "It generates all nonzero residues by powers, giving the multiplicative group one engine.", "steps": ["It generates all nonzero residues by powers, giving the multiplicative group one engine.", "So the answer stands: It generates all nonzero residues by powers, giving the multiplicative group one engine."], "text": "Answer: It generates all nonzero residues by powers, giving the multiplicative group one engine.\nSteps:\n1. It generates all nonzero residues by powers, giving the multiplicative group one engine.\n2. So the answer stands: It generates all nonzero residues by powers, giving the multiplicative group one engine."}
{"id": "mat_0384", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why Ramsey theory?", "choices": null, "answer": "It guarantees order inside chaos, finding monochromatic cliques in all big graphs.", "steps": ["It guarantees order inside chaos, finding monochromatic cliques in all big graphs.", "So the answer stands: It guarantees order inside chaos, finding monochromatic cliques in all big graphs."], "text": "Answer: It guarantees order inside chaos, finding monochromatic cliques in all big graphs.\nSteps:\n1. It guarantees order inside chaos, finding monochromatic cliques in all big graphs.\n2. So the answer stands: It guarantees order inside chaos, finding monochromatic cliques in all big graphs."}
{"id": "mat_0385", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Co je důkaz rozborem případů?", "choices": null, "answer": "Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém.", "steps": ["Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém.", "Odpověď tedy platí: Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém."], "text": "Answer: Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém.\nSteps:\n1. Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém.\n2. Odpověď tedy platí: Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém."}
{"id": "mat_0386", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why check endpoints?", "choices": null, "answer": "Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them.", "steps": ["Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them.", "So the answer stands: Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them."], "text": "Answer: Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them.\nSteps:\n1. Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them.\n2. So the answer stands: Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them."}
{"id": "mat_0387", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Proč Gaussova eliminace?", "choices": null, "answer": "Eliminace odhaluje pivoty a řešení soustav mechanicky.", "steps": ["Eliminace odhaluje pivoty a řešení soustav mechanicky.", "Odpověď tedy platí: Eliminace odhaluje pivoty a řešení soustav mechanicky."], "text": "Answer: Eliminace odhaluje pivoty a řešení soustav mechanicky.\nSteps:\n1. Eliminace odhaluje pivoty a řešení soustav mechanicky.\n2. Odpověď tedy platí: Eliminace odhaluje pivoty a řešení soustav mechanicky."}
{"id": "mat_0388", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is Monty Hall?", "choices": null, "answer": "Switching doors wins twice as often, because the host's reveal concentrates probability.", "steps": ["Switching doors wins twice as often, because the host's reveal concentrates probability.", "So the answer stands: Switching doors wins twice as often, because the host's reveal concentrates probability."], "text": "Answer: Switching doors wins twice as often, because the host's reveal concentrates probability.\nSteps:\n1. Switching doors wins twice as often, because the host's reveal concentrates probability.\n2. So the answer stands: Switching doors wins twice as often, because the host's reveal concentrates probability."}
{"id": "mat_0389", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(7, 6).", "choices": {"A": "40", "B": "42", "C": "44", "D": "46"}, "answer": "The lcm is 42.", "steps": ["7×6/gcd.", "lcm = 42.", "Select option B."], "text": "Answer: 42\nSteps:\n1. 7×6/gcd.\n2. lcm = 42.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0390", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is a partition?", "choices": null, "answer": "A partition splits integers into summands, counted by deep generating functions.", "steps": ["A partition splits integers into summands, counted by deep generating functions.", "So the answer stands: A partition splits integers into summands, counted by deep generating functions."], "text": "Answer: A partition splits integers into summands, counted by deep generating functions.\nSteps:\n1. A partition splits integers into summands, counted by deep generating functions.\n2. So the answer stands: A partition splits integers into summands, counted by deep generating functions."}
{"id": "mat_0391", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what is the contrapositive?", "choices": {"A": "Q to P", "B": "P to P", "C": "Not-Q to not-P", "D": "Not-P to Q"}, "answer": "The answer is Not-Q to not-P.", "steps": ["Equivalent to original.", "So: Not-Q to not-P.", "Select option C."], "text": "Answer: Not-Q to not-P\nSteps:\n1. Equivalent to original.\n2. So: Not-Q to not-P.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0392", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 10²)/(x − 10) as x → 10.", "choices": null, "answer": "Limit: 20.", "steps": ["Factor: (x−10)(x+10).", "Cancel (x−10).", "Plug 10: 10+10 = 20."], "text": "Answer: Limit: 20.\nSteps:\n1. Factor: (x−10)(x+10).\n2. Cancel (x−10).\n3. Plug 10: 10+10 = 20."}
{"id": "mat_0393", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (9,8) · (3,5) step by step.", "choices": null, "answer": "Dot product: 67.", "steps": ["Multiply components.", "9×3 + 8×5.", "= 67."], "text": "Answer: Dot product: 67.\nSteps:\n1. Multiply components.\n2. 9×3 + 8×5.\n3. = 67."}
{"id": "mat_0394", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Co je rozptyl?", "choices": null, "answer": "Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny.", "steps": ["Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny.", "Odpověď tedy platí: Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny."], "text": "Answer: Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny.\nSteps:\n1. Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny.\n2. Odpověď tedy platí: Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny."}
{"id": "mat_0395", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(9, 8) step by step.", "choices": null, "answer": "lcm = 72.", "steps": ["gcd(9,8) = 1.", "9×8/1.", "= 72."], "text": "Answer: lcm = 72.\nSteps:\n1. gcd(9,8) = 1.\n2. 9×8/1.\n3. = 72."}
{"id": "mat_0396", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what proves by two counts?", "choices": {"A": "Double talking", "B": "Double hoping", "C": "Double counting", "D": "Double tasting"}, "answer": "The answer is Double counting.", "steps": ["One set, two views.", "So: Double counting.", "Select option C."], "text": "Answer: Double counting\nSteps:\n1. One set, two views.\n2. So: Double counting.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0397", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 13.", "choices": null, "answer": "Sum: 91.", "steps": ["Pair 1+13, 2+12, ...", "Use 13×14/2.", "= 91."], "text": "Answer: Sum: 91.\nSteps:\n1. Pair 1+13, 2+12, ...\n2. Use 13×14/2.\n3. = 91."}
{"id": "mat_0398", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 4x² + 9x, compute f'(4) step by step.", "choices": null, "answer": "f'(4) = 41.", "steps": ["f'(x) = 8x + 9.", "Plug x = 4.", "8×4 + 9 = 41."], "text": "Answer: f'(4) = 41.\nSteps:\n1. f'(x) = 8x + 9.\n2. Plug x = 4.\n3. 8×4 + 9 = 41."}
{"id": "mat_0399", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[10,1],[2,11]] step by step.", "choices": null, "answer": "Determinant: 108.", "steps": ["ad − bc.", "10×11 − 1×2.", "= 108."], "text": "Answer: Determinant: 108.\nSteps:\n1. ad − bc.\n2. 10×11 − 1×2.\n3. = 108."}
{"id": "mat_0400", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 12-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/12.", "steps": ["Favourable: 1.", "Possible: 12.", "P = 1/12."], "text": "Answer: P = 1/12.\nSteps:\n1. Favourable: 1.\n2. Possible: 12.\n3. P = 1/12."}
{"id": "mat_0401", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(28, 42) step by step.", "choices": null, "answer": "gcd = 14.", "steps": ["Factor: 28=4×7, 42=6×7.", "Common: 7×gcd(4,6).", "= 7×2 = 14."], "text": "Answer: gcd = 14.\nSteps:\n1. Factor: 28=4×7, 42=6×7.\n2. Common: 7×gcd(4,6).\n3. = 7×2 = 14."}
{"id": "mat_0402", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(7,2).", "choices": {"A": "14", "B": "28", "C": "35", "D": "21"}, "answer": "C(7,2) is 21.", "steps": ["7×6/2.", "= 21.", "Select option D."], "text": "Answer: 21\nSteps:\n1. 7×6/2.\n2. = 21.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0403", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what ends a proof?", "choices": {"A": "New question", "B": "Blank page", "C": "Loud sigh", "D": "QED mark"}, "answer": "The answer is QED mark.", "steps": ["Claim now secured.", "So: QED mark.", "Select option D."], "text": "Answer: QED mark\nSteps:\n1. Claim now secured.\n2. So: QED mark.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0404", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 4²)/(x − 4) as x → 4.", "choices": {"A": "8", "B": "6", "C": "10", "D": "12"}, "answer": "The limit is 8.", "steps": ["Factor (x−4)(x+4).", "Cancel, get 2×4 = 8.", "Select option A."], "text": "Answer: 8\nSteps:\n1. Factor (x−4)(x+4).\n2. Cancel, get 2×4 = 8.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0405", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[4,2],[3,7]].", "choices": {"A": "19", "B": "25", "C": "28", "D": "22"}, "answer": "The determinant is 22.", "steps": ["4×7 − 2×3.", "Det = 22.", "Select option D."], "text": "Answer: 22\nSteps:\n1. 4×7 − 2×3.\n2. Det = 22.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0406", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Sáček má 2 červených a 5 modrých kuliček. Jaká je P(červená)?", "choices": {"A": "5/7", "B": "2/8", "C": "2/7", "D": "3/7"}, "answer": "P(červená) je 2/7.", "steps": ["2 z 7.", "P = 2/7.", "Vyber možnost C."], "text": "Answer: 2/7\nSteps:\n1. 2 z 7.\n2. P = 2/7.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0407", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (105) mod 7.", "choices": null, "answer": "Remainder: 0.", "steps": ["Divide 105 by 7.", "7×15 = 105.", "105 − 105 = 0."], "text": "Answer: Remainder: 0.\nSteps:\n1. Divide 105 by 7.\n2. 7×15 = 105.\n3. 105 − 105 = 0."}
{"id": "mat_0408", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(11,3).", "choices": null, "answer": "P(11,3) = 990.", "steps": ["n(n−1)(n−2).", "11×10×9.", "= 990."], "text": "Answer: P(11,3) = 990.\nSteps:\n1. n(n−1)(n−2).\n2. 11×10×9.\n3. = 990."}
{"id": "mat_0409", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Důkaz prochází všechny případy 4 booleovských proměnných. Kolik řádků?", "choices": null, "answer": "Řádků: 16.", "steps": ["2 volby každá.", "2^4 = 16."], "text": "Answer: Řádků: 16.\nSteps:\n1. 2 volby každá.\n2. 2^4 = 16."}
{"id": "mat_0410", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co aproximuje funkce?", "choices": {"A": "Taylorova řada", "B": "náhodné tečky", "C": "divoké hádání", "D": "prázdné řádky"}, "answer": "Odpověď je Taylorova řada.", "steps": ["Polynomy s odhadem chyby.", "Tedy: Taylorova řada.", "Vyber možnost A."], "text": "Answer: Taylorova řada\nSteps:\n1. Polynomy s odhadem chyby.\n2. Tedy: Taylorova řada.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0411", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[11,0],[0,12]].", "choices": null, "answer": "Trace: 23.", "steps": ["Sum diagonal.", "11 + 12.", "= 23."], "text": "Answer: Trace: 23.\nSteps:\n1. Sum diagonal.\n2. 11 + 12.\n3. = 23."}
{"id": "mat_0412", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..11}.", "choices": null, "answer": "E = 6.", "steps": ["Mean of 1..11.", "(11+1)/2.", "E = 6."], "text": "Answer: E = 6.\nSteps:\n1. Mean of 1..11.\n2. (11+1)/2.\n3. E = 6."}
{"id": "mat_0413", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what wraps numbers?", "choices": {"A": "Big hammer", "B": "Long rope", "C": "Modulus clock", "D": "Wide net"}, "answer": "The answer is Modulus clock.", "steps": ["Remainders kept.", "So: Modulus clock.", "Select option C."], "text": "Answer: Modulus clock\nSteps:\n1. Remainders kept.\n2. So: Modulus clock.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0414", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co definuje z minulosti?", "choices": {"A": "rekurence", "B": "proroctví", "C": "fáma", "D": "sen"}, "answer": "Odpověď je rekurence.", "steps": ["Základ plus krok.", "Tedy: rekurence.", "Vyber možnost A."], "text": "Answer: rekurence\nSteps:\n1. Základ plus krok.\n2. Tedy: rekurence.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0415", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 9 by computing its half.", "choices": null, "answer": "Half: 45, so 90 is even.", "steps": ["9² + 9 = 90.", "90 / 2 = 45.", "Integer, so even."], "text": "Answer: Half: 45, so 90 is even.\nSteps:\n1. 9² + 9 = 90.\n2. 90 / 2 = 45.\n3. Integer, so even."}
{"id": "mat_0416", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 11.", "choices": null, "answer": "Integral: 1331.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 11.", "11³ − 0 = 1331."], "text": "Answer: Integral: 1331.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 11.\n3. 11³ − 0 = 1331."}
{"id": "mat_0417", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej stopu matice [[2,1],[2,5]].", "choices": {"A": "5", "B": "7", "C": "9", "D": "11"}, "answer": "Stopa je 7.", "steps": ["2 + 5.", "Stopa = 7.", "Vyber možnost B."], "text": "Answer: 7\nSteps:\n1. 2 + 5.\n2. Stopa = 7.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0418", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 9 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 9/13.", "steps": ["Red: 9.", "Total: 13.", "P = 9/13."], "text": "Answer: P(red) = 9/13.\nSteps:\n1. Red: 9.\n2. Total: 13.\n3. P = 9/13."}
{"id": "mat_0419", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(10, 8) step by step.", "choices": null, "answer": "lcm = 40.", "steps": ["gcd(10,8) = 2.", "10×8/2.", "= 40."], "text": "Answer: lcm = 40.\nSteps:\n1. gcd(10,8) = 2.\n2. 10×8/2.\n3. = 40."}
{"id": "mat_0420", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(6,2).", "choices": {"A": "30", "B": "24", "C": "36", "D": "42"}, "answer": "P(6,2) is 30.", "steps": ["6×5.", "= 30.", "Select option A."], "text": "Answer: 30\nSteps:\n1. 6×5.\n2. = 30.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0421", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Dokaž vzorcem: najdi součet 1 + 2 + ... + 8.", "choices": null, "answer": "Součet: 36.", "steps": ["Páruj 1+8, 2+7, ...", "Použij 8×9/2.", "= 36."], "text": "Answer: Součet: 36.\nSteps:\n1. Páruj 1+8, 2+7, ...\n2. Použij 8×9/2.\n3. = 36."}
{"id": "mat_0422", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: what sums slices?", "choices": {"A": "Derivative", "B": "Definite integral", "C": "Limit point", "D": "Equation set"}, "answer": "The answer is Definite integral.", "steps": ["Area under curve.", "So: Definite integral.", "Select option B."], "text": "Answer: Definite integral\nSteps:\n1. Area under curve.\n2. So: Definite integral.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0423", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[5,1],[2,8]].", "choices": {"A": "13", "B": "11", "C": "15", "D": "17"}, "answer": "The trace is 13.", "steps": ["5 + 8.", "Trace = 13.", "Select option A."], "text": "Answer: 13\nSteps:\n1. 5 + 8.\n2. Trace = 13.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0424", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why simulate?", "choices": null, "answer": "Simulation estimates hard probabilities by repeated random trials on computers.", "steps": ["Simulation estimates hard probabilities by repeated random trials on computers.", "So the answer stands: Simulation estimates hard probabilities by repeated random trials on computers."], "text": "Answer: Simulation estimates hard probabilities by repeated random trials on computers.\nSteps:\n1. Simulation estimates hard probabilities by repeated random trials on computers.\n2. So the answer stands: Simulation estimates hard probabilities by repeated random trials on computers."}
{"id": "mat_0425", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(32, 48) step by step.", "choices": null, "answer": "gcd = 16.", "steps": ["Factor: 32=4×8, 48=6×8.", "Common: 8×gcd(4,6).", "= 8×2 = 16."], "text": "Answer: gcd = 16.\nSteps:\n1. Factor: 32=4×8, 48=6×8.\n2. Common: 8×gcd(4,6).\n3. = 8×2 = 16."}
{"id": "mat_0426", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(13,3).", "choices": null, "answer": "C(13,3) = 286.", "steps": ["n(n−1)(n−2)/6.", "13×12×11/6.", "= 286."], "text": "Answer: C(13,3) = 286.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 13×12×11/6.\n3. = 286."}
{"id": "mat_0427", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 10 boolean variables. How many rows?", "choices": null, "answer": "Rows: 1024.", "steps": ["2 choices each.", "2^10 = 1024."], "text": "Answer: Rows: 1024.\nSteps:\n1. 2 choices each.\n2. 2^10 = 1024."}
{"id": "mat_0428", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: what shows bending?", "choices": {"A": "First guess", "B": "Last digit", "C": "Second derivative", "D": "Middle value"}, "answer": "The answer is Second derivative.", "steps": ["Sign gives shape.", "So: Second derivative.", "Select option C."], "text": "Answer: Second derivative\nSteps:\n1. Sign gives shape.\n2. So: Second derivative.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0429", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (10,9) · (3,5) step by step.", "choices": null, "answer": "Dot product: 75.", "steps": ["Multiply components.", "10×3 + 9×5.", "= 75."], "text": "Answer: Dot product: 75.\nSteps:\n1. Multiply components.\n2. 10×3 + 9×5.\n3. = 75."}
{"id": "mat_0430", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 5 red and 11 blue marbles. What is P(red)?", "choices": {"A": "11/16", "B": "5/16", "C": "5/17", "D": "6/16"}, "answer": "P(red) is 5/16.", "steps": ["5 of 16.", "P = 5/16.", "Select option B."], "text": "Answer: 5/16\nSteps:\n1. 5 of 16.\n2. P = 5/16.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0431", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (116) mod 7.", "choices": null, "answer": "Remainder: 4.", "steps": ["Divide 116 by 7.", "7×16 = 112.", "116 − 112 = 4."], "text": "Answer: Remainder: 4.\nSteps:\n1. Divide 116 by 7.\n2. 7×16 = 112.\n3. 116 − 112 = 4."}
{"id": "mat_0432", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(13,2).", "choices": null, "answer": "C(13,2) = 78.", "steps": ["Formula n(n−1)/2.", "13×12/2.", "= 78."], "text": "Answer: C(13,2) = 78.\nSteps:\n1. Formula n(n−1)/2.\n2. 13×12/2.\n3. = 78."}
{"id": "mat_0433", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Kolik je součet 1 + 2 + ... + 5?", "choices": {"A": "10", "B": "15", "C": "20", "D": "25"}, "answer": "Součet je 15.", "steps": ["5×6/2.", "Součet = 15.", "Vyber možnost B."], "text": "Answer: 15\nSteps:\n1. 5×6/2.\n2. Součet = 15.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0434", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is concavity?", "choices": null, "answer": "Concavity from the second derivative shows where graphs bend up or down around points.", "steps": ["Concavity from the second derivative shows where graphs bend up or down around points.", "So the answer stands: Concavity from the second derivative shows where graphs bend up or down around points."], "text": "Answer: Concavity from the second derivative shows where graphs bend up or down around points.\nSteps:\n1. Concavity from the second derivative shows where graphs bend up or down around points.\n2. So the answer stands: Concavity from the second derivative shows where graphs bend up or down around points."}
{"id": "mat_0435", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is orthogonality?", "choices": null, "answer": "Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions.", "steps": ["Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions.", "So the answer stands: Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions."], "text": "Answer: Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions.\nSteps:\n1. Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions.\n2. So the answer stands: Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions."}
{"id": "mat_0436", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what multiplies for independent?", "choices": {"A": "Outcomes", "B": "Names", "C": "Probabilities", "D": "Colors"}, "answer": "The answer is Probabilities.", "steps": ["Joint equals product.", "So: Probabilities.", "Select option C."], "text": "Answer: Probabilities\nSteps:\n1. Joint equals product.\n2. So: Probabilities.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0437", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what secures RSA?", "choices": {"A": "Big primes secret", "B": "Long keys", "C": "Fast computers", "D": "Hard factoring"}, "answer": "The answer is Hard factoring.", "steps": ["Multiply easy, split hard.", "So: Hard factoring.", "Select option D."], "text": "Answer: Hard factoring\nSteps:\n1. Multiply easy, split hard.\n2. So: Hard factoring.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0438", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(12,3).", "choices": null, "answer": "P(12,3) = 1320.", "steps": ["n(n−1)(n−2).", "12×11×10.", "= 1320."], "text": "Answer: P(12,3) = 1320.\nSteps:\n1. n(n−1)(n−2).\n2. 12×11×10.\n3. = 1320."}
{"id": "mat_0439", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 6 variables have?", "choices": {"A": "64", "B": "62", "C": "66", "D": "68"}, "answer": "The table has 64 rows.", "steps": ["2^6 rows.", "Total: 64.", "Select option A."], "text": "Answer: 64\nSteps:\n1. 2^6 rows.\n2. Total: 64.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0440", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 1x² + 5x, find f'(4).", "choices": {"A": "11", "B": "15", "C": "17", "D": "13"}, "answer": "The derivative is 13.", "steps": ["f'(x) = 2x + 5.", "At 4: 13.", "Select option D."], "text": "Answer: 13\nSteps:\n1. f'(x) = 2x + 5.\n2. At 4: 13.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0441", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why Gram-Schmidt?", "choices": null, "answer": "Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work.", "steps": ["Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work.", "So the answer stands: Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work."], "text": "Answer: Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work.\nSteps:\n1. Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work.\n2. So the answer stands: Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work."}
{"id": "mat_0442", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 13-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/13.", "steps": ["Favourable: 1.", "Possible: 13.", "P = 1/13."], "text": "Answer: P = 1/13.\nSteps:\n1. Favourable: 1.\n2. Possible: 13.\n3. P = 1/13."}
{"id": "mat_0443", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej (72) mod 7.", "choices": null, "answer": "Zbytek: 2.", "steps": ["Děl 72 sedmi.", "7×10 = 70.", "72 − 70 = 2."], "text": "Answer: Zbytek: 2.\nSteps:\n1. Děl 72 sedmi.\n2. 7×10 = 70.\n3. 72 − 70 = 2."}
{"id": "mat_0444", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why Bell numbers?", "choices": null, "answer": "They count set partitions, tallying every way to group objects into blocks.", "steps": ["They count set partitions, tallying every way to group objects into blocks.", "So the answer stands: They count set partitions, tallying every way to group objects into blocks."], "text": "Answer: They count set partitions, tallying every way to group objects into blocks.\nSteps:\n1. They count set partitions, tallying every way to group objects into blocks.\n2. So the answer stands: They count set partitions, tallying every way to group objects into blocks."}
{"id": "mat_0445", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 9?", "choices": {"A": "36", "B": "45", "C": "54", "D": "63"}, "answer": "The sum is 45.", "steps": ["9×10/2.", "Sum = 45.", "Select option B."], "text": "Answer: 45\nSteps:\n1. 9×10/2.\n2. Sum = 45.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0446", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 11²)/(x − 11) as x → 11.", "choices": null, "answer": "Limit: 22.", "steps": ["Factor: (x−11)(x+11).", "Cancel (x−11).", "Plug 11: 11+11 = 22."], "text": "Answer: Limit: 22.\nSteps:\n1. Factor: (x−11)(x+11).\n2. Cancel (x−11).\n3. Plug 11: 11+11 = 22."}
{"id": "mat_0447", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej stopu matice [[4,0],[0,5]].", "choices": null, "answer": "Stopa: 9.", "steps": ["Sečti diagonálu.", "4 + 5.", "= 9."], "text": "Answer: Stopa: 9.\nSteps:\n1. Sečti diagonálu.\n2. 4 + 5.\n3. = 9."}
{"id": "mat_0448", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..12}.", "choices": null, "answer": "E = 6.5.", "steps": ["Mean of 1..12.", "(12+1)/2.", "E = 6.5."], "text": "Answer: E = 6.5.\nSteps:\n1. Mean of 1..12.\n2. (12+1)/2.\n3. E = 6.5."}
{"id": "mat_0449", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(11, 8) step by step.", "choices": null, "answer": "lcm = 88.", "steps": ["gcd(11,8) = 1.", "11×8/1.", "= 88."], "text": "Answer: lcm = 88.\nSteps:\n1. gcd(11,8) = 1.\n2. 11×8/1.\n3. = 88."}
{"id": "mat_0450", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(8,3).", "choices": {"A": "48", "B": "56", "C": "64", "D": "72"}, "answer": "C(8,3) is 56.", "steps": ["8×7×6/6.", "= 56.", "Select option B."], "text": "Answer: 56\nSteps:\n1. 8×7×6/6.\n2. = 56.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0451", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 7.", "choices": {"A": "54", "B": "58", "C": "56", "D": "60"}, "answer": "The value is 56.", "steps": ["7² + 7.", "Value = 56.", "Select option C."], "text": "Answer: 56\nSteps:\n1. 7² + 7.\n2. Value = 56.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0452", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Pro f(x) = 3x² + 2x spočítej f'(3) krok za krokem.", "choices": null, "answer": "f'(3) = 20.", "steps": ["f'(x) = 6x + 2.", "Dosaď x = 3.", "6×3 + 2 = 20."], "text": "Answer: f'(3) = 20.\nSteps:\n1. f'(x) = 6x + 2.\n2. Dosaď x = 3.\n3. 6×3 + 2 = 20."}
{"id": "mat_0453", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is a projection?", "choices": null, "answer": "A projection drops a vector onto a subspace along perpendiculars, finding the closest point.", "steps": ["A projection drops a vector onto a subspace along perpendiculars, finding the closest point.", "So the answer stands: A projection drops a vector onto a subspace along perpendiculars, finding the closest point."], "text": "Answer: A projection drops a vector onto a subspace along perpendiculars, finding the closest point.\nSteps:\n1. A projection drops a vector onto a subspace along perpendiculars, finding the closest point.\n2. So the answer stands: A projection drops a vector onto a subspace along perpendiculars, finding the closest point."}
{"id": "mat_0454", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Hází se férovou 7-stěnnou kostkou. Ukaž P(jedno konkrétní číslo).", "choices": null, "answer": "P = 1/7.", "steps": ["Příznivé: 1.", "Možné: 7.", "P = 1/7."], "text": "Answer: P = 1/7.\nSteps:\n1. Příznivé: 1.\n2. Možné: 7.\n3. P = 1/7."}
{"id": "mat_0455", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Najdi největšího společného dělitele NSD(18, 27).", "choices": {"A": "6", "B": "9", "C": "12", "D": "15"}, "answer": "NSD je 9.", "steps": ["Vytkni 3 dvakrát.", "NSD = 9.", "Vyber možnost B."], "text": "Answer: 9\nSteps:\n1. Vytkni 3 dvakrát.\n2. NSD = 9.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0456", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Co je kombinační číslo?", "choices": null, "answer": "Počítá podmnožiny velikosti r z n objektů.", "steps": ["Počítá podmnožiny velikosti r z n objektů.", "Odpověď tedy platí: Počítá podmnožiny velikosti r z n objektů."], "text": "Answer: Počítá podmnožiny velikosti r z n objektů.\nSteps:\n1. Počítá podmnožiny velikosti r z n objektů.\n2. Odpověď tedy platí: Počítá podmnožiny velikosti r z n objektů."}
{"id": "mat_0457", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is a biconditional proof?", "choices": null, "answer": "An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately.", "steps": ["An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately.", "So the answer stands: An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately."], "text": "Answer: An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately.\nSteps:\n1. An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately.\n2. So the answer stands: An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately."}
{"id": "mat_0458", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 5.", "choices": {"A": "125", "B": "120", "C": "130", "D": "135"}, "answer": "The integral is 125.", "steps": ["Antiderivative x³.", "5³ = 125.", "Select option A."], "text": "Answer: 125\nSteps:\n1. Antiderivative x³.\n2. 5³ = 125.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0459", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (4,5) · (3,4).", "choices": {"A": "28", "B": "32", "C": "36", "D": "40"}, "answer": "The dot product is 32.", "steps": ["4×3 + 5×4.", "Product = 32.", "Select option B."], "text": "Answer: 32\nSteps:\n1. 4×3 + 5×4.\n2. Product = 32.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0460", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 10 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 10/14.", "steps": ["Red: 10.", "Total: 14.", "P = 10/14."], "text": "Answer: P(red) = 10/14.\nSteps:\n1. Red: 10.\n2. Total: 14.\n3. P = 10/14."}
{"id": "mat_0461", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(36, 54) step by step.", "choices": null, "answer": "gcd = 18.", "steps": ["Factor: 36=4×9, 54=6×9.", "Common: 9×gcd(4,6).", "= 9×2 = 18."], "text": "Answer: gcd = 18.\nSteps:\n1. Factor: 36=4×9, 54=6×9.\n2. Common: 9×gcd(4,6).\n3. = 9×2 = 18."}
{"id": "mat_0462", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej kombinační číslo C(5,2).", "choices": {"A": "5", "B": "10", "C": "15", "D": "20"}, "answer": "C(5,2) je 10.", "steps": ["5×4/2.", "= 10.", "Vyber možnost B."], "text": "Answer: 10\nSteps:\n1. 5×4/2.\n2. = 10.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0463", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 14.", "choices": null, "answer": "Sum: 105.", "steps": ["Pair 1+14, 2+13, ...", "Use 14×15/2.", "= 105."], "text": "Answer: Sum: 105.\nSteps:\n1. Pair 1+14, 2+13, ...\n2. Use 14×15/2.\n3. = 105."}
{"id": "mat_0464", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 2x² + 10x, compute f'(2) step by step.", "choices": null, "answer": "f'(2) = 18.", "steps": ["f'(x) = 4x + 10.", "Plug x = 2.", "4×2 + 10 = 18."], "text": "Answer: f'(2) = 18.\nSteps:\n1. f'(x) = 4x + 10.\n2. Plug x = 2.\n3. 4×2 + 10 = 18."}
{"id": "mat_0465", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Najdi skalární součin (1,2) · (3,4).", "choices": {"A": "7", "B": "15", "C": "11", "D": "19"}, "answer": "Skalární součin je 11.", "steps": ["1×3 + 2×4.", "Součin = 11.", "Vyber možnost C."], "text": "Answer: 11\nSteps:\n1. 1×3 + 2×4.\n2. Součin = 11.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0466", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 14-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/14.", "steps": ["Favourable: 1.", "Possible: 14.", "P = 1/14."], "text": "Answer: P = 1/14.\nSteps:\n1. Favourable: 1.\n2. Possible: 14.\n3. P = 1/14."}
{"id": "mat_0467", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej NSN(7, 8) krok za krokem.", "choices": null, "answer": "NSN = 56.", "steps": ["NSD(7,8) = 1.", "7×8/1.", "= 56."], "text": "Answer: NSN = 56.\nSteps:\n1. NSD(7,8) = 1.\n2. 7×8/1.\n3. = 56."}
{"id": "mat_0468", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is Stirling's formula?", "choices": null, "answer": "It approximates n! via roots and powers, taming huge factorials asymptotically.", "steps": ["It approximates n! via roots and powers, taming huge factorials asymptotically.", "So the answer stands: It approximates n! via roots and powers, taming huge factorials asymptotically."], "text": "Answer: It approximates n! via roots and powers, taming huge factorials asymptotically.\nSteps:\n1. It approximates n! via roots and powers, taming huge factorials asymptotically.\n2. So the answer stands: It approximates n! via roots and powers, taming huge factorials asymptotically."}
{"id": "mat_0469", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 10 by computing its half.", "choices": null, "answer": "Half: 55, so 110 is even.", "steps": ["10² + 10 = 110.", "110 / 2 = 55.", "Integer, so even."], "text": "Answer: Half: 55, so 110 is even.\nSteps:\n1. 10² + 10 = 110.\n2. 110 / 2 = 55.\n3. Integer, so even."}
{"id": "mat_0470", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 12.", "choices": null, "answer": "Integral: 1728.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 12.", "12³ − 0 = 1728."], "text": "Answer: Integral: 1728.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 12.\n3. 12³ − 0 = 1728."}
{"id": "mat_0471", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why least squares?", "choices": null, "answer": "Least squares fits lines by minimizing squared error, solving unsolvable systems approximately.", "steps": ["Least squares fits lines by minimizing squared error, solving unsolvable systems approximately.", "So the answer stands: Least squares fits lines by minimizing squared error, solving unsolvable systems approximately."], "text": "Answer: Least squares fits lines by minimizing squared error, solving unsolvable systems approximately.\nSteps:\n1. Least squares fits lines by minimizing squared error, solving unsolvable systems approximately.\n2. So the answer stands: Least squares fits lines by minimizing squared error, solving unsolvable systems approximately."}
{"id": "mat_0472", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what scores extreme data?", "choices": {"A": "Coin age", "B": "Shoe size", "C": "Star sign", "D": "p-value"}, "answer": "The answer is p-value.", "steps": ["Small casts doubt.", "So: p-value.", "Select option D."], "text": "Answer: p-value\nSteps:\n1. Small casts doubt.\n2. So: p-value.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0473", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (127) mod 7.", "choices": null, "answer": "Remainder: 1.", "steps": ["Divide 127 by 7.", "7×18 = 126.", "127 − 126 = 1."], "text": "Answer: Remainder: 1.\nSteps:\n1. Divide 127 by 7.\n2. 7×18 = 126.\n3. 127 − 126 = 1."}
{"id": "mat_0474", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(14,3).", "choices": null, "answer": "C(14,3) = 364.", "steps": ["n(n−1)(n−2)/6.", "14×13×12/6.", "= 364."], "text": "Answer: C(14,3) = 364.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 14×13×12/6.\n3. = 364."}
{"id": "mat_0475", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what flips under negation?", "choices": {"A": "Numbers only", "B": "Names only", "C": "Nothing ever", "D": "And/or quantifiers"}, "answer": "The answer is And/or quantifiers.", "steps": ["De Morgan rules it.", "So: And/or quantifiers.", "Select option D."], "text": "Answer: And/or quantifiers\nSteps:\n1. De Morgan rules it.\n2. So: And/or quantifiers.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0476", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Vyčísli integrál 3x² od 0 do 4.", "choices": null, "answer": "Integrál: 64.", "steps": ["Primitivní: x³.", "Vyčísli 0 až 4.", "4³ − 0 = 64."], "text": "Answer: Integrál: 64.\nSteps:\n1. Primitivní: x³.\n2. Vyčísli 0 až 4.\n3. 4³ − 0 = 64."}
{"id": "mat_0477", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[11,1],[2,12]] step by step.", "choices": null, "answer": "Determinant: 130.", "steps": ["ad − bc.", "11×12 − 1×2.", "= 130."], "text": "Answer: Determinant: 130.\nSteps:\n1. ad − bc.\n2. 11×12 − 1×2.\n3. = 130."}
{"id": "mat_0478", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..13}.", "choices": null, "answer": "E = 7.", "steps": ["Mean of 1..13.", "(13+1)/2.", "E = 7."], "text": "Answer: E = 7.\nSteps:\n1. Mean of 1..13.\n2. (13+1)/2.\n3. E = 7."}
{"id": "mat_0479", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej (107) mod 9.", "choices": {"A": "0", "B": "1", "C": "8", "D": "2"}, "answer": "Zbytek je 8.", "steps": ["107 ÷ 9.", "Zbytek 8.", "Vyber možnost C."], "text": "Answer: 8\nSteps:\n1. 107 ÷ 9.\n2. Zbytek 8.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0480", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet P(5,3).", "choices": null, "answer": "P(5,3) = 60.", "steps": ["n(n−1)(n−2).", "5×4×3.", "= 60."], "text": "Answer: P(5,3) = 60.\nSteps:\n1. n(n−1)(n−2).\n2. 5×4×3.\n3. = 60."}
{"id": "mat_0481", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Ukaž, že N² + N je sudé pro N = 4, výpočtem poloviny.", "choices": null, "answer": "Polovina: 10, číslo 20 je sudé.", "steps": ["4² + 4 = 20.", "20 / 2 = 10.", "Celé číslo, tedy sudé."], "text": "Answer: Polovina: 10, číslo 20 je sudé.\nSteps:\n1. 4² + 4 = 20.\n2. 20 / 2 = 10.\n3. Celé číslo, tedy sudé."}
{"id": "mat_0482", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Vyčísli limitu (x² − 4²)/(x − 4) pro x → 4.", "choices": null, "answer": "Limita: 8.", "steps": ["Rozlož: (x−4)(x+4).", "Pokrať (x−4).", "Dosaď 4: 4+4 = 8."], "text": "Answer: Limita: 8.\nSteps:\n1. Rozlož: (x−4)(x+4).\n2. Pokrať (x−4).\n3. Dosaď 4: 4+4 = 8."}
{"id": "mat_0483", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what keeps direction?", "choices": {"A": "Random vector", "B": "Zero vector", "C": "Eigenvector", "D": "Long vector"}, "answer": "The answer is Eigenvector.", "steps": ["Scaled by eigenvalue.", "So: Eigenvector.", "Select option C."], "text": "Answer: Eigenvector\nSteps:\n1. Scaled by eigenvalue.\n2. So: Eigenvector.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0484", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is covariance?", "choices": null, "answer": "Covariance measures joint variation of two variables, positive when they rise together.", "steps": ["Covariance measures joint variation of two variables, positive when they rise together.", "So the answer stands: Covariance measures joint variation of two variables, positive when they rise together."], "text": "Answer: Covariance measures joint variation of two variables, positive when they rise together.\nSteps:\n1. Covariance measures joint variation of two variables, positive when they rise together.\n2. So the answer stands: Covariance measures joint variation of two variables, positive when they rise together."}
{"id": "mat_0485", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why quadratic residues?", "choices": null, "answer": "They tell which numbers are squares mod p, decided fast by Euler's criterion.", "steps": ["They tell which numbers are squares mod p, decided fast by Euler's criterion.", "So the answer stands: They tell which numbers are squares mod p, decided fast by Euler's criterion."], "text": "Answer: They tell which numbers are squares mod p, decided fast by Euler's criterion.\nSteps:\n1. They tell which numbers are squares mod p, decided fast by Euler's criterion.\n2. So the answer stands: They tell which numbers are squares mod p, decided fast by Euler's criterion."}
{"id": "mat_0486", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej permutační číslo P(4,2).", "choices": {"A": "8", "B": "16", "C": "12", "D": "20"}, "answer": "P(4,2) je 12.", "steps": ["4×3.", "= 12.", "Vyber možnost C."], "text": "Answer: 12\nSteps:\n1. 4×3.\n2. = 12.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0487", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Důkaz prochází všechny případy 5 booleovských proměnných. Kolik řádků?", "choices": null, "answer": "Řádků: 32.", "steps": ["2 volby každá.", "2^5 = 32."], "text": "Answer: Řádků: 32.\nSteps:\n1. 2 volby každá.\n2. 2^5 = 32."}
{"id": "mat_0488", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why find inflection points?", "choices": null, "answer": "Inflection points mark where bending flips, revealing structural changes in the graph.", "steps": ["Inflection points mark where bending flips, revealing structural changes in the graph.", "So the answer stands: Inflection points mark where bending flips, revealing structural changes in the graph."], "text": "Answer: Inflection points mark where bending flips, revealing structural changes in the graph.\nSteps:\n1. Inflection points mark where bending flips, revealing structural changes in the graph.\n2. So the answer stands: Inflection points mark where bending flips, revealing structural changes in the graph."}
{"id": "mat_0489", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[12,0],[0,13]].", "choices": null, "answer": "Trace: 25.", "steps": ["Sum diagonal.", "12 + 13.", "= 25."], "text": "Answer: Trace: 25.\nSteps:\n1. Sum diagonal.\n2. 12 + 13.\n3. = 25."}
{"id": "mat_0490", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why correlation?", "choices": null, "answer": "Correlation scales covariance to [-1,1], giving a universal strength of linear relation.", "steps": ["Correlation scales covariance to [-1,1], giving a universal strength of linear relation.", "So the answer stands: Correlation scales covariance to [-1,1], giving a universal strength of linear relation."], "text": "Answer: Correlation scales covariance to [-1,1], giving a universal strength of linear relation.\nSteps:\n1. Correlation scales covariance to [-1,1], giving a universal strength of linear relation.\n2. So the answer stands: Correlation scales covariance to [-1,1], giving a universal strength of linear relation."}
{"id": "mat_0491", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(36, 54).", "choices": {"A": "18", "B": "15", "C": "21", "D": "24"}, "answer": "The gcd is 18.", "steps": ["Factor 3 out twice.", "gcd = 18.", "Select option A."], "text": "Answer: 18\nSteps:\n1. Factor 3 out twice.\n2. gcd = 18.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0492", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(14,2).", "choices": null, "answer": "C(14,2) = 91.", "steps": ["Formula n(n−1)/2.", "14×13/2.", "= 91."], "text": "Answer: C(14,2) = 91.\nSteps:\n1. Formula n(n−1)/2.\n2. 14×13/2.\n3. = 91."}
{"id": "mat_0493", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 11 boolean variables. How many rows?", "choices": null, "answer": "Rows: 2048.", "steps": ["2 choices each.", "2^11 = 2048."], "text": "Answer: Rows: 2048.\nSteps:\n1. 2 choices each.\n2. 2^11 = 2048."}
{"id": "mat_0494", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 12²)/(x − 12) as x → 12.", "choices": null, "answer": "Limit: 24.", "steps": ["Factor: (x−12)(x+12).", "Cancel (x−12).", "Plug 12: 12+12 = 24."], "text": "Answer: Limit: 24.\nSteps:\n1. Factor: (x−12)(x+12).\n2. Cancel (x−12).\n3. Plug 12: 12+12 = 24."}
{"id": "mat_0495", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is the null space?", "choices": null, "answer": "The null space holds all vectors mapped to zero, measuring how much the map collapses.", "steps": ["The null space holds all vectors mapped to zero, measuring how much the map collapses.", "So the answer stands: The null space holds all vectors mapped to zero, measuring how much the map collapses."], "text": "Answer: The null space holds all vectors mapped to zero, measuring how much the map collapses.\nSteps:\n1. The null space holds all vectors mapped to zero, measuring how much the map collapses.\n2. So the answer stands: The null space holds all vectors mapped to zero, measuring how much the map collapses."}
{"id": "mat_0496", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co vypisuje všechny výsledky?", "choices": {"A": "seznam přání", "B": "pytel hádání", "C": "truhla nadějí", "D": "prostor výsledků"}, "answer": "Odpověď je prostor výsledků.", "steps": ["Jevy jsou podmnožiny.", "Tedy: prostor výsledků.", "Vyber možnost D."], "text": "Answer: prostor výsledků\nSteps:\n1. Jevy jsou podmnožiny.\n2. Tedy: prostor výsledků.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0497", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(12, 8) step by step.", "choices": null, "answer": "lcm = 24.", "steps": ["gcd(12,8) = 4.", "12×8/4.", "= 24."], "text": "Answer: lcm = 24.\nSteps:\n1. gcd(12,8) = 4.\n2. 12×8/4.\n3. = 24."}
{"id": "mat_0498", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what forces sharing?", "choices": {"A": "Big boxes", "B": "Few items", "C": "Pigeonhole", "D": "Kind hosts"}, "answer": "The answer is Pigeonhole.", "steps": ["Items exceed boxes.", "So: Pigeonhole.", "Select option C."], "text": "Answer: Pigeonhole\nSteps:\n1. Items exceed boxes.\n2. So: Pigeonhole.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0499", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 15.", "choices": null, "answer": "Sum: 120.", "steps": ["Pair 1+15, 2+14, ...", "Use 15×16/2.", "= 120."], "text": "Answer: Sum: 120.\nSteps:\n1. Pair 1+15, 2+14, ...\n2. Use 15×16/2.\n3. = 120."}
{"id": "mat_0500", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 5²)/(x − 5) as x → 5.", "choices": {"A": "8", "B": "10", "C": "12", "D": "14"}, "answer": "The limit is 10.", "steps": ["Factor (x−5)(x+5).", "Cancel, get 2×5 = 10.", "Select option B."], "text": "Answer: 10\nSteps:\n1. Factor (x−5)(x+5).\n2. Cancel, get 2×5 = 10.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0501", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what sums the diagonal?", "choices": {"A": "Rank", "B": "Norm", "C": "Size", "D": "Trace"}, "answer": "The answer is Trace.", "steps": ["Eigenvalue sum equal.", "So: Trace.", "Select option D."], "text": "Answer: Trace\nSteps:\n1. Eigenvalue sum equal.\n2. So: Trace.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0502", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is a Markov chain?", "choices": null, "answer": "It jumps between states with fixed probabilities depending only on the current state.", "steps": ["It jumps between states with fixed probabilities depending only on the current state.", "So the answer stands: It jumps between states with fixed probabilities depending only on the current state."], "text": "Answer: It jumps between states with fixed probabilities depending only on the current state.\nSteps:\n1. It jumps between states with fixed probabilities depending only on the current state.\n2. So the answer stands: It jumps between states with fixed probabilities depending only on the current state."}
{"id": "mat_0503", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is Legendre's symbol?", "choices": null, "answer": "It encodes residue status as 1, -1, or 0, compressing square questions to algebra.", "steps": ["It encodes residue status as 1, -1, or 0, compressing square questions to algebra.", "So the answer stands: It encodes residue status as 1, -1, or 0, compressing square questions to algebra."], "text": "Answer: It encodes residue status as 1, -1, or 0, compressing square questions to algebra.\nSteps:\n1. It encodes residue status as 1, -1, or 0, compressing square questions to algebra.\n2. So the answer stands: It encodes residue status as 1, -1, or 0, compressing square questions to algebra."}
{"id": "mat_0504", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what orders flipping one bit?", "choices": {"A": "Wild order", "B": "Blind list", "C": "Vague chain", "D": "Gray code"}, "answer": "The answer is Gray code.", "steps": ["Neighbours differ once.", "So: Gray code.", "Select option D."], "text": "Answer: Gray code\nSteps:\n1. Neighbours differ once.\n2. So: Gray code.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0505", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Choose: what is circular reasoning?", "choices": {"A": "Assuming conclusion", "B": "Round diagrams", "C": "Long proofs", "D": "Many cases"}, "answer": "The answer is Assuming conclusion.", "steps": ["Proves nothing at all.", "So: Assuming conclusion.", "Select option A."], "text": "Answer: Assuming conclusion\nSteps:\n1. Proves nothing at all.\n2. So: Assuming conclusion.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0506", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Pro f(x) = 4x² + 3x spočítej f'(4) krok za krokem.", "choices": null, "answer": "f'(4) = 35.", "steps": ["f'(x) = 8x + 3.", "Dosaď x = 4.", "8×4 + 3 = 35."], "text": "Answer: f'(4) = 35.\nSteps:\n1. f'(x) = 8x + 3.\n2. Dosaď x = 4.\n3. 8×4 + 3 = 35."}
{"id": "mat_0507", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co detekuje regularitu?", "choices": {"A": "velké prvky", "B": "kulatý tvar", "C": "dlouhé jméno", "D": "nenulový determinant"}, "answer": "Odpověď je nenulový determinant.", "steps": ["Nula znamená zhroucení.", "Tedy: nenulový determinant.", "Vyber možnost D."], "text": "Answer: nenulový determinant\nSteps:\n1. Nula znamená zhroucení.\n2. Tedy: nenulový determinant.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0508", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why stationary distributions?", "choices": null, "answer": "They give long-run state fractions, predicting where chains settle over time.", "steps": ["They give long-run state fractions, predicting where chains settle over time.", "So the answer stands: They give long-run state fractions, predicting where chains settle over time."], "text": "Answer: They give long-run state fractions, predicting where chains settle over time.\nSteps:\n1. They give long-run state fractions, predicting where chains settle over time.\n2. So the answer stands: They give long-run state fractions, predicting where chains settle over time."}
{"id": "mat_0509", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Co je NSD?", "choices": null, "answer": "Největší společný dělitel je největší číslo dělící obě čísla.", "steps": ["Největší společný dělitel je největší číslo dělící obě čísla.", "Odpověď tedy platí: Největší společný dělitel je největší číslo dělící obě čísla."], "text": "Answer: Největší společný dělitel je největší číslo dělící obě čísla.\nSteps:\n1. Největší společný dělitel je největší číslo dělící obě čísla.\n2. Odpověď tedy platí: Největší společný dělitel je největší číslo dělící obě čísla."}
{"id": "mat_0510", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why double counting?", "choices": null, "answer": "Counting one set two ways proves identities, a favourite elegant technique.", "steps": ["Counting one set two ways proves identities, a favourite elegant technique.", "So the answer stands: Counting one set two ways proves identities, a favourite elegant technique."], "text": "Answer: Counting one set two ways proves identities, a favourite elegant technique.\nSteps:\n1. Counting one set two ways proves identities, a favourite elegant technique.\n2. So the answer stands: Counting one set two ways proves identities, a favourite elegant technique."}
{"id": "mat_0511", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why use Venn diagrams?", "choices": null, "answer": "Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements.", "steps": ["Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements.", "So the answer stands: Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements."], "text": "Answer: Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements.\nSteps:\n1. Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements.\n2. So the answer stands: Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements."}
{"id": "mat_0512", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: what links both operations?", "choices": {"A": "Guessing rule", "B": "Lucky axiom", "C": "Fundamental theorem", "D": "Long division"}, "answer": "The answer is Fundamental theorem.", "steps": ["Integrals via antiderivatives.", "So: Fundamental theorem.", "Select option C."], "text": "Answer: Fundamental theorem\nSteps:\n1. Integrals via antiderivatives.\n2. So: Fundamental theorem.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0513", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why rank-nullity?", "choices": null, "answer": "Rank plus nullity equals columns, linking image size to collapse in one equation.", "steps": ["Rank plus nullity equals columns, linking image size to collapse in one equation.", "So the answer stands: Rank plus nullity equals columns, linking image size to collapse in one equation."], "text": "Answer: Rank plus nullity equals columns, linking image size to collapse in one equation.\nSteps:\n1. Rank plus nullity equals columns, linking image size to collapse in one equation.\n2. So the answer stands: Rank plus nullity equals columns, linking image size to collapse in one equation."}
{"id": "mat_0514", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co měří rozptyl?", "choices": {"A": "rozptyl", "B": "průměr", "C": "modus", "D": "naděje"}, "answer": "Odpověď je rozptyl.", "steps": ["Čtvercové odchylky průměrovány.", "Tedy: rozptyl.", "Vyber možnost A."], "text": "Answer: rozptyl\nSteps:\n1. Čtvercové odchylky průměrovány.\n2. Tedy: rozptyl.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0515", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why quadratic reciprocity?", "choices": null, "answer": "It swaps primes in Legendre symbols, making huge residue questions computable.", "steps": ["It swaps primes in Legendre symbols, making huge residue questions computable.", "So the answer stands: It swaps primes in Legendre symbols, making huge residue questions computable."], "text": "Answer: It swaps primes in Legendre symbols, making huge residue questions computable.\nSteps:\n1. It swaps primes in Legendre symbols, making huge residue questions computable.\n2. So the answer stands: It swaps primes in Legendre symbols, making huge residue questions computable."}
{"id": "mat_0516", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is a bijection?", "choices": null, "answer": "A bijection pairs sets one-to-one, proving equal size without any counting.", "steps": ["A bijection pairs sets one-to-one, proving equal size without any counting.", "So the answer stands: A bijection pairs sets one-to-one, proving equal size without any counting."], "text": "Answer: A bijection pairs sets one-to-one, proving equal size without any counting.\nSteps:\n1. A bijection pairs sets one-to-one, proving equal size without any counting.\n2. So the answer stands: A bijection pairs sets one-to-one, proving equal size without any counting."}
{"id": "mat_0517", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Co je lemma?", "choices": null, "answer": "Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky.", "steps": ["Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky.", "Odpověď tedy platí: Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky."], "text": "Answer: Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky.\nSteps:\n1. Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky.\n2. Odpověď tedy platí: Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky."}
{"id": "mat_0518", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 3x² + 11x, compute f'(3) step by step.", "choices": null, "answer": "f'(3) = 29.", "steps": ["f'(x) = 6x + 11.", "Plug x = 3.", "6×3 + 11 = 29."], "text": "Answer: f'(3) = 29.\nSteps:\n1. f'(x) = 6x + 11.\n2. Plug x = 3.\n3. 6×3 + 11 = 29."}
{"id": "mat_0519", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is a linear system?", "choices": null, "answer": "A linear system is equations linear in unknowns, solved together by elimination.", "steps": ["A linear system is equations linear in unknowns, solved together by elimination.", "So the answer stands: A linear system is equations linear in unknowns, solved together by elimination."], "text": "Answer: A linear system is equations linear in unknowns, solved together by elimination.\nSteps:\n1. A linear system is equations linear in unknowns, solved together by elimination.\n2. So the answer stands: A linear system is equations linear in unknowns, solved together by elimination."}
{"id": "mat_0520", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Hází se férovou 7-stěnnou kostkou. Jaká je pravděpodobnost jednoho konkrétního čísla?", "choices": {"A": "1/6", "B": "1/7", "C": "1/8", "D": "2/7"}, "answer": "Pravděpodobnost je 1/7.", "steps": ["1 stěna z 7.", "P = 1/7.", "Vyber možnost B."], "text": "Answer: 1/7\nSteps:\n1. 1 stěna z 7.\n2. P = 1/7.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0521", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(40, 60) step by step.", "choices": null, "answer": "gcd = 20.", "steps": ["Factor: 40=4×10, 60=6×10.", "Common: 10×gcd(4,6).", "= 10×2 = 20."], "text": "Answer: gcd = 20.\nSteps:\n1. Factor: 40=4×10, 60=6×10.\n2. Common: 10×gcd(4,6).\n3. = 10×2 = 20."}
{"id": "mat_0522", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(8,2).", "choices": {"A": "28", "B": "20", "C": "36", "D": "44"}, "answer": "C(8,2) is 28.", "steps": ["8×7/2.", "= 28.", "Select option A."], "text": "Answer: 28\nSteps:\n1. 8×7/2.\n2. = 28.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0523", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Dokaž vzorcem: najdi součet 1 + 2 + ... + 9.", "choices": null, "answer": "Součet: 45.", "steps": ["Páruj 1+9, 2+8, ...", "Použij 9×10/2.", "= 45."], "text": "Answer: Součet: 45.\nSteps:\n1. Páruj 1+9, 2+8, ...\n2. Použij 9×10/2.\n3. = 45."}
{"id": "mat_0524", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is a partial derivative?", "choices": null, "answer": "A partial derivative varies one variable while freezing others, extending calculus to surfaces.", "steps": ["A partial derivative varies one variable while freezing others, extending calculus to surfaces.", "So the answer stands: A partial derivative varies one variable while freezing others, extending calculus to surfaces."], "text": "Answer: A partial derivative varies one variable while freezing others, extending calculus to surfaces.\nSteps:\n1. A partial derivative varies one variable while freezing others, extending calculus to surfaces.\n2. So the answer stands: A partial derivative varies one variable while freezing others, extending calculus to surfaces."}
{"id": "mat_0525", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Co je inverzní matice?", "choices": null, "answer": "Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu.", "steps": ["Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu.", "Odpověď tedy platí: Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu."], "text": "Answer: Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu.\nSteps:\n1. Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu.\n2. Odpověď tedy platí: Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu."}
{"id": "mat_0526", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is the Poisson distribution?", "choices": null, "answer": "It counts rare events in fixed intervals, modelling arrivals and decays.", "steps": ["It counts rare events in fixed intervals, modelling arrivals and decays.", "So the answer stands: It counts rare events in fixed intervals, modelling arrivals and decays."], "text": "Answer: It counts rare events in fixed intervals, modelling arrivals and decays.\nSteps:\n1. It counts rare events in fixed intervals, modelling arrivals and decays.\n2. So the answer stands: It counts rare events in fixed intervals, modelling arrivals and decays."}
{"id": "mat_0527", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej NSD(20, 30) krok za krokem.", "choices": null, "answer": "NSD = 10.", "steps": ["Rozlož: 20=4×5, 30=6×5.", "Společné: 5×NSD(4,6).", "= 5×2 = 10."], "text": "Answer: NSD = 10.\nSteps:\n1. Rozlož: 20=4×5, 30=6×5.\n2. Společné: 5×NSD(4,6).\n3. = 5×2 = 10."}
{"id": "mat_0528", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(13,3).", "choices": null, "answer": "P(13,3) = 1716.", "steps": ["n(n−1)(n−2).", "13×12×11.", "= 1716."], "text": "Answer: P(13,3) = 1716.\nSteps:\n1. n(n−1)(n−2).\n2. 13×12×11.\n3. = 1716."}
{"id": "mat_0529", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 7 variables have?", "choices": {"A": "126", "B": "128", "C": "130", "D": "132"}, "answer": "The table has 128 rows.", "steps": ["2^7 rows.", "Total: 128.", "Select option B."], "text": "Answer: 128\nSteps:\n1. 2^7 rows.\n2. Total: 128.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0530", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Co je základní věta kalkulu?", "choices": null, "answer": "Základní věta spojuje derivace a integrály přes primitivní funkce.", "steps": ["Základní věta spojuje derivace a integrály přes primitivní funkce.", "Odpověď tedy platí: Základní věta spojuje derivace a integrály přes primitivní funkce."], "text": "Answer: Základní věta spojuje derivace a integrály přes primitivní funkce.\nSteps:\n1. Základní věta spojuje derivace a integrály přes primitivní funkce.\n2. Odpověď tedy platí: Základní věta spojuje derivace a integrály přes primitivní funkce."}
{"id": "mat_0531", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[5,2],[3,8]].", "choices": {"A": "34", "B": "31", "C": "37", "D": "40"}, "answer": "The determinant is 34.", "steps": ["5×8 − 2×3.", "Det = 34.", "Select option A."], "text": "Answer: 34\nSteps:\n1. 5×8 − 2×3.\n2. Det = 34.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0532", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Najdi střední hodnotu rovnoměrného tahu z {1..5}.", "choices": {"A": "4", "B": "2", "C": "3", "D": "5"}, "answer": "Střední hodnota je 3.", "steps": ["(5+1)/2.", "E = 3.", "Vyber možnost C."], "text": "Answer: 3\nSteps:\n1. (5+1)/2.\n2. E = 3.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0533", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Najdi nejmenší společný násobek NSN(5, 6).", "choices": {"A": "28", "B": "32", "C": "34", "D": "30"}, "answer": "NSN je 30.", "steps": ["5×6/NSD.", "NSN = 30.", "Vyber možnost D."], "text": "Answer: 30\nSteps:\n1. 5×6/NSD.\n2. NSN = 30.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0534", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(7,2).", "choices": {"A": "35", "B": "42", "C": "49", "D": "56"}, "answer": "P(7,2) is 42.", "steps": ["7×6.", "= 42.", "Select option B."], "text": "Answer: 42\nSteps:\n1. 7×6.\n2. = 42.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0535", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 11 by computing its half.", "choices": null, "answer": "Half: 66, so 132 is even.", "steps": ["11² + 11 = 132.", "132 / 2 = 66.", "Integer, so even."], "text": "Answer: Half: 66, so 132 is even.\nSteps:\n1. 11² + 11 = 132.\n2. 132 / 2 = 66.\n3. Integer, so even."}
{"id": "mat_0536", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Proč Taylorovy řady?", "choices": null, "answer": "Taylorovy řady aproximují funkce polynomy s odhadem chyby.", "steps": ["Taylorovy řady aproximují funkce polynomy s odhadem chyby.", "Odpověď tedy platí: Taylorovy řady aproximují funkce polynomy s odhadem chyby."], "text": "Answer: Taylorovy řady aproximují funkce polynomy s odhadem chyby.\nSteps:\n1. Taylorovy řady aproximují funkce polynomy s odhadem chyby.\n2. Odpověď tedy platí: Taylorovy řady aproximují funkce polynomy s odhadem chyby."}
{"id": "mat_0537", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (11,10) · (3,5) step by step.", "choices": null, "answer": "Dot product: 83.", "steps": ["Multiply components.", "11×3 + 10×5.", "= 83."], "text": "Answer: Dot product: 83.\nSteps:\n1. Multiply components.\n2. 11×3 + 10×5.\n3. = 83."}
{"id": "mat_0538", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why exponential distribution?", "choices": null, "answer": "It models waiting times with no memory, fitting lifetimes and queues.", "steps": ["It models waiting times with no memory, fitting lifetimes and queues.", "So the answer stands: It models waiting times with no memory, fitting lifetimes and queues."], "text": "Answer: It models waiting times with no memory, fitting lifetimes and queues.\nSteps:\n1. It models waiting times with no memory, fitting lifetimes and queues.\n2. So the answer stands: It models waiting times with no memory, fitting lifetimes and queues."}
{"id": "mat_0539", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (138) mod 7.", "choices": null, "answer": "Remainder: 5.", "steps": ["Divide 138 by 7.", "7×19 = 133.", "138 − 133 = 5."], "text": "Answer: Remainder: 5.\nSteps:\n1. Divide 138 by 7.\n2. 7×19 = 133.\n3. 138 − 133 = 5."}
{"id": "mat_0540", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(15,3).", "choices": null, "answer": "C(15,3) = 455.", "steps": ["n(n−1)(n−2)/6.", "15×14×13/6.", "= 455."], "text": "Answer: C(15,3) = 455.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 15×14×13/6.\n3. = 455."}
{"id": "mat_0541", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 12 boolean variables. How many rows?", "choices": null, "answer": "Rows: 4096.", "steps": ["2 choices each.", "2^12 = 4096."], "text": "Answer: Rows: 4096.\nSteps:\n1. 2 choices each.\n2. 2^12 = 4096."}
{"id": "mat_0542", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 13.", "choices": null, "answer": "Integral: 2197.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 13.", "13³ − 0 = 2197."], "text": "Answer: Integral: 2197.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 13.\n3. 13³ − 0 = 2197."}
{"id": "mat_0543", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why augmented matrices?", "choices": null, "answer": "Augmenting appends answers to coefficients, keeping elimination tidy in one array.", "steps": ["Augmenting appends answers to coefficients, keeping elimination tidy in one array.", "So the answer stands: Augmenting appends answers to coefficients, keeping elimination tidy in one array."], "text": "Answer: Augmenting appends answers to coefficients, keeping elimination tidy in one array.\nSteps:\n1. Augmenting appends answers to coefficients, keeping elimination tidy in one array.\n2. So the answer stands: Augmenting appends answers to coefficients, keeping elimination tidy in one array."}
{"id": "mat_0544", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 11 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 11/15.", "steps": ["Red: 11.", "Total: 15.", "P = 11/15."], "text": "Answer: P(red) = 11/15.\nSteps:\n1. Red: 11.\n2. Total: 15.\n3. P = 11/15."}
{"id": "mat_0545", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (128) mod 9.", "choices": {"A": "3", "B": "2", "C": "4", "D": "5"}, "answer": "The remainder is 2.", "steps": ["128 ÷ 9.", "Remainder 2.", "Select option B."], "text": "Answer: 2\nSteps:\n1. 128 ÷ 9.\n2. Remainder 2.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0546", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(9,3).", "choices": {"A": "75", "B": "93", "C": "84", "D": "102"}, "answer": "C(9,3) is 84.", "steps": ["9×8×7/6.", "= 84.", "Select option C."], "text": "Answer: 84\nSteps:\n1. 9×8×7/6.\n2. = 84.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0547", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is De Morgan's law?", "choices": null, "answer": "De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B).", "steps": ["De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B).", "So the answer stands: De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B)."], "text": "Answer: De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B).\nSteps:\n1. De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B).\n2. So the answer stands: De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B)."}
{"id": "mat_0548", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: what varies one variable?", "choices": {"A": "Total guess", "B": "Full stop", "C": "Side note", "D": "Partial derivative"}, "answer": "The answer is Partial derivative.", "steps": ["Others frozen.", "So: Partial derivative.", "Select option D."], "text": "Answer: Partial derivative\nSteps:\n1. Others frozen.\n2. So: Partial derivative.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0549", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[6,1],[2,9]].", "choices": {"A": "13", "B": "15", "C": "17", "D": "19"}, "answer": "The trace is 15.", "steps": ["6 + 9.", "Trace = 15.", "Select option B."], "text": "Answer: 15\nSteps:\n1. 6 + 9.\n2. Trace = 15.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0550", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Proč normální rozdělení?", "choices": null, "answer": "Průměry mnoha veličin jsou normální, modeluje chyby a variace.", "steps": ["Průměry mnoha veličin jsou normální, modeluje chyby a variace.", "Odpověď tedy platí: Průměry mnoha veličin jsou normální, modeluje chyby a variace."], "text": "Answer: Průměry mnoha veličin jsou normální, modeluje chyby a variace.\nSteps:\n1. Průměry mnoha veličin jsou normální, modeluje chyby a variace.\n2. Odpověď tedy platí: Průměry mnoha veličin jsou normální, modeluje chyby a variace."}
{"id": "mat_0551", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(13, 8) step by step.", "choices": null, "answer": "lcm = 104.", "steps": ["gcd(13,8) = 1.", "13×8/1.", "= 104."], "text": "Answer: lcm = 104.\nSteps:\n1. gcd(13,8) = 1.\n2. 13×8/1.\n3. = 104."}
{"id": "mat_0552", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Proč Pascalův trojúhelník?", "choices": null, "answer": "Staví řádky sčítáním a ukazuje symetrie.", "steps": ["Staví řádky sčítáním a ukazuje symetrie.", "Odpověď tedy platí: Staví řádky sčítáním a ukazuje symetrie."], "text": "Answer: Staví řádky sčítáním a ukazuje symetrie.\nSteps:\n1. Staví řádky sčítáním a ukazuje symetrie.\n2. Odpověď tedy platí: Staví řádky sčítáním a ukazuje symetrie."}
{"id": "mat_0553", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why learn truth tables?", "choices": null, "answer": "Truth tables list every combination of truth values, so they verify logical equivalences mechanically.", "steps": ["Truth tables list every combination of truth values, so they verify logical equivalences mechanically.", "So the answer stands: Truth tables list every combination of truth values, so they verify logical equivalences mechanically."], "text": "Answer: Truth tables list every combination of truth values, so they verify logical equivalences mechanically.\nSteps:\n1. Truth tables list every combination of truth values, so they verify logical equivalences mechanically.\n2. So the answer stands: Truth tables list every combination of truth values, so they verify logical equivalences mechanically."}
{"id": "mat_0554", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why use Lagrange multipliers?", "choices": null, "answer": "They find extrema under constraints by aligning gradients, solving optimization with limits.", "steps": ["They find extrema under constraints by aligning gradients, solving optimization with limits.", "So the answer stands: They find extrema under constraints by aligning gradients, solving optimization with limits."], "text": "Answer: They find extrema under constraints by aligning gradients, solving optimization with limits.\nSteps:\n1. They find extrema under constraints by aligning gradients, solving optimization with limits.\n2. So the answer stands: They find extrema under constraints by aligning gradients, solving optimization with limits."}
{"id": "mat_0555", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is Cramer's rule?", "choices": null, "answer": "It solves systems via determinant ratios, elegant for theory though slow in practice.", "steps": ["It solves systems via determinant ratios, elegant for theory though slow in practice.", "So the answer stands: It solves systems via determinant ratios, elegant for theory though slow in practice."], "text": "Answer: It solves systems via determinant ratios, elegant for theory though slow in practice.\nSteps:\n1. It solves systems via determinant ratios, elegant for theory though slow in practice.\n2. So the answer stands: It solves systems via determinant ratios, elegant for theory though slow in practice."}
{"id": "mat_0556", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 15-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/15.", "steps": ["Favourable: 1.", "Possible: 15.", "P = 1/15."], "text": "Answer: P = 1/15.\nSteps:\n1. Favourable: 1.\n2. Possible: 15.\n3. P = 1/15."}
{"id": "mat_0557", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Proč Euklidův algoritmus?", "choices": null, "answer": "Opakované zbytky rychle zmenšují dvojici.", "steps": ["Opakované zbytky rychle zmenšují dvojici.", "Odpověď tedy platí: Opakované zbytky rychle zmenšují dvojici."], "text": "Answer: Opakované zbytky rychle zmenšují dvojici.\nSteps:\n1. Opakované zbytky rychle zmenšují dvojici.\n2. Odpověď tedy platí: Opakované zbytky rychle zmenšují dvojici."}
{"id": "mat_0558", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why rook theory?", "choices": null, "answer": "Rook placements count permutations with restrictions, via inclusion-exclusion boards.", "steps": ["Rook placements count permutations with restrictions, via inclusion-exclusion boards.", "So the answer stands: Rook placements count permutations with restrictions, via inclusion-exclusion boards."], "text": "Answer: Rook placements count permutations with restrictions, via inclusion-exclusion boards.\nSteps:\n1. Rook placements count permutations with restrictions, via inclusion-exclusion boards.\n2. So the answer stands: Rook placements count permutations with restrictions, via inclusion-exclusion boards."}
{"id": "mat_0559", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 16.", "choices": null, "answer": "Sum: 136.", "steps": ["Pair 1+16, 2+15, ...", "Use 16×17/2.", "= 136."], "text": "Answer: Sum: 136.\nSteps:\n1. Pair 1+16, 2+15, ...\n2. Use 16×17/2.\n3. = 136."}
{"id": "mat_0560", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 2x² + 6x, find f'(1).", "choices": {"A": "10", "B": "8", "C": "12", "D": "14"}, "answer": "The derivative is 10.", "steps": ["f'(x) = 4x + 6.", "At 1: 10.", "Select option A."], "text": "Answer: 10\nSteps:\n1. f'(x) = 4x + 6.\n2. At 1: 10.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0561", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[12,1],[2,13]] step by step.", "choices": null, "answer": "Determinant: 154.", "steps": ["ad − bc.", "12×13 − 1×2.", "= 154."], "text": "Answer: Determinant: 154.\nSteps:\n1. ad − bc.\n2. 12×13 − 1×2.\n3. = 154."}
{"id": "mat_0562", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Spočítej E rovnoměrného tahu z {1..6}.", "choices": null, "answer": "E = 3,5.", "steps": ["Průměr 1..6.", "(6+1)/2.", "E = 3,5."], "text": "Answer: E = 3,5.\nSteps:\n1. Průměr 1..6.\n2. (6+1)/2.\n3. E = 3,5."}
{"id": "mat_0563", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is a perfect number?", "choices": null, "answer": "It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious.", "steps": ["It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious.", "So the answer stands: It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious."], "text": "Answer: It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious.\nSteps:\n1. It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious.\n2. So the answer stands: It equals the sum of its proper divisors, like 6 = 1+2+3, rare and mysterious."}
{"id": "mat_0564", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej kombinační číslo C(6,3).", "choices": {"A": "14", "B": "26", "C": "32", "D": "20"}, "answer": "C(6,3) je 20.", "steps": ["6×5×4/6.", "= 20.", "Vyber možnost D."], "text": "Answer: 20\nSteps:\n1. 6×5×4/6.\n2. = 20.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0565", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 12 by computing its half.", "choices": null, "answer": "Half: 78, so 156 is even.", "steps": ["12² + 12 = 156.", "156 / 2 = 78.", "Integer, so even."], "text": "Answer: Half: 78, so 156 is even.\nSteps:\n1. 12² + 12 = 156.\n2. 156 / 2 = 78.\n3. Integer, so even."}
{"id": "mat_0566", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 6.", "choices": {"A": "210", "B": "216", "C": "222", "D": "228"}, "answer": "The integral is 216.", "steps": ["Antiderivative x³.", "6³ = 216.", "Select option B."], "text": "Answer: 216\nSteps:\n1. Antiderivative x³.\n2. 6³ = 216.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0567", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[13,0],[0,14]].", "choices": null, "answer": "Trace: 27.", "steps": ["Sum diagonal.", "13 + 14.", "= 27."], "text": "Answer: Trace: 27.\nSteps:\n1. Sum diagonal.\n2. 13 + 14.\n3. = 27."}
{"id": "mat_0568", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..14}.", "choices": null, "answer": "E = 7.5.", "steps": ["Mean of 1..14.", "(14+1)/2.", "E = 7.5."], "text": "Answer: E = 7.5.\nSteps:\n1. Mean of 1..14.\n2. (14+1)/2.\n3. E = 7.5."}
{"id": "mat_0569", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Co je modulární aritmetika?", "choices": null, "answer": "Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin.", "steps": ["Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin.", "Odpověď tedy platí: Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin."], "text": "Answer: Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin.\nSteps:\n1. Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin.\n2. Odpověď tedy platí: Počítá se zbytky po dělení, čísla se obtáčejí kolem hodin."}
{"id": "mat_0570", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co počítá podmnožiny?", "choices": {"A": "kombinační číslo", "B": "n+r", "C": "n*r", "D": "n+r!"}, "answer": "Odpověď je kombinační číslo.", "steps": ["Pořadí vyděleno pryč.", "Tedy: kombinační číslo.", "Vyber možnost A."], "text": "Answer: kombinační číslo\nSteps:\n1. Pořadí vyděleno pryč.\n2. Tedy: kombinační číslo.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0571", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 13 boolean variables. How many rows?", "choices": null, "answer": "Rows: 8192.", "steps": ["2 choices each.", "2^13 = 8192."], "text": "Answer: Rows: 8192.\nSteps:\n1. 2 choices each.\n2. 2^13 = 8192."}
{"id": "mat_0572", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is a differential equation?", "choices": null, "answer": "It relates a function to its derivatives, and solving it finds the function from its change law.", "steps": ["It relates a function to its derivatives, and solving it finds the function from its change law.", "So the answer stands: It relates a function to its derivatives, and solving it finds the function from its change law."], "text": "Answer: It relates a function to its derivatives, and solving it finds the function from its change law.\nSteps:\n1. It relates a function to its derivatives, and solving it finds the function from its change law.\n2. So the answer stands: It relates a function to its derivatives, and solving it finds the function from its change law."}
{"id": "mat_0573", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (12,11) · (3,5) step by step.", "choices": null, "answer": "Dot product: 91.", "steps": ["Multiply components.", "12×3 + 11×5.", "= 91."], "text": "Answer: Dot product: 91.\nSteps:\n1. Multiply components.\n2. 12×3 + 11×5.\n3. = 91."}
{"id": "mat_0574", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is a joint distribution?", "choices": null, "answer": "It gives probabilities of value pairs, capturing how two variables vary together.", "steps": ["It gives probabilities of value pairs, capturing how two variables vary together.", "So the answer stands: It gives probabilities of value pairs, capturing how two variables vary together."], "text": "Answer: It gives probabilities of value pairs, capturing how two variables vary together.\nSteps:\n1. It gives probabilities of value pairs, capturing how two variables vary together.\n2. So the answer stands: It gives probabilities of value pairs, capturing how two variables vary together."}
{"id": "mat_0575", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co nutí nová prvočísla?", "choices": {"A": "Euklidův argument", "B": "velké počítače", "C": "dlouhé seznamy", "D": "hlasitá tvrzení"}, "answer": "Odpověď je Euklidův argument.", "steps": ["Součin plus jedna.", "Tedy: Euklidův argument.", "Vyber možnost A."], "text": "Answer: Euklidův argument\nSteps:\n1. Součin plus jedna.\n2. Tedy: Euklidův argument.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0576", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Co je binomická věta?", "choices": null, "answer": "Rozvíjí (a+b)^n do členů s kombinačními čísly.", "steps": ["Rozvíjí (a+b)^n do členů s kombinačními čísly.", "Odpověď tedy platí: Rozvíjí (a+b)^n do členů s kombinačními čísly."], "text": "Answer: Rozvíjí (a+b)^n do členů s kombinačními čísly.\nSteps:\n1. Rozvíjí (a+b)^n do členů s kombinačními čísly.\n2. Odpověď tedy platí: Rozvíjí (a+b)^n do členů s kombinačními čísly."}
{"id": "mat_0577", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 10?", "choices": {"A": "45", "B": "65", "C": "55", "D": "75"}, "answer": "The sum is 55.", "steps": ["10×11/2.", "Sum = 55.", "Select option C."], "text": "Answer: 55\nSteps:\n1. 10×11/2.\n2. Sum = 55.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0578", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 6²)/(x − 6) as x → 6.", "choices": {"A": "10", "B": "14", "C": "12", "D": "16"}, "answer": "The limit is 12.", "steps": ["Factor (x−6)(x+6).", "Cancel, get 2×6 = 12.", "Select option C."], "text": "Answer: 12\nSteps:\n1. Factor (x−6)(x+6).\n2. Cancel, get 2×6 = 12.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0579", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[13,1],[2,14]] step by step.", "choices": null, "answer": "Determinant: 180.", "steps": ["ad − bc.", "13×14 − 1×2.", "= 180."], "text": "Answer: Determinant: 180.\nSteps:\n1. ad − bc.\n2. 13×14 − 1×2.\n3. = 180."}
{"id": "mat_0580", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 12 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 12/16.", "steps": ["Red: 12.", "Total: 16.", "P = 12/16."], "text": "Answer: P(red) = 12/16.\nSteps:\n1. Red: 12.\n2. Total: 16.\n3. P = 12/16."}
{"id": "mat_0581", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why Mersenne primes?", "choices": null, "answer": "Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers.", "steps": ["Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers.", "So the answer stands: Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers."], "text": "Answer: Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers.\nSteps:\n1. Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers.\n2. So the answer stands: Numbers 2^p-1 yield huge primes, driving record prime hunts and perfect numbers."}
{"id": "mat_0582", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(15,2).", "choices": null, "answer": "C(15,2) = 105.", "steps": ["Formula n(n−1)/2.", "15×14/2.", "= 105."], "text": "Answer: C(15,2) = 105.\nSteps:\n1. Formula n(n−1)/2.\n2. 15×14/2.\n3. = 105."}
{"id": "mat_0583", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is modus ponens?", "choices": null, "answer": "Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning.", "steps": ["Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning.", "So the answer stands: Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning."], "text": "Answer: Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning.\nSteps:\n1. Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning.\n2. So the answer stands: Modus ponens infers Q from P and P-implies-Q, and it is the basic step of deductive reasoning."}
{"id": "mat_0584", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why separate variables?", "choices": null, "answer": "Separation splits dy/dx equations into integrable halves, solving many first-order equations.", "steps": ["Separation splits dy/dx equations into integrable halves, solving many first-order equations.", "So the answer stands: Separation splits dy/dx equations into integrable halves, solving many first-order equations."], "text": "Answer: Separation splits dy/dx equations into integrable halves, solving many first-order equations.\nSteps:\n1. Separation splits dy/dx equations into integrable halves, solving many first-order equations.\n2. So the answer stands: Separation splits dy/dx equations into integrable halves, solving many first-order equations."}
{"id": "mat_0585", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej (3,2) · (3,5) krok za krokem.", "choices": null, "answer": "Skalární součin: 19.", "steps": ["Násob složky.", "3×3 + 2×5.", "= 19."], "text": "Answer: Skalární součin: 19.\nSteps:\n1. Násob složky.\n2. 3×3 + 2×5.\n3. = 19."}
{"id": "mat_0586", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Sáček má 4 červených a 4 modré kuličky. Ukaž P(červená).", "choices": null, "answer": "P(červená) = 4/8.", "steps": ["Červené: 4.", "Celkem: 8.", "P = 4/8."], "text": "Answer: P(červená) = 4/8.\nSteps:\n1. Červené: 4.\n2. Celkem: 8.\n3. P = 4/8."}
{"id": "mat_0587", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Proč kongruence?", "choices": null, "answer": "Kongruence zachovávají součty a součiny, velké úlohy zmenšují.", "steps": ["Kongruence zachovávají součty a součiny, velké úlohy zmenšují.", "Odpověď tedy platí: Kongruence zachovávají součty a součiny, velké úlohy zmenšují."], "text": "Answer: Kongruence zachovávají součty a součiny, velké úlohy zmenšují.\nSteps:\n1. Kongruence zachovávají součty a součiny, velké úlohy zmenšují.\n2. Odpověď tedy platí: Kongruence zachovávají součty a součiny, velké úlohy zmenšují."}
{"id": "mat_0588", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is the multinomial?", "choices": null, "answer": "It extends nCr to many groups at once, dividing n! by each group factorial.", "steps": ["It extends nCr to many groups at once, dividing n! by each group factorial.", "So the answer stands: It extends nCr to many groups at once, dividing n! by each group factorial."], "text": "Answer: It extends nCr to many groups at once, dividing n! by each group factorial.\nSteps:\n1. It extends nCr to many groups at once, dividing n! by each group factorial.\n2. So the answer stands: It extends nCr to many groups at once, dividing n! by each group factorial."}
{"id": "mat_0589", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why avoid circular proofs?", "choices": null, "answer": "Circular proofs assume the conclusion, so they prove nothing even when every step looks valid.", "steps": ["Circular proofs assume the conclusion, so they prove nothing even when every step looks valid.", "So the answer stands: Circular proofs assume the conclusion, so they prove nothing even when every step looks valid."], "text": "Answer: Circular proofs assume the conclusion, so they prove nothing even when every step looks valid.\nSteps:\n1. Circular proofs assume the conclusion, so they prove nothing even when every step looks valid.\n2. So the answer stands: Circular proofs assume the conclusion, so they prove nothing even when every step looks valid."}
{"id": "mat_0590", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 13²)/(x − 13) as x → 13.", "choices": null, "answer": "Limit: 26.", "steps": ["Factor: (x−13)(x+13).", "Cancel (x−13).", "Plug 13: 13+13 = 26."], "text": "Answer: Limit: 26.\nSteps:\n1. Factor: (x−13)(x+13).\n2. Cancel (x−13).\n3. Plug 13: 13+13 = 26."}
{"id": "mat_0591", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why study linear maps?", "choices": null, "answer": "Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly.", "steps": ["Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly.", "So the answer stands: Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly."], "text": "Answer: Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly.\nSteps:\n1. Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly.\n2. So the answer stands: Linear maps preserve structure simply, modelling rotations, scalings, and projections exactly."}
{"id": "mat_0592", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 10-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/10", "B": "1/9", "C": "1/11", "D": "2/10"}, "answer": "The probability is 1/10.", "steps": ["1 face of 10.", "P = 1/10.", "Select option A."], "text": "Answer: 1/10\nSteps:\n1. 1 face of 10.\n2. P = 1/10.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0593", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(8, 6).", "choices": {"A": "22", "B": "26", "C": "24", "D": "28"}, "answer": "The lcm is 24.", "steps": ["8×6/gcd.", "lcm = 24.", "Select option C."], "text": "Answer: 24\nSteps:\n1. 8×6/gcd.\n2. lcm = 24.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0594", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(14,3).", "choices": null, "answer": "P(14,3) = 2184.", "steps": ["n(n−1)(n−2).", "14×13×12.", "= 2184."], "text": "Answer: P(14,3) = 2184.\nSteps:\n1. n(n−1)(n−2).\n2. 14×13×12.\n3. = 2184."}
{"id": "mat_0595", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej N² + N pro N = 3.", "choices": {"A": "10", "B": "14", "C": "12", "D": "16"}, "answer": "Hodnota je 12.", "steps": ["3² + 3.", "Hodnota = 12.", "Vyber možnost C."], "text": "Answer: 12\nSteps:\n1. 3² + 3.\n2. Hodnota = 12.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0596", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 4x² + 12x, compute f'(4) step by step.", "choices": null, "answer": "f'(4) = 44.", "steps": ["f'(x) = 8x + 12.", "Plug x = 4.", "8×4 + 12 = 44."], "text": "Answer: f'(4) = 44.\nSteps:\n1. f'(x) = 8x + 12.\n2. Plug x = 4.\n3. 8×4 + 12 = 44."}
{"id": "mat_0597", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (5,6) · (3,4).", "choices": {"A": "35", "B": "43", "C": "39", "D": "47"}, "answer": "The dot product is 39.", "steps": ["5×3 + 6×4.", "Product = 39.", "Select option C."], "text": "Answer: 39\nSteps:\n1. 5×3 + 6×4.\n2. Product = 39.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0598", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Hází se férovou 8-stěnnou kostkou. Ukaž P(jedno konkrétní číslo).", "choices": null, "answer": "P = 1/8.", "steps": ["Příznivé: 1.", "Možné: 8.", "P = 1/8."], "text": "Answer: P = 1/8.\nSteps:\n1. Příznivé: 1.\n2. Možné: 8.\n3. P = 1/8."}
{"id": "mat_0599", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(44, 66) step by step.", "choices": null, "answer": "gcd = 22.", "steps": ["Factor: 44=4×11, 66=6×11.", "Common: 11×gcd(4,6).", "= 11×2 = 22."], "text": "Answer: gcd = 22.\nSteps:\n1. Factor: 44=4×11, 66=6×11.\n2. Common: 11×gcd(4,6).\n3. = 11×2 = 22."}
{"id": "mat_0600", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why exponential formula?", "choices": null, "answer": "It builds labelled structures from components, exponentiating their generating series.", "steps": ["It builds labelled structures from components, exponentiating their generating series.", "So the answer stands: It builds labelled structures from components, exponentiating their generating series."], "text": "Answer: It builds labelled structures from components, exponentiating their generating series.\nSteps:\n1. It builds labelled structures from components, exponentiating their generating series.\n2. So the answer stands: It builds labelled structures from components, exponentiating their generating series."}
{"id": "mat_0601", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is strong induction?", "choices": null, "answer": "Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors.", "steps": ["Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors.", "So the answer stands: Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors."], "text": "Answer: Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors.\nSteps:\n1. Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors.\n2. So the answer stands: Strong induction assumes all earlier cases to prove the next, which suits claims needing many predecessors."}
{"id": "mat_0602", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: what tests series?", "choices": {"A": "Coin toss", "B": "Mood ring", "C": "Dice roll", "D": "Ratio test"}, "answer": "The answer is Ratio test.", "steps": ["Below one converges.", "So: Ratio test.", "Select option D."], "text": "Answer: Ratio test\nSteps:\n1. Below one converges.\n2. So: Ratio test.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0603", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[14,0],[0,15]].", "choices": null, "answer": "Trace: 29.", "steps": ["Sum diagonal.", "14 + 15.", "= 29."], "text": "Answer: Trace: 29.\nSteps:\n1. Sum diagonal.\n2. 14 + 15.\n3. = 29."}
{"id": "mat_0604", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why marginalize?", "choices": null, "answer": "Summing out variables isolates one distribution, focusing analysis where needed.", "steps": ["Summing out variables isolates one distribution, focusing analysis where needed.", "So the answer stands: Summing out variables isolates one distribution, focusing analysis where needed."], "text": "Answer: Summing out variables isolates one distribution, focusing analysis where needed.\nSteps:\n1. Summing out variables isolates one distribution, focusing analysis where needed.\n2. So the answer stands: Summing out variables isolates one distribution, focusing analysis where needed."}
{"id": "mat_0605", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what counts coprimes?", "choices": {"A": "Lucky guess", "B": "Coin toss", "C": "Dice roll", "D": "Totient phi"}, "answer": "The answer is Totient phi.", "steps": ["Below n, coprime.", "So: Totient phi.", "Select option D."], "text": "Answer: Totient phi\nSteps:\n1. Below n, coprime.\n2. So: Totient phi.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0606", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is Burnside's lemma?", "choices": null, "answer": "It counts orbits under symmetry by averaging fixed points over the group.", "steps": ["It counts orbits under symmetry by averaging fixed points over the group.", "So the answer stands: It counts orbits under symmetry by averaging fixed points over the group."], "text": "Answer: It counts orbits under symmetry by averaging fixed points over the group.\nSteps:\n1. It counts orbits under symmetry by averaging fixed points over the group.\n2. So the answer stands: It counts orbits under symmetry by averaging fixed points over the group."}
{"id": "mat_0607", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why prove divisibility rules?", "choices": null, "answer": "Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure.", "steps": ["Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure.", "So the answer stands: Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure."], "text": "Answer: Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure.\nSteps:\n1. Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure.\n2. So the answer stands: Divisibility proofs reveal why digit tricks work, grounding mental arithmetic in modular structure."}
{"id": "mat_0608", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 14.", "choices": null, "answer": "Integral: 2744.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 14.", "14³ − 0 = 2744."], "text": "Answer: Integral: 2744.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 14.\n3. 14³ − 0 = 2744."}
{"id": "mat_0609", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (13,12) · (3,5) step by step.", "choices": null, "answer": "Dot product: 99.", "steps": ["Multiply components.", "13×3 + 12×5.", "= 99."], "text": "Answer: Dot product: 99.\nSteps:\n1. Multiply components.\n2. 13×3 + 12×5.\n3. = 99."}
{"id": "mat_0610", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 16-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/16.", "steps": ["Favourable: 1.", "Possible: 16.", "P = 1/16."], "text": "Answer: P = 1/16.\nSteps:\n1. Favourable: 1.\n2. Possible: 16.\n3. P = 1/16."}
{"id": "mat_0611", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (149) mod 7.", "choices": null, "answer": "Remainder: 2.", "steps": ["Divide 149 by 7.", "7×21 = 147.", "149 − 147 = 2."], "text": "Answer: Remainder: 2.\nSteps:\n1. Divide 149 by 7.\n2. 7×21 = 147.\n3. 149 − 147 = 2."}
{"id": "mat_0612", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet C(7,3).", "choices": null, "answer": "C(7,3) = 35.", "steps": ["n(n−1)(n−2)/6.", "7×6×5/6.", "= 35."], "text": "Answer: C(7,3) = 35.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 7×6×5/6.\n3. = 35."}
{"id": "mat_0613", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is the pigeonhole principle?", "choices": null, "answer": "If more items than boxes exist, some box holds two, and this counting fact proves surprising claims.", "steps": ["If more items than boxes exist, some box holds two, and this counting fact proves surprising claims.", "So the answer stands: If more items than boxes exist, some box holds two, and this counting fact proves surprising claims."], "text": "Answer: If more items than boxes exist, some box holds two, and this counting fact proves surprising claims.\nSteps:\n1. If more items than boxes exist, some box holds two, and this counting fact proves surprising claims.\n2. So the answer stands: If more items than boxes exist, some box holds two, and this counting fact proves surprising claims."}
{"id": "mat_0614", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Pro f(x) = 2x² + 3x vypočítej f'(2).", "choices": {"A": "9", "B": "11", "C": "13", "D": "15"}, "answer": "Derivace je 11.", "steps": ["f'(x) = 4x + 3.", "V 2: 11.", "Vyber možnost B."], "text": "Answer: 11\nSteps:\n1. f'(x) = 4x + 3.\n2. V 2: 11.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0615", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is the kernel?", "choices": null, "answer": "The kernel is the null space of a map, capturing exactly what information is lost.", "steps": ["The kernel is the null space of a map, capturing exactly what information is lost.", "So the answer stands: The kernel is the null space of a map, capturing exactly what information is lost."], "text": "Answer: The kernel is the null space of a map, capturing exactly what information is lost.\nSteps:\n1. The kernel is the null space of a map, capturing exactly what information is lost.\n2. So the answer stands: The kernel is the null space of a map, capturing exactly what information is lost."}
{"id": "mat_0616", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Co je zákon velkých čísel?", "choices": null, "answer": "Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů.", "steps": ["Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů.", "Odpověď tedy platí: Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů."], "text": "Answer: Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů.\nSteps:\n1. Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů.\n2. Odpověď tedy platí: Výběrové průměry se blíží střední hodnotě s rostoucím počtem pokusů."}
{"id": "mat_0617", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(14, 8) step by step.", "choices": null, "answer": "lcm = 56.", "steps": ["gcd(14,8) = 2.", "14×8/2.", "= 56."], "text": "Answer: lcm = 56.\nSteps:\n1. gcd(14,8) = 2.\n2. 14×8/2.\n3. = 56."}
{"id": "mat_0618", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what expands (a+b)^n?", "choices": {"A": "Wild guess", "B": "Long division", "C": "Blind hope", "D": "Binomial theorem"}, "answer": "The answer is Binomial theorem.", "steps": ["nCr-weighted terms.", "So: Binomial theorem.", "Select option D."], "text": "Answer: Binomial theorem\nSteps:\n1. nCr-weighted terms.\n2. So: Binomial theorem.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0619", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 17.", "choices": null, "answer": "Sum: 153.", "steps": ["Pair 1+17, 2+16, ...", "Use 17×18/2.", "= 153."], "text": "Answer: Sum: 153.\nSteps:\n1. Pair 1+17, 2+16, ...\n2. Use 17×18/2.\n3. = 153."}
{"id": "mat_0620", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 14²)/(x − 14) as x → 14.", "choices": null, "answer": "Limit: 28.", "steps": ["Factor: (x−14)(x+14).", "Cancel (x−14).", "Plug 14: 14+14 = 28."], "text": "Answer: Limit: 28.\nSteps:\n1. Factor: (x−14)(x+14).\n2. Cancel (x−14).\n3. Plug 14: 14+14 = 28."}
{"id": "mat_0621", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej det [[4,1],[2,5]] krok za krokem.", "choices": null, "answer": "Determinant: 18.", "steps": ["ad − bc.", "4×5 − 1×2.", "= 18."], "text": "Answer: Determinant: 18.\nSteps:\n1. ad − bc.\n2. 4×5 − 1×2.\n3. = 18."}
{"id": "mat_0622", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..8}.", "choices": {"A": "5.5", "B": "4.5", "C": "3.5", "D": "6.5"}, "answer": "The expected value is 4.5.", "steps": ["(8+1)/2.", "E = 4.5.", "Select option B."], "text": "Answer: 4.5\nSteps:\n1. (8+1)/2.\n2. E = 4.5.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0623", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(48, 72) step by step.", "choices": null, "answer": "gcd = 24.", "steps": ["Factor: 48=4×12, 72=6×12.", "Common: 12×gcd(4,6).", "= 12×2 = 24."], "text": "Answer: gcd = 24.\nSteps:\n1. Factor: 48=4×12, 72=6×12.\n2. Common: 12×gcd(4,6).\n3. = 12×2 = 24."}
{"id": "mat_0624", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(16,3).", "choices": null, "answer": "C(16,3) = 560.", "steps": ["n(n−1)(n−2)/6.", "16×15×14/6.", "= 560."], "text": "Answer: C(16,3) = 560.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 16×15×14/6.\n3. = 560."}
{"id": "mat_0625", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 13 by computing its half.", "choices": null, "answer": "Half: 91, so 182 is even.", "steps": ["13² + 13 = 182.", "182 / 2 = 91.", "Integer, so even."], "text": "Answer: Half: 91, so 182 is even.\nSteps:\n1. 13² + 13 = 182.\n2. 182 / 2 = 91.\n3. Integer, so even."}
{"id": "mat_0626", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej integrál 3x² od 0 do 3.", "choices": {"A": "24", "B": "30", "C": "27", "D": "33"}, "answer": "Integrál je 27.", "steps": ["Primitivní x³.", "3³ = 27.", "Vyber možnost C."], "text": "Answer: 27\nSteps:\n1. Primitivní x³.\n2. 3³ = 27.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0627", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what gives coordinates?", "choices": {"A": "A guess", "B": "A hope", "C": "A dream", "D": "A basis"}, "answer": "The answer is A basis.", "steps": ["Independent spanning set.", "So: A basis.", "Select option D."], "text": "Answer: A basis\nSteps:\n1. Independent spanning set.\n2. So: A basis.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0628", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Spočítej E rovnoměrného tahu z {1..7}.", "choices": null, "answer": "E = 4.", "steps": ["Průměr 1..7.", "(7+1)/2.", "E = 4."], "text": "Answer: E = 4.\nSteps:\n1. Průměr 1..7.\n2. (7+1)/2.\n3. E = 4."}
{"id": "mat_0629", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Co je malá Fermatova věta?", "choices": null, "answer": "Pro prvočíslo p je a^(p-1) rovno 1 mod p.", "steps": ["Pro prvočíslo p je a^(p-1) rovno 1 mod p.", "Odpověď tedy platí: Pro prvočíslo p je a^(p-1) rovno 1 mod p."], "text": "Answer: Pro prvočíslo p je a^(p-1) rovno 1 mod p.\nSteps:\n1. Pro prvočíslo p je a^(p-1) rovno 1 mod p.\n2. Odpověď tedy platí: Pro prvočíslo p je a^(p-1) rovno 1 mod p."}
{"id": "mat_0630", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why Pólya counting?", "choices": null, "answer": "It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly.", "steps": ["It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly.", "So the answer stands: It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly."], "text": "Answer: It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly.\nSteps:\n1. It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly.\n2. So the answer stands: It counts colorings up to symmetry via cycle index, generalizing Burnside elegantly."}
{"id": "mat_0631", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 14 boolean variables. How many rows?", "choices": null, "answer": "Rows: 16384.", "steps": ["2 choices each.", "2^14 = 16384."], "text": "Answer: Rows: 16384.\nSteps:\n1. 2 choices each.\n2. 2^14 = 16384."}
{"id": "mat_0632", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Choose: what guarantees matched rates?", "choices": {"A": "Mean value theorem", "B": "Lucky guess", "C": "Long proof", "D": "Big book"}, "answer": "The answer is Mean value theorem.", "steps": ["Some point matches.", "So: Mean value theorem.", "Select option A."], "text": "Answer: Mean value theorem\nSteps:\n1. Some point matches.\n2. So: Mean value theorem.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0633", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[14,1],[2,15]] step by step.", "choices": null, "answer": "Determinant: 208.", "steps": ["ad − bc.", "14×15 − 1×2.", "= 208."], "text": "Answer: Determinant: 208.\nSteps:\n1. ad − bc.\n2. 14×15 − 1×2.\n3. = 208."}
{"id": "mat_0634", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..15}.", "choices": null, "answer": "E = 8.", "steps": ["Mean of 1..15.", "(15+1)/2.", "E = 8."], "text": "Answer: E = 8.\nSteps:\n1. Mean of 1..15.\n2. (15+1)/2.\n3. E = 8."}
{"id": "mat_0635", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Choose: what sums 1/n^s?", "choices": {"A": "Riemann zeta", "B": "Wild series", "C": "Blind sum", "D": "Vague total"}, "answer": "The answer is Riemann zeta.", "steps": ["Zeros rule primes.", "So: Riemann zeta.", "Select option A."], "text": "Answer: Riemann zeta\nSteps:\n1. Zeros rule primes.\n2. So: Riemann zeta.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0636", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is a Latin square?", "choices": null, "answer": "Rows and columns each permute symbols, the combinatorics behind sudoku puzzles.", "steps": ["Rows and columns each permute symbols, the combinatorics behind sudoku puzzles.", "So the answer stands: Rows and columns each permute symbols, the combinatorics behind sudoku puzzles."], "text": "Answer: Rows and columns each permute symbols, the combinatorics behind sudoku puzzles.\nSteps:\n1. Rows and columns each permute symbols, the combinatorics behind sudoku puzzles.\n2. So the answer stands: Rows and columns each permute symbols, the combinatorics behind sudoku puzzles."}
{"id": "mat_0637", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Proč se vyhnout kruhovým důkazům?", "choices": null, "answer": "Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic.", "steps": ["Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic.", "Odpověď tedy platí: Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic."], "text": "Answer: Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic.\nSteps:\n1. Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic.\n2. Odpověď tedy platí: Kruhový důkaz předpokládá závěr, a nedokazuje tedy nic."}
{"id": "mat_0638", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 2x² + 13x, compute f'(2) step by step.", "choices": null, "answer": "f'(2) = 21.", "steps": ["f'(x) = 4x + 13.", "Plug x = 2.", "4×2 + 13 = 21."], "text": "Answer: f'(2) = 21.\nSteps:\n1. f'(x) = 4x + 13.\n2. Plug x = 2.\n3. 4×2 + 13 = 21."}
{"id": "mat_0639", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[15,0],[0,16]].", "choices": null, "answer": "Trace: 31.", "steps": ["Sum diagonal.", "15 + 16.", "= 31."], "text": "Answer: Trace: 31.\nSteps:\n1. Sum diagonal.\n2. 15 + 16.\n3. = 31."}
{"id": "mat_0640", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 13 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 13/17.", "steps": ["Red: 13.", "Total: 17.", "P = 13/17."], "text": "Answer: P(red) = 13/17.\nSteps:\n1. Red: 13.\n2. Total: 17.\n3. P = 13/17."}
{"id": "mat_0641", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (160) mod 7.", "choices": null, "answer": "Remainder: 6.", "steps": ["Divide 160 by 7.", "7×22 = 154.", "160 − 154 = 6."], "text": "Answer: Remainder: 6.\nSteps:\n1. Divide 160 by 7.\n2. 7×22 = 154.\n3. 160 − 154 = 6."}
{"id": "mat_0642", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet C(7,2).", "choices": null, "answer": "C(7,2) = 21.", "steps": ["Vzorec n(n−1)/2.", "7×6/2.", "= 21."], "text": "Answer: C(7,2) = 21.\nSteps:\n1. Vzorec n(n−1)/2.\n2. 7×6/2.\n3. = 21."}
{"id": "mat_0643", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 8.", "choices": {"A": "70", "B": "74", "C": "76", "D": "72"}, "answer": "The value is 72.", "steps": ["8² + 8.", "Value = 72.", "Select option D."], "text": "Answer: 72\nSteps:\n1. 8² + 8.\n2. Value = 72.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0644", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Vyčísli integrál 3x² od 0 do 5.", "choices": null, "answer": "Integrál: 125.", "steps": ["Primitivní: x³.", "Vyčísli 0 až 5.", "5³ − 0 = 125."], "text": "Answer: Integrál: 125.\nSteps:\n1. Primitivní: x³.\n2. Vyčísli 0 až 5.\n3. 5³ − 0 = 125."}
{"id": "mat_0645", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (14,13) · (3,5) step by step.", "choices": null, "answer": "Dot product: 107.", "steps": ["Multiply components.", "14×3 + 13×5.", "= 107."], "text": "Answer: Dot product: 107.\nSteps:\n1. Multiply components.\n2. 14×3 + 13×5.\n3. = 107."}
{"id": "mat_0646", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 17-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/17.", "steps": ["Favourable: 1.", "Possible: 17.", "P = 1/17."], "text": "Answer: P = 1/17.\nSteps:\n1. Favourable: 1.\n2. Possible: 17.\n3. P = 1/17."}
{"id": "mat_0647", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(42, 63).", "choices": {"A": "18", "B": "21", "C": "24", "D": "27"}, "answer": "The gcd is 21.", "steps": ["Factor 3 out twice.", "gcd = 21.", "Select option B."], "text": "Answer: 21\nSteps:\n1. Factor 3 out twice.\n2. gcd = 21.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0648", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(16,2).", "choices": null, "answer": "C(16,2) = 120.", "steps": ["Formula n(n−1)/2.", "16×15/2.", "= 120."], "text": "Answer: C(16,2) = 120.\nSteps:\n1. Formula n(n−1)/2.\n2. 16×15/2.\n3. = 120."}
{"id": "mat_0649", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 8 variables have?", "choices": {"A": "254", "B": "258", "C": "256", "D": "260"}, "answer": "The table has 256 rows.", "steps": ["2^8 rows.", "Total: 256.", "Select option C."], "text": "Answer: 256\nSteps:\n1. 2^8 rows.\n2. Total: 256.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0650", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is Euler's method?", "choices": null, "answer": "It steps along tangent slopes to approximate solutions numerically when formulas fail.", "steps": ["It steps along tangent slopes to approximate solutions numerically when formulas fail.", "So the answer stands: It steps along tangent slopes to approximate solutions numerically when formulas fail."], "text": "Answer: It steps along tangent slopes to approximate solutions numerically when formulas fail.\nSteps:\n1. It steps along tangent slopes to approximate solutions numerically when formulas fail.\n2. So the answer stands: It steps along tangent slopes to approximate solutions numerically when formulas fail."}
{"id": "mat_0651", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[15,1],[2,16]] step by step.", "choices": null, "answer": "Determinant: 238.", "steps": ["ad − bc.", "15×16 − 1×2.", "= 238."], "text": "Answer: Determinant: 238.\nSteps:\n1. ad − bc.\n2. 15×16 − 1×2.\n3. = 238."}
{"id": "mat_0652", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is conditional expectation?", "choices": null, "answer": "It averages one variable given another's value, the basis of regression prediction.", "steps": ["It averages one variable given another's value, the basis of regression prediction.", "So the answer stands: It averages one variable given another's value, the basis of regression prediction."], "text": "Answer: It averages one variable given another's value, the basis of regression prediction.\nSteps:\n1. It averages one variable given another's value, the basis of regression prediction.\n2. So the answer stands: It averages one variable given another's value, the basis of regression prediction."}
{"id": "mat_0653", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(15, 8) step by step.", "choices": null, "answer": "lcm = 120.", "steps": ["gcd(15,8) = 1.", "15×8/1.", "= 120."], "text": "Answer: lcm = 120.\nSteps:\n1. gcd(15,8) = 1.\n2. 15×8/1.\n3. = 120."}
{"id": "mat_0654", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why block designs?", "choices": null, "answer": "They arrange balanced experiments, with each pair meeting exactly lambda times.", "steps": ["They arrange balanced experiments, with each pair meeting exactly lambda times.", "So the answer stands: They arrange balanced experiments, with each pair meeting exactly lambda times."], "text": "Answer: They arrange balanced experiments, with each pair meeting exactly lambda times.\nSteps:\n1. They arrange balanced experiments, with each pair meeting exactly lambda times.\n2. So the answer stands: They arrange balanced experiments, with each pair meeting exactly lambda times."}
{"id": "mat_0655", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 11?", "choices": {"A": "55", "B": "77", "C": "88", "D": "66"}, "answer": "The sum is 66.", "steps": ["11×12/2.", "Sum = 66.", "Select option D."], "text": "Answer: 66\nSteps:\n1. 11×12/2.\n2. Sum = 66.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0656", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why study convergence?", "choices": null, "answer": "Convergence decides whether infinite processes settle, separating meaningful sums from nonsense.", "steps": ["Convergence decides whether infinite processes settle, separating meaningful sums from nonsense.", "So the answer stands: Convergence decides whether infinite processes settle, separating meaningful sums from nonsense."], "text": "Answer: Convergence decides whether infinite processes settle, separating meaningful sums from nonsense.\nSteps:\n1. Convergence decides whether infinite processes settle, separating meaningful sums from nonsense.\n2. So the answer stands: Convergence decides whether infinite processes settle, separating meaningful sums from nonsense."}
{"id": "mat_0657", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[16,0],[0,17]].", "choices": null, "answer": "Trace: 33.", "steps": ["Sum diagonal.", "16 + 17.", "= 33."], "text": "Answer: Trace: 33.\nSteps:\n1. Sum diagonal.\n2. 16 + 17.\n3. = 33."}
{"id": "mat_0658", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..16}.", "choices": null, "answer": "E = 8.5.", "steps": ["Mean of 1..16.", "(16+1)/2.", "E = 8.5."], "text": "Answer: E = 8.5.\nSteps:\n1. Mean of 1..16.\n2. (16+1)/2.\n3. E = 8.5."}
{"id": "mat_0659", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is the Goldbach conjecture?", "choices": null, "answer": "It claims every even number above 2 is a prime sum, verified far but unproven.", "steps": ["It claims every even number above 2 is a prime sum, verified far but unproven.", "So the answer stands: It claims every even number above 2 is a prime sum, verified far but unproven."], "text": "Answer: It claims every even number above 2 is a prime sum, verified far but unproven.\nSteps:\n1. It claims every even number above 2 is a prime sum, verified far but unproven.\n2. So the answer stands: It claims every even number above 2 is a prime sum, verified far but unproven."}
{"id": "mat_0660", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet P(6,3).", "choices": null, "answer": "P(6,3) = 120.", "steps": ["n(n−1)(n−2).", "6×5×4.", "= 120."], "text": "Answer: P(6,3) = 120.\nSteps:\n1. n(n−1)(n−2).\n2. 6×5×4.\n3. = 120."}
{"id": "mat_0661", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why use parity arguments?", "choices": null, "answer": "Parity splits numbers into even and odd, and tracking it often forces contradictions quickly.", "steps": ["Parity splits numbers into even and odd, and tracking it often forces contradictions quickly.", "So the answer stands: Parity splits numbers into even and odd, and tracking it often forces contradictions quickly."], "text": "Answer: Parity splits numbers into even and odd, and tracking it often forces contradictions quickly.\nSteps:\n1. Parity splits numbers into even and odd, and tracking it often forces contradictions quickly.\n2. So the answer stands: Parity splits numbers into even and odd, and tracking it often forces contradictions quickly."}
{"id": "mat_0662", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 3x² + 7x, find f'(2).", "choices": {"A": "17", "B": "19", "C": "21", "D": "23"}, "answer": "The derivative is 19.", "steps": ["f'(x) = 6x + 7.", "At 2: 19.", "Select option B."], "text": "Answer: 19\nSteps:\n1. f'(x) = 6x + 7.\n2. At 2: 19.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0663", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej stopu matice [[5,0],[0,6]].", "choices": null, "answer": "Stopa: 11.", "steps": ["Sečti diagonálu.", "5 + 6.", "= 11."], "text": "Answer: Stopa: 11.\nSteps:\n1. Sečti diagonálu.\n2. 5 + 6.\n3. = 11."}
{"id": "mat_0664", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 14 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 14/18.", "steps": ["Red: 14.", "Total: 18.", "P = 14/18."], "text": "Answer: P(red) = 14/18.\nSteps:\n1. Red: 14.\n2. Total: 18.\n3. P = 14/18."}
{"id": "mat_0665", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(52, 78) step by step.", "choices": null, "answer": "gcd = 26.", "steps": ["Factor: 52=4×13, 78=6×13.", "Common: 13×gcd(4,6).", "= 13×2 = 26."], "text": "Answer: gcd = 26.\nSteps:\n1. Factor: 52=4×13, 78=6×13.\n2. Common: 13×gcd(4,6).\n3. = 13×2 = 26."}
{"id": "mat_0666", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet C(8,3).", "choices": null, "answer": "C(8,3) = 56.", "steps": ["n(n−1)(n−2)/6.", "8×7×6/6.", "= 56."], "text": "Answer: C(8,3) = 56.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 8×7×6/6.\n3. = 56."}
{"id": "mat_0667", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 18.", "choices": null, "answer": "Sum: 171.", "steps": ["Pair 1+18, 2+17, ...", "Use 18×19/2.", "= 171."], "text": "Answer: Sum: 171.\nSteps:\n1. Pair 1+18, 2+17, ...\n2. Use 18×19/2.\n3. = 171."}
{"id": "mat_0668", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "concept", "question": "Co je l'Hôpitalovo pravidlo?", "choices": null, "answer": "Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele.", "steps": ["Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele.", "Odpověď tedy platí: Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele."], "text": "Answer: Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele.\nSteps:\n1. Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele.\n2. Odpověď tedy platí: Řeší neurčitosti 0/0 derivováním čitatele i jmenovatele."}
{"id": "mat_0669", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (15,14) · (3,5) step by step.", "choices": null, "answer": "Dot product: 115.", "steps": ["Multiply components.", "15×3 + 14×5.", "= 115."], "text": "Answer: Dot product: 115.\nSteps:\n1. Multiply components.\n2. 15×3 + 14×5.\n3. = 115."}
{"id": "mat_0670", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 6 red and 13 blue marbles. What is P(red)?", "choices": {"A": "13/19", "B": "6/20", "C": "6/19", "D": "7/19"}, "answer": "P(red) is 6/19.", "steps": ["6 of 19.", "P = 6/19.", "Select option C."], "text": "Answer: 6/19\nSteps:\n1. 6 of 19.\n2. P = 6/19.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0671", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (135) mod 9.", "choices": {"A": "1", "B": "2", "C": "0", "D": "3"}, "answer": "The remainder is 0.", "steps": ["135 ÷ 9.", "Remainder 0.", "Select option C."], "text": "Answer: 0\nSteps:\n1. 135 ÷ 9.\n2. Remainder 0.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0672", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co páruje množiny stejně?", "choices": {"A": "divoká mapa", "B": "bijekce", "C": "slepé hádání", "D": "mlhavá naděje"}, "answer": "Odpověď je bijekce.", "steps": ["Inverze dokazuje velikost.", "Tedy: bijekce.", "Vyber možnost B."], "text": "Answer: bijekce\nSteps:\n1. Inverze dokazuje velikost.\n2. Tedy: bijekce.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0673", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 9.", "choices": {"A": "90", "B": "88", "C": "92", "D": "94"}, "answer": "The value is 90.", "steps": ["9² + 9.", "Value = 90.", "Select option A."], "text": "Answer: 90\nSteps:\n1. 9² + 9.\n2. Value = 90.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0674", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is an improper integral?", "choices": null, "answer": "It integrates over infinite bounds or singularities via limits, extending area to the unbounded.", "steps": ["It integrates over infinite bounds or singularities via limits, extending area to the unbounded.", "So the answer stands: It integrates over infinite bounds or singularities via limits, extending area to the unbounded."], "text": "Answer: It integrates over infinite bounds or singularities via limits, extending area to the unbounded.\nSteps:\n1. It integrates over infinite bounds or singularities via limits, extending area to the unbounded.\n2. So the answer stands: It integrates over infinite bounds or singularities via limits, extending area to the unbounded."}
{"id": "mat_0675", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Proč vlastní čísla?", "choices": null, "answer": "Vlastní čísla odhalují směry natažení zobrazení.", "steps": ["Vlastní čísla odhalují směry natažení zobrazení.", "Odpověď tedy platí: Vlastní čísla odhalují směry natažení zobrazení."], "text": "Answer: Vlastní čísla odhalují směry natažení zobrazení.\nSteps:\n1. Vlastní čísla odhalují směry natažení zobrazení.\n2. Odpověď tedy platí: Vlastní čísla odhalují směry natažení zobrazení."}
{"id": "mat_0676", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what averages long-run?", "choices": {"A": "First outcome", "B": "Last digit", "C": "Wild guess", "D": "Expected value"}, "answer": "The answer is Expected value.", "steps": ["Weighted by chance.", "So: Expected value.", "Select option D."], "text": "Answer: Expected value\nSteps:\n1. Weighted by chance.\n2. So: Expected value.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0677", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why twin primes?", "choices": null, "answer": "Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream.", "steps": ["Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream.", "So the answer stands: Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream."], "text": "Answer: Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream.\nSteps:\n1. Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream.\n2. So the answer stands: Pairs differing by 2 may be infinite, and bounded gaps progress nears the dream."}
{"id": "mat_0678", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Choose: what adds regardless?", "choices": {"A": "Linearity of expectation", "B": "Wild hope", "C": "Blind sum", "D": "Vague wish"}, "answer": "The answer is Linearity of expectation.", "steps": ["Dependence ignored.", "So: Linearity of expectation.", "Select option A."], "text": "Answer: Linearity of expectation\nSteps:\n1. Dependence ignored.\n2. So: Linearity of expectation.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0679", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 14 by computing its half.", "choices": null, "answer": "Half: 105, so 210 is even.", "steps": ["14² + 14 = 210.", "210 / 2 = 105.", "Integer, so even."], "text": "Answer: Half: 105, so 210 is even.\nSteps:\n1. 14² + 14 = 210.\n2. 210 / 2 = 105.\n3. Integer, so even."}
{"id": "mat_0680", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Vyčísli limitu (x² − 5²)/(x − 5) pro x → 5.", "choices": null, "answer": "Limita: 10.", "steps": ["Rozlož: (x−5)(x+5).", "Pokrať (x−5).", "Dosaď 5: 5+5 = 10."], "text": "Answer: Limita: 10.\nSteps:\n1. Rozlož: (x−5)(x+5).\n2. Pokrať (x−5).\n3. Dosaď 5: 5+5 = 10."}
{"id": "mat_0681", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[16,1],[2,17]] step by step.", "choices": null, "answer": "Determinant: 270.", "steps": ["ad − bc.", "16×17 − 1×2.", "= 270."], "text": "Answer: Determinant: 270.\nSteps:\n1. ad − bc.\n2. 16×17 − 1×2.\n3. = 270."}
{"id": "mat_0682", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 18-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/18.", "steps": ["Favourable: 1.", "Possible: 18.", "P = 1/18."], "text": "Answer: P = 1/18.\nSteps:\n1. Favourable: 1.\n2. Possible: 18.\n3. P = 1/18."}
{"id": "mat_0683", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is the prime number theorem?", "choices": null, "answer": "Primes thin like n/log n, describing their large-scale density precisely.", "steps": ["Primes thin like n/log n, describing their large-scale density precisely.", "So the answer stands: Primes thin like n/log n, describing their large-scale density precisely."], "text": "Answer: Primes thin like n/log n, describing their large-scale density precisely.\nSteps:\n1. Primes thin like n/log n, describing their large-scale density precisely.\n2. So the answer stands: Primes thin like n/log n, describing their large-scale density precisely."}
{"id": "mat_0684", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(9,2).", "choices": {"A": "27", "B": "36", "C": "45", "D": "54"}, "answer": "C(9,2) is 36.", "steps": ["9×8/2.", "= 36.", "Select option B."], "text": "Answer: 36\nSteps:\n1. 9×8/2.\n2. = 36.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0685", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Ukaž, že N² + N je sudé pro N = 5, výpočtem poloviny.", "choices": null, "answer": "Polovina: 15, číslo 30 je sudé.", "steps": ["5² + 5 = 30.", "30 / 2 = 15.", "Celé číslo, tedy sudé."], "text": "Answer: Polovina: 15, číslo 30 je sudé.\nSteps:\n1. 5² + 5 = 30.\n2. 30 / 2 = 15.\n3. Celé číslo, tedy sudé."}
{"id": "mat_0686", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 15.", "choices": null, "answer": "Integral: 3375.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 15.", "15³ − 0 = 3375."], "text": "Answer: Integral: 3375.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 15.\n3. 15³ − 0 = 3375."}
{"id": "mat_0687", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co měří úhly?", "choices": {"A": "skalární součin", "B": "divoký odhad", "C": "slepý součet", "D": "slepá naděje"}, "answer": "Odpověď je skalární součin.", "steps": ["Nula znamená kolmost.", "Tedy: skalární součin.", "Vyber možnost A."], "text": "Answer: skalární součin\nSteps:\n1. Nula znamená kolmost.\n2. Tedy: skalární součin.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0688", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why regression?", "choices": null, "answer": "Regression fits lines minimizing squared error, predicting responses from predictors.", "steps": ["Regression fits lines minimizing squared error, predicting responses from predictors.", "So the answer stands: Regression fits lines minimizing squared error, predicting responses from predictors."], "text": "Answer: Regression fits lines minimizing squared error, predicting responses from predictors.\nSteps:\n1. Regression fits lines minimizing squared error, predicting responses from predictors.\n2. So the answer stands: Regression fits lines minimizing squared error, predicting responses from predictors."}
{"id": "mat_0689", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (171) mod 7.", "choices": null, "answer": "Remainder: 3.", "steps": ["Divide 171 by 7.", "7×24 = 168.", "171 − 168 = 3."], "text": "Answer: Remainder: 3.\nSteps:\n1. Divide 171 by 7.\n2. 7×24 = 168.\n3. 171 − 168 = 3."}
{"id": "mat_0690", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(15,3).", "choices": null, "answer": "P(15,3) = 2730.", "steps": ["n(n−1)(n−2).", "15×14×13.", "= 2730."], "text": "Answer: P(15,3) = 2730.\nSteps:\n1. n(n−1)(n−2).\n2. 15×14×13.\n3. = 2730."}
{"id": "mat_0691", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Co je De Morganův zákon?", "choices": null, "answer": "De Morganovy zákony převracejí a/nebo při negaci.", "steps": ["De Morganovy zákony převracejí a/nebo při negaci.", "Odpověď tedy platí: De Morganovy zákony převracejí a/nebo při negaci."], "text": "Answer: De Morganovy zákony převracejí a/nebo při negaci.\nSteps:\n1. De Morganovy zákony převracejí a/nebo při negaci.\n2. Odpověď tedy platí: De Morganovy zákony převracejí a/nebo při negaci."}
{"id": "mat_0692", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 7.", "choices": {"A": "336", "B": "350", "C": "343", "D": "357"}, "answer": "The integral is 343.", "steps": ["Antiderivative x³.", "7³ = 343.", "Select option C."], "text": "Answer: 343\nSteps:\n1. Antiderivative x³.\n2. 7³ = 343.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0693", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej (4,3) · (3,5) krok za krokem.", "choices": null, "answer": "Skalární součin: 27.", "steps": ["Násob složky.", "4×3 + 3×5.", "= 27."], "text": "Answer: Skalární součin: 27.\nSteps:\n1. Násob složky.\n2. 4×3 + 3×5.\n3. = 27."}
{"id": "mat_0694", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Choose: what has no memory?", "choices": {"A": "Independent trials", "B": "Old coins", "C": "Tired dice", "D": "Worn cards"}, "answer": "The answer is Independent trials.", "steps": ["Streaks mean nothing.", "So: Independent trials.", "Select option A."], "text": "Answer: Independent trials\nSteps:\n1. Streaks mean nothing.\n2. So: Independent trials.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0695", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(16, 8) step by step.", "choices": null, "answer": "lcm = 16.", "steps": ["gcd(16,8) = 8.", "16×8/8.", "= 16."], "text": "Answer: lcm = 16.\nSteps:\n1. gcd(16,8) = 8.\n2. 16×8/8.\n3. = 16."}
{"id": "mat_0696", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is extremal combinatorics?", "choices": null, "answer": "It finds maxima under forbidden patterns, like most edges without triangles.", "steps": ["It finds maxima under forbidden patterns, like most edges without triangles.", "So the answer stands: It finds maxima under forbidden patterns, like most edges without triangles."], "text": "Answer: It finds maxima under forbidden patterns, like most edges without triangles.\nSteps:\n1. It finds maxima under forbidden patterns, like most edges without triangles.\n2. So the answer stands: It finds maxima under forbidden patterns, like most edges without triangles."}
{"id": "mat_0697", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is an invariant?", "choices": null, "answer": "An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable.", "steps": ["An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable.", "So the answer stands: An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable."], "text": "Answer: An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable.\nSteps:\n1. An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable.\n2. So the answer stands: An invariant is a quantity unchanged by allowed moves, so differing invariants prove targets unreachable."}
{"id": "mat_0698", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 7²)/(x − 7) as x → 7.", "choices": {"A": "12", "B": "16", "C": "18", "D": "14"}, "answer": "The limit is 14.", "steps": ["Factor (x−7)(x+7).", "Cancel, get 2×7 = 14.", "Select option D."], "text": "Answer: 14\nSteps:\n1. Factor (x−7)(x+7).\n2. Cancel, get 2×7 = 14.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0699", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Choose: what splits any matrix?", "choices": {"A": "SVD", "B": "Lucky cut", "C": "Blind split", "D": "Wild guess"}, "answer": "The answer is SVD.", "steps": ["Rotations plus scalings.", "So: SVD.", "Select option A."], "text": "Answer: SVD\nSteps:\n1. Rotations plus scalings.\n2. So: SVD.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0700", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..17}.", "choices": null, "answer": "E = 9.", "steps": ["Mean of 1..17.", "(17+1)/2.", "E = 9."], "text": "Answer: E = 9.\nSteps:\n1. Mean of 1..17.\n2. (17+1)/2.\n3. E = 9."}
{"id": "mat_0701", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(9, 6).", "choices": {"A": "16", "B": "20", "C": "22", "D": "18"}, "answer": "The lcm is 18.", "steps": ["9×6/gcd.", "lcm = 18.", "Select option D."], "text": "Answer: 18\nSteps:\n1. 9×6/gcd.\n2. lcm = 18.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0702", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet C(8,2).", "choices": null, "answer": "C(8,2) = 28.", "steps": ["Vzorec n(n−1)/2.", "8×7/2.", "= 28."], "text": "Answer: C(8,2) = 28.\nSteps:\n1. Vzorec n(n−1)/2.\n2. 8×7/2.\n3. = 28."}
{"id": "mat_0703", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why formalize induction?", "choices": null, "answer": "Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning.", "steps": ["Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning.", "So the answer stands: Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning."], "text": "Answer: Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning.\nSteps:\n1. Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning.\n2. So the answer stands: Formal induction states base and step explicitly, catching hidden gaps in recursive reasoning."}
{"id": "mat_0704", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why parametrize curves?", "choices": null, "answer": "Parametrization traces curves by a parameter, simplifying motion and arc-length questions.", "steps": ["Parametrization traces curves by a parameter, simplifying motion and arc-length questions.", "So the answer stands: Parametrization traces curves by a parameter, simplifying motion and arc-length questions."], "text": "Answer: Parametrization traces curves by a parameter, simplifying motion and arc-length questions.\nSteps:\n1. Parametrization traces curves by a parameter, simplifying motion and arc-length questions.\n2. So the answer stands: Parametrization traces curves by a parameter, simplifying motion and arc-length questions."}
{"id": "mat_0705", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[17,0],[0,18]].", "choices": null, "answer": "Trace: 35.", "steps": ["Sum diagonal.", "17 + 18.", "= 35."], "text": "Answer: Trace: 35.\nSteps:\n1. Sum diagonal.\n2. 17 + 18.\n3. = 35."}
{"id": "mat_0706", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 11-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/10", "B": "1/11", "C": "1/12", "D": "2/11"}, "answer": "The probability is 1/11.", "steps": ["1 face of 11.", "P = 1/11.", "Select option B."], "text": "Answer: 1/11\nSteps:\n1. 1 face of 11.\n2. P = 1/11.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0707", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(56, 84) step by step.", "choices": null, "answer": "gcd = 28.", "steps": ["Factor: 56=4×14, 84=6×14.", "Common: 14×gcd(4,6).", "= 14×2 = 28."], "text": "Answer: gcd = 28.\nSteps:\n1. Factor: 56=4×14, 84=6×14.\n2. Common: 14×gcd(4,6).\n3. = 14×2 = 28."}
{"id": "mat_0708", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Proč metoda hvězdiček?", "choices": null, "answer": "Počítá rozdělení stejných předmětů do krabic.", "steps": ["Počítá rozdělení stejných předmětů do krabic.", "Odpověď tedy platí: Počítá rozdělení stejných předmětů do krabic."], "text": "Answer: Počítá rozdělení stejných předmětů do krabic.\nSteps:\n1. Počítá rozdělení stejných předmětů do krabic.\n2. Odpověď tedy platí: Počítá rozdělení stejných předmětů do krabic."}
{"id": "mat_0709", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 15 boolean variables. How many rows?", "choices": null, "answer": "Rows: 32768.", "steps": ["2 choices each.", "2^15 = 32768."], "text": "Answer: Rows: 32768.\nSteps:\n1. 2 choices each.\n2. 2^15 = 32768."}
{"id": "mat_0710", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is arc length?", "choices": null, "answer": "Arc length integrates speed along a curve, giving the exact distance travelled.", "steps": ["Arc length integrates speed along a curve, giving the exact distance travelled.", "So the answer stands: Arc length integrates speed along a curve, giving the exact distance travelled."], "text": "Answer: Arc length integrates speed along a curve, giving the exact distance travelled.\nSteps:\n1. Arc length integrates speed along a curve, giving the exact distance travelled.\n2. So the answer stands: Arc length integrates speed along a curve, giving the exact distance travelled."}
{"id": "mat_0711", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej det [[5,1],[2,6]] krok za krokem.", "choices": null, "answer": "Determinant: 28.", "steps": ["ad − bc.", "5×6 − 1×2.", "= 28."], "text": "Answer: Determinant: 28.\nSteps:\n1. ad − bc.\n2. 5×6 − 1×2.\n3. = 28."}
{"id": "mat_0712", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 15 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 15/19.", "steps": ["Red: 15.", "Total: 19.", "P = 15/19."], "text": "Answer: P(red) = 15/19.\nSteps:\n1. Red: 15.\n2. Total: 19.\n3. P = 15/19."}
{"id": "mat_0713", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why sieve of Eratosthenes?", "choices": null, "answer": "It crosses multiples to list primes fast, the classic prime-finding engine.", "steps": ["It crosses multiples to list primes fast, the classic prime-finding engine.", "So the answer stands: It crosses multiples to list primes fast, the classic prime-finding engine."], "text": "Answer: It crosses multiples to list primes fast, the classic prime-finding engine.\nSteps:\n1. It crosses multiples to list primes fast, the classic prime-finding engine.\n2. So the answer stands: It crosses multiples to list primes fast, the classic prime-finding engine."}
{"id": "mat_0714", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(17,3).", "choices": null, "answer": "C(17,3) = 680.", "steps": ["n(n−1)(n−2)/6.", "17×16×15/6.", "= 680."], "text": "Answer: C(17,3) = 680.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 17×16×15/6.\n3. = 680."}
{"id": "mat_0715", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is proof by exhaustion?", "choices": null, "answer": "Exhaustion checks every case by hand or computer, and it works when the case list is finite.", "steps": ["Exhaustion checks every case by hand or computer, and it works when the case list is finite.", "So the answer stands: Exhaustion checks every case by hand or computer, and it works when the case list is finite."], "text": "Answer: Exhaustion checks every case by hand or computer, and it works when the case list is finite.\nSteps:\n1. Exhaustion checks every case by hand or computer, and it works when the case list is finite.\n2. So the answer stands: Exhaustion checks every case by hand or computer, and it works when the case list is finite."}
{"id": "mat_0716", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why learn gradient?", "choices": null, "answer": "The gradient points to steepest ascent, guiding optimization and physics on surfaces.", "steps": ["The gradient points to steepest ascent, guiding optimization and physics on surfaces.", "So the answer stands: The gradient points to steepest ascent, guiding optimization and physics on surfaces."], "text": "Answer: The gradient points to steepest ascent, guiding optimization and physics on surfaces.\nSteps:\n1. The gradient points to steepest ascent, guiding optimization and physics on surfaces.\n2. So the answer stands: The gradient points to steepest ascent, guiding optimization and physics on surfaces."}
{"id": "mat_0717", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (16,15) · (3,5) step by step.", "choices": null, "answer": "Dot product: 123.", "steps": ["Multiply components.", "16×3 + 15×5.", "= 123."], "text": "Answer: Dot product: 123.\nSteps:\n1. Multiply components.\n2. 16×3 + 15×5.\n3. = 123."}
{"id": "mat_0718", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 19-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/19.", "steps": ["Favourable: 1.", "Possible: 19.", "P = 1/19."], "text": "Answer: P = 1/19.\nSteps:\n1. Favourable: 1.\n2. Possible: 19.\n3. P = 1/19."}
{"id": "mat_0719", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is trial division?", "choices": null, "answer": "It tests divisors up to sqrt(n), simple and sure for small primality checks.", "steps": ["It tests divisors up to sqrt(n), simple and sure for small primality checks.", "So the answer stands: It tests divisors up to sqrt(n), simple and sure for small primality checks."], "text": "Answer: It tests divisors up to sqrt(n), simple and sure for small primality checks.\nSteps:\n1. It tests divisors up to sqrt(n), simple and sure for small primality checks.\n2. So the answer stands: It tests divisors up to sqrt(n), simple and sure for small primality checks."}
{"id": "mat_0720", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why probabilistic method?", "choices": null, "answer": "Random choices prove existence, showing desired objects occur with positive chance.", "steps": ["Random choices prove existence, showing desired objects occur with positive chance.", "So the answer stands: Random choices prove existence, showing desired objects occur with positive chance."], "text": "Answer: Random choices prove existence, showing desired objects occur with positive chance.\nSteps:\n1. Random choices prove existence, showing desired objects occur with positive chance.\n2. So the answer stands: Random choices prove existence, showing desired objects occur with positive chance."}
{"id": "mat_0721", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why prefer short proofs?", "choices": null, "answer": "Short proofs are easier to check and teach, though finding them often takes more insight.", "steps": ["Short proofs are easier to check and teach, though finding them often takes more insight.", "So the answer stands: Short proofs are easier to check and teach, though finding them often takes more insight."], "text": "Answer: Short proofs are easier to check and teach, though finding them often takes more insight.\nSteps:\n1. Short proofs are easier to check and teach, though finding them often takes more insight.\n2. So the answer stands: Short proofs are easier to check and teach, though finding them often takes more insight."}
{"id": "mat_0722", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 1x² + 8x, find f'(3).", "choices": {"A": "12", "B": "16", "C": "14", "D": "18"}, "answer": "The derivative is 14.", "steps": ["f'(x) = 2x + 8.", "At 3: 14.", "Select option C."], "text": "Answer: 14\nSteps:\n1. f'(x) = 2x + 8.\n2. At 3: 14.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0723", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[6,2],[3,9]].", "choices": {"A": "45", "B": "48", "C": "51", "D": "54"}, "answer": "The determinant is 48.", "steps": ["6×9 − 2×3.", "Det = 48.", "Select option B."], "text": "Answer: 48\nSteps:\n1. 6×9 − 2×3.\n2. Det = 48.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0724", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..18}.", "choices": null, "answer": "E = 9.5.", "steps": ["Mean of 1..18.", "(18+1)/2.", "E = 9.5."], "text": "Answer: E = 9.5.\nSteps:\n1. Mean of 1..18.\n2. (18+1)/2.\n3. E = 9.5."}
{"id": "mat_0725", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(48, 72).", "choices": {"A": "21", "B": "27", "C": "24", "D": "30"}, "answer": "The gcd is 24.", "steps": ["Factor 3 out twice.", "gcd = 24.", "Select option C."], "text": "Answer: 24\nSteps:\n1. Factor 3 out twice.\n2. gcd = 24.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0726", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(17,2).", "choices": null, "answer": "C(17,2) = 136.", "steps": ["Formula n(n−1)/2.", "17×16/2.", "= 136."], "text": "Answer: C(17,2) = 136.\nSteps:\n1. Formula n(n−1)/2.\n2. 17×16/2.\n3. = 136."}
{"id": "mat_0727", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 9 variables have?", "choices": {"A": "510", "B": "514", "C": "516", "D": "512"}, "answer": "The table has 512 rows.", "steps": ["2^9 rows.", "Total: 512.", "Select option D."], "text": "Answer: 512\nSteps:\n1. 2^9 rows.\n2. Total: 512.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0728", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is a double integral?", "choices": null, "answer": "It sums over regions in the plane, computing volumes under surfaces slice by slice.", "steps": ["It sums over regions in the plane, computing volumes under surfaces slice by slice.", "So the answer stands: It sums over regions in the plane, computing volumes under surfaces slice by slice."], "text": "Answer: It sums over regions in the plane, computing volumes under surfaces slice by slice.\nSteps:\n1. It sums over regions in the plane, computing volumes under surfaces slice by slice.\n2. So the answer stands: It sums over regions in the plane, computing volumes under surfaces slice by slice."}
{"id": "mat_0729", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[7,1],[2,10]].", "choices": {"A": "15", "B": "19", "C": "17", "D": "21"}, "answer": "The trace is 17.", "steps": ["7 + 10.", "Trace = 17.", "Select option C."], "text": "Answer: 17\nSteps:\n1. 7 + 10.\n2. Trace = 17.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0730", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 16 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 16/20.", "steps": ["Red: 16.", "Total: 20.", "P = 16/20."], "text": "Answer: P(red) = 16/20.\nSteps:\n1. Red: 16.\n2. Total: 20.\n3. P = 16/20."}
{"id": "mat_0731", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (142) mod 9.", "choices": {"A": "8", "B": "0", "C": "1", "D": "7"}, "answer": "The remainder is 7.", "steps": ["142 ÷ 9.", "Remainder 7.", "Select option D."], "text": "Answer: 7\nSteps:\n1. 142 ÷ 9.\n2. Remainder 7.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0732", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(16,3).", "choices": null, "answer": "P(16,3) = 3360.", "steps": ["n(n−1)(n−2).", "16×15×14.", "= 3360."], "text": "Answer: P(16,3) = 3360.\nSteps:\n1. n(n−1)(n−2).\n2. 16×15×14.\n3. = 3360."}
{"id": "mat_0733", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co kotví indukci?", "choices": {"A": "složitá notace", "B": "dlouhé řetězy", "C": "velká čísla", "D": "základní případ"}, "answer": "Odpověď je základní případ.", "steps": ["Bez základu řetěz nic nedokazuje.", "Tedy: základní případ.", "Vyber možnost D."], "text": "Answer: základní případ\nSteps:\n1. Bez základu řetěz nic nedokazuje.\n2. Tedy: základní případ.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0734", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 15²)/(x − 15) as x → 15.", "choices": null, "answer": "Limit: 30.", "steps": ["Factor: (x−15)(x+15).", "Cancel (x−15).", "Plug 15: 15+15 = 30."], "text": "Answer: Limit: 30.\nSteps:\n1. Factor: (x−15)(x+15).\n2. Cancel (x−15).\n3. Plug 15: 15+15 = 30."}
{"id": "mat_0735", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why change basis?", "choices": null, "answer": "New bases simplify matrices, revealing hidden diagonal or triangular structure.", "steps": ["New bases simplify matrices, revealing hidden diagonal or triangular structure.", "So the answer stands: New bases simplify matrices, revealing hidden diagonal or triangular structure."], "text": "Answer: New bases simplify matrices, revealing hidden diagonal or triangular structure.\nSteps:\n1. New bases simplify matrices, revealing hidden diagonal or triangular structure.\n2. So the answer stands: New bases simplify matrices, revealing hidden diagonal or triangular structure."}
{"id": "mat_0736", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is overfitting?", "choices": null, "answer": "Overfitting memorizes noise instead of patterns, failing on new data despite perfect training.", "steps": ["Overfitting memorizes noise instead of patterns, failing on new data despite perfect training.", "So the answer stands: Overfitting memorizes noise instead of patterns, failing on new data despite perfect training."], "text": "Answer: Overfitting memorizes noise instead of patterns, failing on new data despite perfect training.\nSteps:\n1. Overfitting memorizes noise instead of patterns, failing on new data despite perfect training.\n2. So the answer stands: Overfitting memorizes noise instead of patterns, failing on new data despite perfect training."}
{"id": "mat_0737", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (182) mod 7.", "choices": null, "answer": "Remainder: 0.", "steps": ["Divide 182 by 7.", "7×26 = 182.", "182 − 182 = 0."], "text": "Answer: Remainder: 0.\nSteps:\n1. Divide 182 by 7.\n2. 7×26 = 182.\n3. 182 − 182 = 0."}
{"id": "mat_0738", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(8,2).", "choices": {"A": "48", "B": "64", "C": "56", "D": "72"}, "answer": "P(8,2) is 56.", "steps": ["8×7.", "= 56.", "Select option C."], "text": "Answer: 56\nSteps:\n1. 8×7.\n2. = 56.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0739", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 12?", "choices": {"A": "78", "B": "66", "C": "90", "D": "102"}, "answer": "The sum is 78.", "steps": ["12×13/2.", "Sum = 78.", "Select option A."], "text": "Answer: 78\nSteps:\n1. 12×13/2.\n2. Sum = 78.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0740", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej limitu (x² − 3²)/(x − 3) pro x → 3.", "choices": {"A": "4", "B": "8", "C": "10", "D": "6"}, "answer": "Limita je 6.", "steps": ["Rozlož (x−3)(x+3).", "Pokrať, dostaneš 2×3 = 6.", "Vyber možnost D."], "text": "Answer: 6\nSteps:\n1. Rozlož (x−3)(x+3).\n2. Pokrať, dostaneš 2×3 = 6.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0741", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (6,7) · (3,4).", "choices": {"A": "42", "B": "50", "C": "54", "D": "46"}, "answer": "The dot product is 46.", "steps": ["6×3 + 7×4.", "Product = 46.", "Select option D."], "text": "Answer: 46\nSteps:\n1. 6×3 + 7×4.\n2. Product = 46.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0742", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Sáček má 3 červených a 7 modrých kuliček. Jaká je P(červená)?", "choices": {"A": "7/10", "B": "3/11", "C": "4/10", "D": "3/10"}, "answer": "P(červená) je 3/10.", "steps": ["3 z 10.", "P = 3/10.", "Vyber možnost D."], "text": "Answer: 3/10\nSteps:\n1. 3 z 10.\n2. P = 3/10.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0743", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(17, 8) step by step.", "choices": null, "answer": "lcm = 136.", "steps": ["gcd(17,8) = 1.", "17×8/1.", "= 136."], "text": "Answer: lcm = 136.\nSteps:\n1. gcd(17,8) = 1.\n2. 17×8/1.\n3. = 136."}
{"id": "mat_0744", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(10,3).", "choices": {"A": "110", "B": "130", "C": "140", "D": "120"}, "answer": "C(10,3) is 120.", "steps": ["10×9×8/6.", "= 120.", "Select option D."], "text": "Answer: 120\nSteps:\n1. 10×9×8/6.\n2. = 120.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0745", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is a non-constructive proof?", "choices": null, "answer": "A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction.", "steps": ["A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction.", "So the answer stands: A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction."], "text": "Answer: A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction.\nSteps:\n1. A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction.\n2. So the answer stands: A non-constructive proof shows existence without exhibiting the object, often via counting or contradiction."}
{"id": "mat_0746", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 3x² + 14x, compute f'(3) step by step.", "choices": null, "answer": "f'(3) = 32.", "steps": ["f'(x) = 6x + 14.", "Plug x = 3.", "6×3 + 14 = 32."], "text": "Answer: f'(3) = 32.\nSteps:\n1. f'(x) = 6x + 14.\n2. Plug x = 3.\n3. 6×3 + 14 = 32."}
{"id": "mat_0747", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej determinant matice [[2,2],[3,5]].", "choices": {"A": "1", "B": "4", "C": "7", "D": "10"}, "answer": "Determinant je 4.", "steps": ["2×5 − 2×3.", "Det = 4.", "Vyber možnost B."], "text": "Answer: 4\nSteps:\n1. 2×5 − 2×3.\n2. Det = 4.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0748", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why cross-validate?", "choices": null, "answer": "It tests models on held-out data, honestly estimating real-world performance.", "steps": ["It tests models on held-out data, honestly estimating real-world performance.", "So the answer stands: It tests models on held-out data, honestly estimating real-world performance."], "text": "Answer: It tests models on held-out data, honestly estimating real-world performance.\nSteps:\n1. It tests models on held-out data, honestly estimating real-world performance.\n2. So the answer stands: It tests models on held-out data, honestly estimating real-world performance."}
{"id": "mat_0749", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(10, 6).", "choices": {"A": "30", "B": "28", "C": "32", "D": "34"}, "answer": "The lcm is 30.", "steps": ["10×6/gcd.", "lcm = 30.", "Select option A."], "text": "Answer: 30\nSteps:\n1. 10×6/gcd.\n2. lcm = 30.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0750", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(10,2).", "choices": {"A": "35", "B": "55", "C": "45", "D": "65"}, "answer": "C(10,2) is 45.", "steps": ["10×9/2.", "= 45.", "Select option C."], "text": "Answer: 45\nSteps:\n1. 10×9/2.\n2. = 45.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0751", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why study proof techniques?", "choices": null, "answer": "Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics.", "steps": ["Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics.", "So the answer stands: Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics."], "text": "Answer: Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics.\nSteps:\n1. Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics.\n2. So the answer stands: Techniques transfer across fields, so mastering induction and contradiction unlocks all of mathematics."}
{"id": "mat_0752", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 8.", "choices": {"A": "504", "B": "520", "C": "528", "D": "512"}, "answer": "The integral is 512.", "steps": ["Antiderivative x³.", "8³ = 512.", "Select option D."], "text": "Answer: 512\nSteps:\n1. Antiderivative x³.\n2. 8³ = 512.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0753", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is similarity?", "choices": null, "answer": "Similar matrices represent one map in different bases, sharing eigenvalues and determinant.", "steps": ["Similar matrices represent one map in different bases, sharing eigenvalues and determinant.", "So the answer stands: Similar matrices represent one map in different bases, sharing eigenvalues and determinant."], "text": "Answer: Similar matrices represent one map in different bases, sharing eigenvalues and determinant.\nSteps:\n1. Similar matrices represent one map in different bases, sharing eigenvalues and determinant.\n2. So the answer stands: Similar matrices represent one map in different bases, sharing eigenvalues and determinant."}
{"id": "mat_0754", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 20-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/20.", "steps": ["Favourable: 1.", "Possible: 20.", "P = 1/20."], "text": "Answer: P = 1/20.\nSteps:\n1. Favourable: 1.\n2. Possible: 20.\n3. P = 1/20."}
{"id": "mat_0755", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(54, 81).", "choices": {"A": "24", "B": "30", "C": "33", "D": "27"}, "answer": "The gcd is 27.", "steps": ["Factor 3 out twice.", "gcd = 27.", "Select option D."], "text": "Answer: 27\nSteps:\n1. Factor 3 out twice.\n2. gcd = 27.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0756", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is linearity of expectation?", "choices": null, "answer": "Expectations add regardless of dependence, simplifying random counts hugely.", "steps": ["Expectations add regardless of dependence, simplifying random counts hugely.", "So the answer stands: Expectations add regardless of dependence, simplifying random counts hugely."], "text": "Answer: Expectations add regardless of dependence, simplifying random counts hugely.\nSteps:\n1. Expectations add regardless of dependence, simplifying random counts hugely.\n2. So the answer stands: Expectations add regardless of dependence, simplifying random counts hugely."}
{"id": "mat_0757", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Důkaz prochází všechny případy 6 booleovských proměnných. Kolik řádků?", "choices": null, "answer": "Řádků: 64.", "steps": ["2 volby každá.", "2^6 = 64."], "text": "Answer: Řádků: 64.\nSteps:\n1. 2 volby každá.\n2. 2^6 = 64."}
{"id": "mat_0758", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 8²)/(x − 8) as x → 8.", "choices": {"A": "16", "B": "14", "C": "18", "D": "20"}, "answer": "The limit is 16.", "steps": ["Factor (x−8)(x+8).", "Cancel, get 2×8 = 16.", "Select option A."], "text": "Answer: 16\nSteps:\n1. Factor (x−8)(x+8).\n2. Cancel, get 2×8 = 16.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0759", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[17,1],[2,18]] step by step.", "choices": null, "answer": "Determinant: 304.", "steps": ["ad − bc.", "17×18 − 1×2.", "= 304."], "text": "Answer: Determinant: 304.\nSteps:\n1. ad − bc.\n2. 17×18 − 1×2.\n3. = 304."}
{"id": "mat_0760", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..9}.", "choices": {"A": "6", "B": "4", "C": "5", "D": "7"}, "answer": "The expected value is 5.", "steps": ["(9+1)/2.", "E = 5.", "Select option C."], "text": "Answer: 5\nSteps:\n1. (9+1)/2.\n2. E = 5.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0761", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "concept", "question": "Co je Eulerova funkce?", "choices": null, "answer": "phi(n) počítá čísla menší než n nesoudělná s ním.", "steps": ["phi(n) počítá čísla menší než n nesoudělná s ním.", "Odpověď tedy platí: phi(n) počítá čísla menší než n nesoudělná s ním."], "text": "Answer: phi(n) počítá čísla menší než n nesoudělná s ním.\nSteps:\n1. phi(n) počítá čísla menší než n nesoudělná s ním.\n2. Odpověď tedy platí: phi(n) počítá čísla menší než n nesoudělná s ním."}
{"id": "mat_0762", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(18,3).", "choices": null, "answer": "C(18,3) = 816.", "steps": ["n(n−1)(n−2)/6.", "18×17×16/6.", "= 816."], "text": "Answer: C(18,3) = 816.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 18×17×16/6.\n3. = 816."}
{"id": "mat_0763", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 19.", "choices": null, "answer": "Sum: 190.", "steps": ["Pair 1+19, 2+18, ...", "Use 19×20/2.", "= 190."], "text": "Answer: Sum: 190.\nSteps:\n1. Pair 1+19, 2+18, ...\n2. Use 19×20/2.\n3. = 190."}
{"id": "mat_0764", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 16.", "choices": null, "answer": "Integral: 4096.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 16.", "16³ − 0 = 4096."], "text": "Answer: Integral: 4096.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 16.\n3. 16³ − 0 = 4096."}
{"id": "mat_0765", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej stopu matice [[6,0],[0,7]].", "choices": null, "answer": "Stopa: 13.", "steps": ["Sečti diagonálu.", "6 + 7.", "= 13."], "text": "Answer: Stopa: 13.\nSteps:\n1. Sečti diagonálu.\n2. 6 + 7.\n3. = 13."}
{"id": "mat_0766", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Sáček má 5 červených a 4 modré kuličky. Ukaž P(červená).", "choices": null, "answer": "P(červená) = 5/9.", "steps": ["Červené: 5.", "Celkem: 9.", "P = 5/9."], "text": "Answer: P(červená) = 5/9.\nSteps:\n1. Červené: 5.\n2. Celkem: 9.\n3. P = 5/9."}
{"id": "mat_0767", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(60, 90) step by step.", "choices": null, "answer": "gcd = 30.", "steps": ["Factor: 60=4×15, 90=6×15.", "Common: 15×gcd(4,6).", "= 15×2 = 30."], "text": "Answer: gcd = 30.\nSteps:\n1. Factor: 60=4×15, 90=6×15.\n2. Common: 15×gcd(4,6).\n3. = 15×2 = 30."}
{"id": "mat_0768", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(18,2).", "choices": null, "answer": "C(18,2) = 153.", "steps": ["Formula n(n−1)/2.", "18×17/2.", "= 153."], "text": "Answer: C(18,2) = 153.\nSteps:\n1. Formula n(n−1)/2.\n2. 18×17/2.\n3. = 153."}
{"id": "mat_0769", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 15 by computing its half.", "choices": null, "answer": "Half: 120, so 240 is even.", "steps": ["15² + 15 = 240.", "240 / 2 = 120.", "Integer, so even."], "text": "Answer: Half: 120, so 240 is even.\nSteps:\n1. 15² + 15 = 240.\n2. 240 / 2 = 120.\n3. Integer, so even."}
{"id": "mat_0770", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 2x² + 9x, find f'(4).", "choices": {"A": "23", "B": "27", "C": "29", "D": "25"}, "answer": "The derivative is 25.", "steps": ["f'(x) = 4x + 9.", "At 4: 25.", "Select option D."], "text": "Answer: 25\nSteps:\n1. f'(x) = 4x + 9.\n2. At 4: 25.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0771", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej (5,4) · (3,5) krok za krokem.", "choices": null, "answer": "Skalární součin: 35.", "steps": ["Násob složky.", "5×3 + 4×5.", "= 35."], "text": "Answer: Skalární součin: 35.\nSteps:\n1. Násob složky.\n2. 5×3 + 4×5.\n3. = 35."}
{"id": "mat_0772", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 7 red and 15 blue marbles. What is P(red)?", "choices": {"A": "15/22", "B": "7/23", "C": "8/22", "D": "7/22"}, "answer": "P(red) is 7/22.", "steps": ["7 of 22.", "P = 7/22.", "Select option D."], "text": "Answer: 7/22\nSteps:\n1. 7 of 22.\n2. P = 7/22.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0773", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (193) mod 7.", "choices": null, "answer": "Remainder: 4.", "steps": ["Divide 193 by 7.", "7×27 = 189.", "193 − 189 = 4."], "text": "Answer: Remainder: 4.\nSteps:\n1. Divide 193 by 7.\n2. 7×27 = 189.\n3. 193 − 189 = 4."}
{"id": "mat_0774", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(17,3).", "choices": null, "answer": "P(17,3) = 4080.", "steps": ["n(n−1)(n−2).", "17×16×15.", "= 4080."], "text": "Answer: P(17,3) = 4080.\nSteps:\n1. n(n−1)(n−2).\n2. 17×16×15.\n3. = 4080."}
{"id": "mat_0775", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 16 boolean variables. How many rows?", "choices": null, "answer": "Rows: 65536.", "steps": ["2 choices each.", "2^16 = 65536."], "text": "Answer: Rows: 65536.\nSteps:\n1. 2 choices each.\n2. 2^16 = 65536."}
{"id": "mat_0776", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 16²)/(x − 16) as x → 16.", "choices": null, "answer": "Limit: 32.", "steps": ["Factor: (x−16)(x+16).", "Cancel (x−16).", "Plug 16: 16+16 = 32."], "text": "Answer: Limit: 32.\nSteps:\n1. Factor: (x−16)(x+16).\n2. Cancel (x−16).\n3. Plug 16: 16+16 = 32."}
{"id": "mat_0777", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[18,0],[0,19]].", "choices": null, "answer": "Trace: 37.", "steps": ["Sum diagonal.", "18 + 19.", "= 37."], "text": "Answer: Trace: 37.\nSteps:\n1. Sum diagonal.\n2. 18 + 19.\n3. = 37."}
{"id": "mat_0778", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..19}.", "choices": null, "answer": "E = 10.", "steps": ["Mean of 1..19.", "(19+1)/2.", "E = 10."], "text": "Answer: E = 10.\nSteps:\n1. Mean of 1..19.\n2. (19+1)/2.\n3. E = 10."}
{"id": "mat_0779", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej (83) mod 7.", "choices": null, "answer": "Zbytek: 6.", "steps": ["Děl 83 sedmi.", "7×11 = 77.", "83 − 77 = 6."], "text": "Answer: Zbytek: 6.\nSteps:\n1. Děl 83 sedmi.\n2. 7×11 = 77.\n3. 83 − 77 = 6."}
{"id": "mat_0780", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(9,2).", "choices": {"A": "63", "B": "81", "C": "90", "D": "72"}, "answer": "P(9,2) is 72.", "steps": ["9×8.", "= 72.", "Select option D."], "text": "Answer: 72\nSteps:\n1. 9×8.\n2. = 72.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0781", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 20.", "choices": null, "answer": "Sum: 210.", "steps": ["Pair 1+20, 2+19, ...", "Use 20×21/2.", "= 210."], "text": "Answer: Sum: 210.\nSteps:\n1. Pair 1+20, 2+19, ...\n2. Use 20×21/2.\n3. = 210."}
{"id": "mat_0782", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why change coordinates?", "choices": null, "answer": "Polar or spherical coordinates match round symmetry, making hard integrals easy.", "steps": ["Polar or spherical coordinates match round symmetry, making hard integrals easy.", "So the answer stands: Polar or spherical coordinates match round symmetry, making hard integrals easy."], "text": "Answer: Polar or spherical coordinates match round symmetry, making hard integrals easy.\nSteps:\n1. Polar or spherical coordinates match round symmetry, making hard integrals easy.\n2. So the answer stands: Polar or spherical coordinates match round symmetry, making hard integrals easy."}
{"id": "mat_0783", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[7,2],[3,10]].", "choices": {"A": "61", "B": "67", "C": "64", "D": "70"}, "answer": "The determinant is 64.", "steps": ["7×10 − 2×3.", "Det = 64.", "Select option C."], "text": "Answer: 64\nSteps:\n1. 7×10 − 2×3.\n2. Det = 64.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0784", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 17 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 17/21.", "steps": ["Red: 17.", "Total: 21.", "P = 17/21."], "text": "Answer: P(red) = 17/21.\nSteps:\n1. Red: 17.\n2. Total: 21.\n3. P = 17/21."}
{"id": "mat_0785", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why Miller-Rabin?", "choices": null, "answer": "It tests primality fast with random bases, practical for cryptographic-size numbers.", "steps": ["It tests primality fast with random bases, practical for cryptographic-size numbers.", "So the answer stands: It tests primality fast with random bases, practical for cryptographic-size numbers."], "text": "Answer: It tests primality fast with random bases, practical for cryptographic-size numbers.\nSteps:\n1. It tests primality fast with random bases, practical for cryptographic-size numbers.\n2. So the answer stands: It tests primality fast with random bases, practical for cryptographic-size numbers."}
{"id": "mat_0786", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why derangements approximate?", "choices": null, "answer": "Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums.", "steps": ["Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums.", "So the answer stands: Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums."], "text": "Answer: Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums.\nSteps:\n1. Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums.\n2. So the answer stands: Derangement share nears 1/e, a lovely limit from inclusion-exclusion sums."}
{"id": "mat_0787", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is the contrapositive of P to Q?", "choices": null, "answer": "The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication.", "steps": ["The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication.", "So the answer stands: The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication."], "text": "Answer: The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication.\nSteps:\n1. The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication.\n2. So the answer stands: The contrapositive is not-Q to not-P, and it is logically equivalent to the original implication."}
{"id": "mat_0788", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 9.", "choices": {"A": "729", "B": "720", "C": "738", "D": "747"}, "answer": "The integral is 729.", "steps": ["Antiderivative x³.", "9³ = 729.", "Select option A."], "text": "Answer: 729\nSteps:\n1. Antiderivative x³.\n2. 9³ = 729.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0789", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Co je báze?", "choices": null, "answer": "Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice.", "steps": ["Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice.", "Odpověď tedy platí: Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice."], "text": "Answer: Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice.\nSteps:\n1. Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice.\n2. Odpověď tedy platí: Báze je nezávislá množina napínající prostor, dává jednoznačné souřadnice."}
{"id": "mat_0790", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 21-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/21.", "steps": ["Favourable: 1.", "Possible: 21.", "P = 1/21."], "text": "Answer: P = 1/21.\nSteps:\n1. Favourable: 1.\n2. Possible: 21.\n3. P = 1/21."}
{"id": "mat_0791", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(18, 8) step by step.", "choices": null, "answer": "lcm = 72.", "steps": ["gcd(18,8) = 2.", "18×8/2.", "= 72."], "text": "Answer: lcm = 72.\nSteps:\n1. gcd(18,8) = 2.\n2. 18×8/2.\n3. = 72."}
{"id": "mat_0792", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(19,3).", "choices": null, "answer": "C(19,3) = 969.", "steps": ["n(n−1)(n−2)/6.", "19×18×17/6.", "= 969."], "text": "Answer: C(19,3) = 969.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 19×18×17/6.\n3. = 969."}
{"id": "mat_0793", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Dokaž vzorcem: najdi součet 1 + 2 + ... + 10.", "choices": null, "answer": "Součet: 55.", "steps": ["Páruj 1+10, 2+9, ...", "Použij 10×11/2.", "= 55."], "text": "Answer: Součet: 55.\nSteps:\n1. Páruj 1+10, 2+9, ...\n2. Použij 10×11/2.\n3. = 55."}
{"id": "mat_0794", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 9²)/(x − 9) as x → 9.", "choices": {"A": "16", "B": "18", "C": "20", "D": "22"}, "answer": "The limit is 18.", "steps": ["Factor (x−9)(x+9).", "Cancel, get 2×9 = 18.", "Select option B."], "text": "Answer: 18\nSteps:\n1. Factor (x−9)(x+9).\n2. Cancel, get 2×9 = 18.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0795", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why compute characteristic polynomial?", "choices": null, "answer": "Its roots are the eigenvalues, turning eigen-search into polynomial factoring.", "steps": ["Its roots are the eigenvalues, turning eigen-search into polynomial factoring.", "So the answer stands: Its roots are the eigenvalues, turning eigen-search into polynomial factoring."], "text": "Answer: Its roots are the eigenvalues, turning eigen-search into polynomial factoring.\nSteps:\n1. Its roots are the eigenvalues, turning eigen-search into polynomial factoring.\n2. So the answer stands: Its roots are the eigenvalues, turning eigen-search into polynomial factoring."}
{"id": "mat_0796", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 12-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/11", "B": "1/13", "C": "1/12", "D": "2/12"}, "answer": "The probability is 1/12.", "steps": ["1 face of 12.", "P = 1/12.", "Select option C."], "text": "Answer: 1/12\nSteps:\n1. 1 face of 12.\n2. P = 1/12.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0797", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(64, 96) step by step.", "choices": null, "answer": "gcd = 32.", "steps": ["Factor: 64=4×16, 96=6×16.", "Common: 16×gcd(4,6).", "= 16×2 = 32."], "text": "Answer: gcd = 32.\nSteps:\n1. Factor: 64=4×16, 96=6×16.\n2. Common: 16×gcd(4,6).\n3. = 16×2 = 32."}
{"id": "mat_0798", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(19,2).", "choices": null, "answer": "C(19,2) = 171.", "steps": ["Formula n(n−1)/2.", "19×18/2.", "= 171."], "text": "Answer: C(19,2) = 171.\nSteps:\n1. Formula n(n−1)/2.\n2. 19×18/2.\n3. = 171."}
{"id": "mat_0799", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 16 by computing its half.", "choices": null, "answer": "Half: 136, so 272 is even.", "steps": ["16² + 16 = 272.", "272 / 2 = 136.", "Integer, so even."], "text": "Answer: Half: 136, so 272 is even.\nSteps:\n1. 16² + 16 = 272.\n2. 272 / 2 = 136.\n3. Integer, so even."}
{"id": "mat_0800", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 4x² + 15x, compute f'(4) step by step.", "choices": null, "answer": "f'(4) = 47.", "steps": ["f'(x) = 8x + 15.", "Plug x = 4.", "8×4 + 15 = 47."], "text": "Answer: f'(4) = 47.\nSteps:\n1. f'(x) = 8x + 15.\n2. Plug x = 4.\n3. 8×4 + 15 = 47."}
{"id": "mat_0801", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (17,16) · (3,5) step by step.", "choices": null, "answer": "Dot product: 131.", "steps": ["Multiply components.", "17×3 + 16×5.", "= 131."], "text": "Answer: Dot product: 131.\nSteps:\n1. Multiply components.\n2. 17×3 + 16×5.\n3. = 131."}
{"id": "mat_0802", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is a prior?", "choices": null, "answer": "A prior encodes beliefs before data, and Bayes updates it into the posterior.", "steps": ["A prior encodes beliefs before data, and Bayes updates it into the posterior.", "So the answer stands: A prior encodes beliefs before data, and Bayes updates it into the posterior."], "text": "Answer: A prior encodes beliefs before data, and Bayes updates it into the posterior.\nSteps:\n1. A prior encodes beliefs before data, and Bayes updates it into the posterior.\n2. So the answer stands: A prior encodes beliefs before data, and Bayes updates it into the posterior."}
{"id": "mat_0803", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (149) mod 9.", "choices": {"A": "5", "B": "6", "C": "7", "D": "8"}, "answer": "The remainder is 5.", "steps": ["149 ÷ 9.", "Remainder 5.", "Select option A."], "text": "Answer: 5\nSteps:\n1. 149 ÷ 9.\n2. Remainder 5.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0804", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Co je princip inkluze?", "choices": null, "answer": "Opravuje dvojí započtení střídavými součty.", "steps": ["Opravuje dvojí započtení střídavými součty.", "Odpověď tedy platí: Opravuje dvojí započtení střídavými součty."], "text": "Answer: Opravuje dvojí započtení střídavými součty.\nSteps:\n1. Opravuje dvojí započtení střídavými součty.\n2. Odpověď tedy platí: Opravuje dvojí započtení střídavými součty."}
{"id": "mat_0805", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why negate quantifiers carefully?", "choices": null, "answer": "Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely.", "steps": ["Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely.", "So the answer stands: Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely."], "text": "Answer: Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely.\nSteps:\n1. Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely.\n2. So the answer stands: Negating for-all yields exists-not, so sloppy negation flips the claim's meaning entirely."}
{"id": "mat_0806", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 17.", "choices": null, "answer": "Integral: 4913.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 17.", "17³ − 0 = 4913."], "text": "Answer: Integral: 4913.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 17.\n3. 17³ − 0 = 4913."}
{"id": "mat_0807", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[8,1],[2,11]].", "choices": {"A": "17", "B": "21", "C": "23", "D": "19"}, "answer": "The trace is 19.", "steps": ["8 + 11.", "Trace = 19.", "Select option D."], "text": "Answer: 19\nSteps:\n1. 8 + 11.\n2. Trace = 19.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0808", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Hází se férovou 9-stěnnou kostkou. Ukaž P(jedno konkrétní číslo).", "choices": null, "answer": "P = 1/9.", "steps": ["Příznivé: 1.", "Možné: 9.", "P = 1/9."], "text": "Answer: P = 1/9.\nSteps:\n1. Příznivé: 1.\n2. Možné: 9.\n3. P = 1/9."}
{"id": "mat_0809", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (204) mod 7.", "choices": null, "answer": "Remainder: 1.", "steps": ["Divide 204 by 7.", "7×29 = 203.", "204 − 203 = 1."], "text": "Answer: Remainder: 1.\nSteps:\n1. Divide 204 by 7.\n2. 7×29 = 203.\n3. 204 − 203 = 1."}
{"id": "mat_0810", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Proč Dirichletův princip?", "choices": null, "answer": "Více předmětů než krabic vynutí sdílení.", "steps": ["Více předmětů než krabic vynutí sdílení.", "Odpověď tedy platí: Více předmětů než krabic vynutí sdílení."], "text": "Answer: Více předmětů než krabic vynutí sdílení.\nSteps:\n1. Více předmětů než krabic vynutí sdílení.\n2. Odpověď tedy platí: Více předmětů než krabic vynutí sdílení."}
{"id": "mat_0811", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 10.", "choices": {"A": "108", "B": "110", "C": "112", "D": "114"}, "answer": "The value is 110.", "steps": ["10² + 10.", "Value = 110.", "Select option B."], "text": "Answer: 110\nSteps:\n1. 10² + 10.\n2. Value = 110.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0812", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 17²)/(x − 17) as x → 17.", "choices": null, "answer": "Limit: 34.", "steps": ["Factor: (x−17)(x+17).", "Cancel (x−17).", "Plug 17: 17+17 = 34."], "text": "Answer: Limit: 34.\nSteps:\n1. Factor: (x−17)(x+17).\n2. Cancel (x−17).\n3. Plug 17: 17+17 = 34."}
{"id": "mat_0813", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is the trace?", "choices": null, "answer": "The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change.", "steps": ["The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change.", "So the answer stands: The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change."], "text": "Answer: The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change.\nSteps:\n1. The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change.\n2. So the answer stands: The trace sums diagonal entries, equal to the eigenvalue sum and invariant under basis change."}
{"id": "mat_0814", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..10}.", "choices": {"A": "6.5", "B": "4.5", "C": "7.5", "D": "5.5"}, "answer": "The expected value is 5.5.", "steps": ["(10+1)/2.", "E = 5.5.", "Select option D."], "text": "Answer: 5.5\nSteps:\n1. (10+1)/2.\n2. E = 5.5.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0815", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(19, 8) step by step.", "choices": null, "answer": "lcm = 152.", "steps": ["gcd(19,8) = 1.", "19×8/1.", "= 152."], "text": "Answer: lcm = 152.\nSteps:\n1. gcd(19,8) = 1.\n2. 19×8/1.\n3. = 152."}
{"id": "mat_0816", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Co je permutace?", "choices": null, "answer": "Permutace řadí objekty, n různých jich má právě n!.", "steps": ["Permutace řadí objekty, n různých jich má právě n!.", "Odpověď tedy platí: Permutace řadí objekty, n různých jich má právě n!."], "text": "Answer: Permutace řadí objekty, n různých jich má právě n!.\nSteps:\n1. Permutace řadí objekty, n různých jich má právě n!.\n2. Odpověď tedy platí: Permutace řadí objekty, n různých jich má právě n!."}
{"id": "mat_0817", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 10 variables have?", "choices": {"A": "1024", "B": "1022", "C": "1026", "D": "1028"}, "answer": "The table has 1024 rows.", "steps": ["2^10 rows.", "Total: 1024.", "Select option A."], "text": "Answer: 1024\nSteps:\n1. 2^10 rows.\n2. Total: 1024.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0818", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is divergence?", "choices": null, "answer": "Divergence measures net outflow of a field at a point, flagging sources and sinks.", "steps": ["Divergence measures net outflow of a field at a point, flagging sources and sinks.", "So the answer stands: Divergence measures net outflow of a field at a point, flagging sources and sinks."], "text": "Answer: Divergence measures net outflow of a field at a point, flagging sources and sinks.\nSteps:\n1. Divergence measures net outflow of a field at a point, flagging sources and sinks.\n2. So the answer stands: Divergence measures net outflow of a field at a point, flagging sources and sinks."}
{"id": "mat_0819", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (7,8) · (3,4).", "choices": {"A": "53", "B": "49", "C": "57", "D": "61"}, "answer": "The dot product is 53.", "steps": ["7×3 + 8×4.", "Product = 53.", "Select option A."], "text": "Answer: 53\nSteps:\n1. 7×3 + 8×4.\n2. Product = 53.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0820", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..20}.", "choices": null, "answer": "E = 10.5.", "steps": ["Mean of 1..20.", "(20+1)/2.", "E = 10.5."], "text": "Answer: E = 10.5.\nSteps:\n1. Mean of 1..20.\n2. (20+1)/2.\n3. E = 10.5."}
{"id": "mat_0821", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is RSA?", "choices": null, "answer": "RSA multiplies primes for public keys, secure because factoring stays hard.", "steps": ["RSA multiplies primes for public keys, secure because factoring stays hard.", "So the answer stands: RSA multiplies primes for public keys, secure because factoring stays hard."], "text": "Answer: RSA multiplies primes for public keys, secure because factoring stays hard.\nSteps:\n1. RSA multiplies primes for public keys, secure because factoring stays hard.\n2. So the answer stands: RSA multiplies primes for public keys, secure because factoring stays hard."}
{"id": "mat_0822", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet P(7,3).", "choices": null, "answer": "P(7,3) = 210.", "steps": ["n(n−1)(n−2).", "7×6×5.", "= 210."], "text": "Answer: P(7,3) = 210.\nSteps:\n1. n(n−1)(n−2).\n2. 7×6×5.\n3. = 210."}
{"id": "mat_0823", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "worked_example", "question": "Ukaž, že N² + N je sudé pro N = 6, výpočtem poloviny.", "choices": null, "answer": "Polovina: 21, číslo 42 je sudé.", "steps": ["6² + 6 = 42.", "42 / 2 = 21.", "Celé číslo, tedy sudé."], "text": "Answer: Polovina: 21, číslo 42 je sudé.\nSteps:\n1. 6² + 6 = 42.\n2. 42 / 2 = 21.\n3. Celé číslo, tedy sudé."}
{"id": "mat_0824", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Pro f(x) = 2x² + 4x spočítej f'(2) krok za krokem.", "choices": null, "answer": "f'(2) = 12.", "steps": ["f'(x) = 4x + 4.", "Dosaď x = 2.", "4×2 + 4 = 12."], "text": "Answer: f'(2) = 12.\nSteps:\n1. f'(x) = 4x + 4.\n2. Dosaď x = 2.\n3. 4×2 + 4 = 12."}
{"id": "mat_0825", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[8,2],[3,11]].", "choices": {"A": "79", "B": "85", "C": "88", "D": "82"}, "answer": "The determinant is 82.", "steps": ["8×11 − 2×3.", "Det = 82.", "Select option D."], "text": "Answer: 82\nSteps:\n1. 8×11 − 2×3.\n2. Det = 82.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0826", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 8 red and 17 blue marbles. What is P(red)?", "choices": {"A": "8/25", "B": "17/25", "C": "8/26", "D": "9/25"}, "answer": "P(red) is 8/25.", "steps": ["8 of 25.", "P = 8/25.", "Select option A."], "text": "Answer: 8/25\nSteps:\n1. 8 of 25.\n2. P = 8/25.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0827", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why discrete logs?", "choices": null, "answer": "They invert exponentiation mod p, hard enough to secure key exchange protocols.", "steps": ["They invert exponentiation mod p, hard enough to secure key exchange protocols.", "So the answer stands: They invert exponentiation mod p, hard enough to secure key exchange protocols."], "text": "Answer: They invert exponentiation mod p, hard enough to secure key exchange protocols.\nSteps:\n1. They invert exponentiation mod p, hard enough to secure key exchange protocols.\n2. So the answer stands: They invert exponentiation mod p, hard enough to secure key exchange protocols."}
{"id": "mat_0828", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(18,3).", "choices": null, "answer": "P(18,3) = 4896.", "steps": ["n(n−1)(n−2).", "18×17×16.", "= 4896."], "text": "Answer: P(18,3) = 4896.\nSteps:\n1. n(n−1)(n−2).\n2. 18×17×16.\n3. = 4896."}
{"id": "mat_0829", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co dokazuje existenci?", "choices": {"A": "svědek či počítání", "B": "dlouhá jména", "C": "velké fonty", "D": "samotná naděje"}, "answer": "Odpověď je svědek či počítání.", "steps": ["Sestroj nebo spočítej.", "Tedy: svědek či počítání.", "Vyber možnost A."], "text": "Answer: svědek či počítání\nSteps:\n1. Sestroj nebo spočítej.\n2. Tedy: svědek či počítání.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0830", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 2x² + 16x, compute f'(2) step by step.", "choices": null, "answer": "f'(2) = 24.", "steps": ["f'(x) = 4x + 16.", "Plug x = 2.", "4×2 + 16 = 24."], "text": "Answer: f'(2) = 24.\nSteps:\n1. f'(x) = 4x + 16.\n2. Plug x = 2.\n3. 4×2 + 16 = 24."}
{"id": "mat_0831", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[9,1],[2,12]].", "choices": {"A": "21", "B": "19", "C": "23", "D": "25"}, "answer": "The trace is 21.", "steps": ["9 + 12.", "Trace = 21.", "Select option A."], "text": "Answer: 21\nSteps:\n1. 9 + 12.\n2. Trace = 21.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0832", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 13-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/12", "B": "1/14", "C": "2/13", "D": "1/13"}, "answer": "The probability is 1/13.", "steps": ["1 face of 13.", "P = 1/13.", "Select option D."], "text": "Answer: 1/13\nSteps:\n1. 1 face of 13.\n2. P = 1/13.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0833", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is the sigma function?", "choices": null, "answer": "sigma(n) sums all divisors, multiplicative over prime powers for easy formulas.", "steps": ["sigma(n) sums all divisors, multiplicative over prime powers for easy formulas.", "So the answer stands: sigma(n) sums all divisors, multiplicative over prime powers for easy formulas."], "text": "Answer: sigma(n) sums all divisors, multiplicative over prime powers for easy formulas.\nSteps:\n1. sigma(n) sums all divisors, multiplicative over prime powers for easy formulas.\n2. So the answer stands: sigma(n) sums all divisors, multiplicative over prime powers for easy formulas."}
{"id": "mat_0834", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(20,3).", "choices": null, "answer": "C(20,3) = 1140.", "steps": ["n(n−1)(n−2)/6.", "20×19×18/6.", "= 1140."], "text": "Answer: C(20,3) = 1140.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 20×19×18/6.\n3. = 1140."}
{"id": "mat_0835", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is a tautology?", "choices": null, "answer": "A tautology is true under every assignment, like P-or-not-P, and truth tables detect it.", "steps": ["A tautology is true under every assignment, like P-or-not-P, and truth tables detect it.", "So the answer stands: A tautology is true under every assignment, like P-or-not-P, and truth tables detect it."], "text": "Answer: A tautology is true under every assignment, like P-or-not-P, and truth tables detect it.\nSteps:\n1. A tautology is true under every assignment, like P-or-not-P, and truth tables detect it.\n2. So the answer stands: A tautology is true under every assignment, like P-or-not-P, and truth tables detect it."}
{"id": "mat_0836", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 3x² + 10x, find f'(1).", "choices": {"A": "16", "B": "14", "C": "18", "D": "20"}, "answer": "The derivative is 16.", "steps": ["f'(x) = 6x + 10.", "At 1: 16.", "Select option A."], "text": "Answer: 16\nSteps:\n1. f'(x) = 6x + 10.\n2. At 1: 16.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0837", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (8,9) · (3,4).", "choices": {"A": "56", "B": "60", "C": "64", "D": "68"}, "answer": "The dot product is 60.", "steps": ["8×3 + 9×4.", "Product = 60.", "Select option B."], "text": "Answer: 60\nSteps:\n1. 8×3 + 9×4.\n2. Product = 60.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0838", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..11}.", "choices": {"A": "6", "B": "7", "C": "5", "D": "8"}, "answer": "The expected value is 6.", "steps": ["(11+1)/2.", "E = 6.", "Select option A."], "text": "Answer: 6\nSteps:\n1. (11+1)/2.\n2. E = 6.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0839", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why multiplicative functions?", "choices": null, "answer": "They split over coprime products, reducing all values to prime-power cases.", "steps": ["They split over coprime products, reducing all values to prime-power cases.", "So the answer stands: They split over coprime products, reducing all values to prime-power cases."], "text": "Answer: They split over coprime products, reducing all values to prime-power cases.\nSteps:\n1. They split over coprime products, reducing all values to prime-power cases.\n2. So the answer stands: They split over coprime products, reducing all values to prime-power cases."}
{"id": "mat_0840", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(11,3).", "choices": {"A": "165", "B": "154", "C": "176", "D": "187"}, "answer": "C(11,3) is 165.", "steps": ["11×10×9/6.", "= 165.", "Select option A."], "text": "Answer: 165\nSteps:\n1. 11×10×9/6.\n2. = 165.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0841", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 13?", "choices": {"A": "78", "B": "91", "C": "104", "D": "117"}, "answer": "The sum is 91.", "steps": ["13×14/2.", "Sum = 91.", "Select option B."], "text": "Answer: 91\nSteps:\n1. 13×14/2.\n2. Sum = 91.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0842", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 18.", "choices": null, "answer": "Integral: 5832.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 18.", "18³ − 0 = 5832."], "text": "Answer: Integral: 5832.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 18.\n3. 18³ − 0 = 5832."}
{"id": "mat_0843", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why symmetric matrices?", "choices": null, "answer": "Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry.", "steps": ["Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry.", "So the answer stands: Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry."], "text": "Answer: Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry.\nSteps:\n1. Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry.\n2. So the answer stands: Symmetric matrices have real eigenvalues and orthogonal eigenvectors, ideal for geometry."}
{"id": "mat_0844", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 9 red and 19 blue marbles. What is P(red)?", "choices": {"A": "19/28", "B": "9/28", "C": "9/29", "D": "10/28"}, "answer": "P(red) is 9/28.", "steps": ["9 of 28.", "P = 9/28.", "Select option B."], "text": "Answer: 9/28\nSteps:\n1. 9 of 28.\n2. P = 9/28.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0845", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(11, 6).", "choices": {"A": "64", "B": "66", "C": "68", "D": "70"}, "answer": "The lcm is 66.", "steps": ["11×6/gcd.", "lcm = 66.", "Select option B."], "text": "Answer: 66\nSteps:\n1. 11×6/gcd.\n2. lcm = 66.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0846", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(11,2).", "choices": {"A": "44", "B": "66", "C": "77", "D": "55"}, "answer": "C(11,2) is 55.", "steps": ["11×10/2.", "= 55.", "Select option D."], "text": "Answer: 55\nSteps:\n1. 11×10/2.\n2. = 55.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0847", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 17 boolean variables. How many rows?", "choices": null, "answer": "Rows: 131072.", "steps": ["2 choices each.", "2^17 = 131072."], "text": "Answer: Rows: 131072.\nSteps:\n1. 2 choices each.\n2. 2^17 = 131072."}
{"id": "mat_0848", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co řeší 0/0?", "choices": {"A": "l'Hôpitalovo pravidlo", "B": "hádání", "C": "ignorování", "D": "mazání"}, "answer": "Odpověď je l'Hôpitalovo pravidlo.", "steps": ["Derivuj čitatel i jmenovatel.", "Tedy: l'Hôpitalovo pravidlo.", "Vyber možnost A."], "text": "Answer: l'Hôpitalovo pravidlo\nSteps:\n1. Derivuj čitatel i jmenovatel.\n2. Tedy: l'Hôpitalovo pravidlo.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0849", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is positive definite?", "choices": null, "answer": "Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima.", "steps": ["Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima.", "So the answer stands: Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima."], "text": "Answer: Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima.\nSteps:\n1. Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima.\n2. So the answer stands: Positive definite matrices give positive energy for all nonzero vectors, guaranteeing minima."}
{"id": "mat_0850", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 14-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/14", "B": "1/13", "C": "1/15", "D": "2/14"}, "answer": "The probability is 1/14.", "steps": ["1 face of 14.", "P = 1/14.", "Select option A."], "text": "Answer: 1/14\nSteps:\n1. 1 face of 14.\n2. P = 1/14.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0851", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is Möbius inversion?", "choices": null, "answer": "It flips divisor sums via mu, recovering hidden functions from accumulated totals.", "steps": ["It flips divisor sums via mu, recovering hidden functions from accumulated totals.", "So the answer stands: It flips divisor sums via mu, recovering hidden functions from accumulated totals."], "text": "Answer: It flips divisor sums via mu, recovering hidden functions from accumulated totals.\nSteps:\n1. It flips divisor sums via mu, recovering hidden functions from accumulated totals.\n2. So the answer stands: It flips divisor sums via mu, recovering hidden functions from accumulated totals."}
{"id": "mat_0852", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "worked_example", "question": "Ukaž výpočet C(9,3).", "choices": null, "answer": "C(9,3) = 84.", "steps": ["n(n−1)(n−2)/6.", "9×8×7/6.", "= 84."], "text": "Answer: C(9,3) = 84.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 9×8×7/6.\n3. = 84."}
{"id": "mat_0853", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 11.", "choices": {"A": "130", "B": "134", "C": "132", "D": "136"}, "answer": "The value is 132.", "steps": ["11² + 11.", "Value = 132.", "Select option C."], "text": "Answer: 132\nSteps:\n1. 11² + 11.\n2. Value = 132.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0854", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Vyčísli integrál 3x² od 0 do 6.", "choices": null, "answer": "Integrál: 216.", "steps": ["Primitivní: x³.", "Vyčísli 0 až 6.", "6³ − 0 = 216."], "text": "Answer: Integrál: 216.\nSteps:\n1. Primitivní: x³.\n2. Vyčísli 0 až 6.\n3. 6³ − 0 = 216."}
{"id": "mat_0855", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej stopu matice [[3,1],[2,6]].", "choices": {"A": "7", "B": "11", "C": "9", "D": "13"}, "answer": "Stopa je 9.", "steps": ["3 + 6.", "Stopa = 9.", "Vyber možnost C."], "text": "Answer: 9\nSteps:\n1. 3 + 6.\n2. Stopa = 9.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0856", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..12}.", "choices": {"A": "7.5", "B": "6.5", "C": "5.5", "D": "8.5"}, "answer": "The expected value is 6.5.", "steps": ["(12+1)/2.", "E = 6.5.", "Select option B."], "text": "Answer: 6.5\nSteps:\n1. (12+1)/2.\n2. E = 6.5.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0857", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(68, 102) step by step.", "choices": null, "answer": "gcd = 34.", "steps": ["Factor: 68=4×17, 102=6×17.", "Common: 17×gcd(4,6).", "= 17×2 = 34."], "text": "Answer: gcd = 34.\nSteps:\n1. Factor: 68=4×17, 102=6×17.\n2. Common: 17×gcd(4,6).\n3. = 17×2 = 34."}
{"id": "mat_0858", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(10,2).", "choices": {"A": "90", "B": "80", "C": "100", "D": "110"}, "answer": "P(10,2) is 90.", "steps": ["10×9.", "= 90.", "Select option A."], "text": "Answer: 90\nSteps:\n1. 10×9.\n2. = 90.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0859", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 21.", "choices": null, "answer": "Sum: 231.", "steps": ["Pair 1+21, 2+20, ...", "Use 21×22/2.", "= 231."], "text": "Answer: Sum: 231.\nSteps:\n1. Pair 1+21, 2+20, ...\n2. Use 21×22/2.\n3. = 231."}
{"id": "mat_0860", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 18²)/(x − 18) as x → 18.", "choices": null, "answer": "Limit: 36.", "steps": ["Factor: (x−18)(x+18).", "Cancel (x−18).", "Plug 18: 18+18 = 36."], "text": "Answer: Limit: 36.\nSteps:\n1. Factor: (x−18)(x+18).\n2. Cancel (x−18).\n3. Plug 18: 18+18 = 36."}
{"id": "mat_0861", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[18,1],[2,19]] step by step.", "choices": null, "answer": "Determinant: 340.", "steps": ["ad − bc.", "18×19 − 1×2.", "= 340."], "text": "Answer: Determinant: 340.\nSteps:\n1. ad − bc.\n2. 18×19 − 1×2.\n3. = 340."}
{"id": "mat_0862", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why conjugate priors?", "choices": null, "answer": "They keep posteriors in one family, making Bayesian updates algebraic and fast.", "steps": ["They keep posteriors in one family, making Bayesian updates algebraic and fast.", "So the answer stands: They keep posteriors in one family, making Bayesian updates algebraic and fast."], "text": "Answer: They keep posteriors in one family, making Bayesian updates algebraic and fast.\nSteps:\n1. They keep posteriors in one family, making Bayesian updates algebraic and fast.\n2. So the answer stands: They keep posteriors in one family, making Bayesian updates algebraic and fast."}
{"id": "mat_0863", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (215) mod 7.", "choices": null, "answer": "Remainder: 5.", "steps": ["Divide 215 by 7.", "7×30 = 210.", "215 − 210 = 5."], "text": "Answer: Remainder: 5.\nSteps:\n1. Divide 215 by 7.\n2. 7×30 = 210.\n3. 215 − 210 = 5."}
{"id": "mat_0864", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(20,2).", "choices": null, "answer": "C(20,2) = 190.", "steps": ["Formula n(n−1)/2.", "20×19/2.", "= 190."], "text": "Answer: C(20,2) = 190.\nSteps:\n1. Formula n(n−1)/2.\n2. 20×19/2.\n3. = 190."}
{"id": "mat_0865", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 17 by computing its half.", "choices": null, "answer": "Half: 153, so 306 is even.", "steps": ["17² + 17 = 306.", "306 / 2 = 153.", "Integer, so even."], "text": "Answer: Half: 153, so 306 is even.\nSteps:\n1. 17² + 17 = 306.\n2. 306 / 2 = 153.\n3. Integer, so even."}
{"id": "mat_0866", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 3x² + 17x, compute f'(3) step by step.", "choices": null, "answer": "f'(3) = 35.", "steps": ["f'(x) = 6x + 17.", "Plug x = 3.", "6×3 + 17 = 35."], "text": "Answer: f'(3) = 35.\nSteps:\n1. f'(x) = 6x + 17.\n2. Plug x = 3.\n3. 6×3 + 17 = 35."}
{"id": "mat_0867", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[19,0],[0,20]].", "choices": null, "answer": "Trace: 39.", "steps": ["Sum diagonal.", "19 + 20.", "= 39."], "text": "Answer: Trace: 39.\nSteps:\n1. Sum diagonal.\n2. 19 + 20.\n3. = 39."}
{"id": "mat_0868", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 18 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 18/22.", "steps": ["Red: 18.", "Total: 22.", "P = 18/22."], "text": "Answer: P(red) = 18/22.\nSteps:\n1. Red: 18.\n2. Total: 22.\n3. P = 18/22."}
{"id": "mat_0869", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "worked_example", "question": "Spočítej NSN(8, 8) krok za krokem.", "choices": null, "answer": "NSN = 8.", "steps": ["NSD(8,8) = 8.", "8×8/8.", "= 8."], "text": "Answer: NSN = 8.\nSteps:\n1. NSD(8,8) = 8.\n2. 8×8/8.\n3. = 8."}
{"id": "mat_0870", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(19,3).", "choices": null, "answer": "P(19,3) = 5814.", "steps": ["n(n−1)(n−2).", "19×18×17.", "= 5814."], "text": "Answer: P(19,3) = 5814.\nSteps:\n1. n(n−1)(n−2).\n2. 19×18×17.\n3. = 5814."}
{"id": "mat_0871", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why use proof assistants?", "choices": null, "answer": "Proof assistants check every step mechanically, catching human errors in long complex proofs.", "steps": ["Proof assistants check every step mechanically, catching human errors in long complex proofs.", "So the answer stands: Proof assistants check every step mechanically, catching human errors in long complex proofs."], "text": "Answer: Proof assistants check every step mechanically, catching human errors in long complex proofs.\nSteps:\n1. Proof assistants check every step mechanically, catching human errors in long complex proofs.\n2. So the answer stands: Proof assistants check every step mechanically, catching human errors in long complex proofs."}
{"id": "mat_0872", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why study line integrals?", "choices": null, "answer": "Line integrals accumulate fields along paths, computing work done by forces exactly.", "steps": ["Line integrals accumulate fields along paths, computing work done by forces exactly.", "So the answer stands: Line integrals accumulate fields along paths, computing work done by forces exactly."], "text": "Answer: Line integrals accumulate fields along paths, computing work done by forces exactly.\nSteps:\n1. Line integrals accumulate fields along paths, computing work done by forces exactly.\n2. So the answer stands: Line integrals accumulate fields along paths, computing work done by forces exactly."}
{"id": "mat_0873", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute (18,17) · (3,5) step by step.", "choices": null, "answer": "Dot product: 139.", "steps": ["Multiply components.", "18×3 + 17×5.", "= 139."], "text": "Answer: Dot product: 139.\nSteps:\n1. Multiply components.\n2. 18×3 + 17×5.\n3. = 139."}
{"id": "mat_0874", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is maximum likelihood?", "choices": null, "answer": "It picks parameters making observed data most probable, the core fitting principle.", "steps": ["It picks parameters making observed data most probable, the core fitting principle.", "So the answer stands: It picks parameters making observed data most probable, the core fitting principle."], "text": "Answer: It picks parameters making observed data most probable, the core fitting principle.\nSteps:\n1. It picks parameters making observed data most probable, the core fitting principle.\n2. So the answer stands: It picks parameters making observed data most probable, the core fitting principle."}
{"id": "mat_0875", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why Dirichlet series?", "choices": null, "answer": "They encode arithmetic functions as series, linking numbers to complex analysis.", "steps": ["They encode arithmetic functions as series, linking numbers to complex analysis.", "So the answer stands: They encode arithmetic functions as series, linking numbers to complex analysis."], "text": "Answer: They encode arithmetic functions as series, linking numbers to complex analysis.\nSteps:\n1. They encode arithmetic functions as series, linking numbers to complex analysis.\n2. So the answer stands: They encode arithmetic functions as series, linking numbers to complex analysis."}
{"id": "mat_0876", "subject": "math", "topic": "combinatorics", "lang": "cs", "kind": "concept", "question": "Co je rekurence?", "choices": null, "answer": "Rekurence definuje členy z předchozích.", "steps": ["Rekurence definuje členy z předchozích.", "Odpověď tedy platí: Rekurence definuje členy z předchozích."], "text": "Answer: Rekurence definuje členy z předchozích.\nSteps:\n1. Rekurence definuje členy z předchozích.\n2. Odpověď tedy platí: Rekurence definuje členy z předchozích."}
{"id": "mat_0877", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is mathematical rigor?", "choices": null, "answer": "Rigor means every step is justified, leaving no gaps between assumptions and conclusion.", "steps": ["Rigor means every step is justified, leaving no gaps between assumptions and conclusion.", "So the answer stands: Rigor means every step is justified, leaving no gaps between assumptions and conclusion."], "text": "Answer: Rigor means every step is justified, leaving no gaps between assumptions and conclusion.\nSteps:\n1. Rigor means every step is justified, leaving no gaps between assumptions and conclusion.\n2. So the answer stands: Rigor means every step is justified, leaving no gaps between assumptions and conclusion."}
{"id": "mat_0878", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Vyčísli limitu (x² − 6²)/(x − 6) pro x → 6.", "choices": null, "answer": "Limita: 12.", "steps": ["Rozlož: (x−6)(x+6).", "Pokrať (x−6).", "Dosaď 6: 6+6 = 12."], "text": "Answer: Limita: 12.\nSteps:\n1. Rozlož: (x−6)(x+6).\n2. Pokrať (x−6).\n3. Dosaď 6: 6+6 = 12."}
{"id": "mat_0879", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[9,2],[3,12]].", "choices": {"A": "102", "B": "99", "C": "105", "D": "108"}, "answer": "The determinant is 102.", "steps": ["9×12 − 2×3.", "Det = 102.", "Select option A."], "text": "Answer: 102\nSteps:\n1. 9×12 − 2×3.\n2. Det = 102.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0880", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Spočítej E rovnoměrného tahu z {1..8}.", "choices": null, "answer": "E = 4,5.", "steps": ["Průměr 1..8.", "(8+1)/2.", "E = 4,5."], "text": "Answer: E = 4,5.\nSteps:\n1. Průměr 1..8.\n2. (8+1)/2.\n3. E = 4,5."}
{"id": "mat_0881", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(60, 90).", "choices": {"A": "30", "B": "27", "C": "33", "D": "36"}, "answer": "The gcd is 30.", "steps": ["Factor 3 out twice.", "gcd = 30.", "Select option A."], "text": "Answer: 30\nSteps:\n1. Factor 3 out twice.\n2. gcd = 30.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0882", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is the twelvefold way?", "choices": null, "answer": "It classifies counting by labelled/unlabelled balls and boxes into twelve cases.", "steps": ["It classifies counting by labelled/unlabelled balls and boxes into twelve cases.", "So the answer stands: It classifies counting by labelled/unlabelled balls and boxes into twelve cases."], "text": "Answer: It classifies counting by labelled/unlabelled balls and boxes into twelve cases.\nSteps:\n1. It classifies counting by labelled/unlabelled balls and boxes into twelve cases.\n2. So the answer stands: It classifies counting by labelled/unlabelled balls and boxes into twelve cases."}
{"id": "mat_0883", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 18 boolean variables. How many rows?", "choices": null, "answer": "Rows: 262144.", "steps": ["2 choices each.", "2^18 = 262144."], "text": "Answer: Rows: 262144.\nSteps:\n1. 2 choices each.\n2. 2^18 = 262144."}
{"id": "mat_0884", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is Green's theorem?", "choices": null, "answer": "It converts a loop integral into a region integral, trading boundary work for interior sums.", "steps": ["It converts a loop integral into a region integral, trading boundary work for interior sums.", "So the answer stands: It converts a loop integral into a region integral, trading boundary work for interior sums."], "text": "Answer: It converts a loop integral into a region integral, trading boundary work for interior sums.\nSteps:\n1. It converts a loop integral into a region integral, trading boundary work for interior sums.\n2. So the answer stands: It converts a loop integral into a region integral, trading boundary work for interior sums."}
{"id": "mat_0885", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[10,1],[2,13]].", "choices": {"A": "21", "B": "23", "C": "25", "D": "27"}, "answer": "The trace is 23.", "steps": ["10 + 13.", "Trace = 23.", "Select option B."], "text": "Answer: 23\nSteps:\n1. 10 + 13.\n2. Trace = 23.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0886", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why unbiased estimators?", "choices": null, "answer": "Unbiased estimators average to the truth, avoiding systematic error across samples.", "steps": ["Unbiased estimators average to the truth, avoiding systematic error across samples.", "So the answer stands: Unbiased estimators average to the truth, avoiding systematic error across samples."], "text": "Answer: Unbiased estimators average to the truth, avoiding systematic error across samples.\nSteps:\n1. Unbiased estimators average to the truth, avoiding systematic error across samples.\n2. So the answer stands: Unbiased estimators average to the truth, avoiding systematic error across samples."}
{"id": "mat_0887", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is the Riemann zeta?", "choices": null, "answer": "zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations.", "steps": ["zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations.", "So the answer stands: zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations."], "text": "Answer: zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations.\nSteps:\n1. zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations.\n2. So the answer stands: zeta(s) sums 1/n^s, and its zeros conjecturally rule prime fluctuations."}
{"id": "mat_0888", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(21,3).", "choices": null, "answer": "C(21,3) = 1330.", "steps": ["n(n−1)(n−2)/6.", "21×20×19/6.", "= 1330."], "text": "Answer: C(21,3) = 1330.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 21×20×19/6.\n3. = 1330."}
{"id": "mat_0889", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "concept", "question": "Proč pravdivostní tabulky?", "choices": null, "answer": "Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky.", "steps": ["Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky.", "Odpověď tedy platí: Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky."], "text": "Answer: Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky.\nSteps:\n1. Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky.\n2. Odpověď tedy platí: Tabulky vypisují všechny kombinace hodnot a ověřují ekvivalence mechanicky."}
{"id": "mat_0890", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why approximate integrals?", "choices": null, "answer": "Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error.", "steps": ["Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error.", "So the answer stands: Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error."], "text": "Answer: Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error.\nSteps:\n1. Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error.\n2. So the answer stands: Numerical rules like Simpson's estimate integrals lacking formulas, with controlled error."}
{"id": "mat_0891", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (9,10) · (3,4).", "choices": {"A": "63", "B": "71", "C": "67", "D": "75"}, "answer": "The dot product is 67.", "steps": ["9×3 + 10×4.", "Product = 67.", "Select option C."], "text": "Answer: 67\nSteps:\n1. 9×3 + 10×4.\n2. Product = 67.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0892", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 10 red and 21 blue marbles. What is P(red)?", "choices": {"A": "21/31", "B": "10/32", "C": "10/31", "D": "11/31"}, "answer": "P(red) is 10/31.", "steps": ["10 of 31.", "P = 10/31.", "Select option C."], "text": "Answer: 10/31\nSteps:\n1. 10 of 31.\n2. P = 10/31.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0893", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why continued fractions?", "choices": null, "answer": "They give best rational approximations, exposing hidden structure of irrationals.", "steps": ["They give best rational approximations, exposing hidden structure of irrationals.", "So the answer stands: They give best rational approximations, exposing hidden structure of irrationals."], "text": "Answer: They give best rational approximations, exposing hidden structure of irrationals.\nSteps:\n1. They give best rational approximations, exposing hidden structure of irrationals.\n2. So the answer stands: They give best rational approximations, exposing hidden structure of irrationals."}
{"id": "mat_0894", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why parking functions?", "choices": null, "answer": "They count preference lists letting all cars park, equalling (n+1)^(n-1).", "steps": ["They count preference lists letting all cars park, equalling (n+1)^(n-1).", "So the answer stands: They count preference lists letting all cars park, equalling (n+1)^(n-1)."], "text": "Answer: They count preference lists letting all cars park, equalling (n+1)^(n-1).\nSteps:\n1. They count preference lists letting all cars park, equalling (n+1)^(n-1).\n2. So the answer stands: They count preference lists letting all cars park, equalling (n+1)^(n-1)."}
{"id": "mat_0895", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 22.", "choices": null, "answer": "Sum: 253.", "steps": ["Pair 1+22, 2+21, ...", "Use 22×23/2.", "= 253."], "text": "Answer: Sum: 253.\nSteps:\n1. Pair 1+22, 2+21, ...\n2. Use 22×23/2.\n3. = 253."}
{"id": "mat_0896", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is the mean value theorem?", "choices": null, "answer": "It guarantees a point where instantaneous and average rates match on every smooth interval.", "steps": ["It guarantees a point where instantaneous and average rates match on every smooth interval.", "So the answer stands: It guarantees a point where instantaneous and average rates match on every smooth interval."], "text": "Answer: It guarantees a point where instantaneous and average rates match on every smooth interval.\nSteps:\n1. It guarantees a point where instantaneous and average rates match on every smooth interval.\n2. So the answer stands: It guarantees a point where instantaneous and average rates match on every smooth interval."}
{"id": "mat_0897", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the determinant of [[10,2],[3,13]].", "choices": {"A": "121", "B": "124", "C": "127", "D": "130"}, "answer": "The determinant is 124.", "steps": ["10×13 − 2×3.", "Det = 124.", "Select option B."], "text": "Answer: 124\nSteps:\n1. 10×13 − 2×3.\n2. Det = 124.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0898", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is the birthday paradox?", "choices": null, "answer": "Just 23 people give even odds of a shared birthday, since pairs grow quadratically.", "steps": ["Just 23 people give even odds of a shared birthday, since pairs grow quadratically.", "So the answer stands: Just 23 people give even odds of a shared birthday, since pairs grow quadratically."], "text": "Answer: Just 23 people give even odds of a shared birthday, since pairs grow quadratically.\nSteps:\n1. Just 23 people give even odds of a shared birthday, since pairs grow quadratically.\n2. So the answer stands: Just 23 people give even odds of a shared birthday, since pairs grow quadratically."}
{"id": "mat_0899", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (156) mod 9.", "choices": {"A": "4", "B": "3", "C": "5", "D": "6"}, "answer": "The remainder is 3.", "steps": ["156 ÷ 9.", "Remainder 3.", "Select option B."], "text": "Answer: 3\nSteps:\n1. 156 ÷ 9.\n2. Remainder 3.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0900", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(21,2).", "choices": null, "answer": "C(21,2) = 210.", "steps": ["Formula n(n−1)/2.", "21×20/2.", "= 210."], "text": "Answer: C(21,2) = 210.\nSteps:\n1. Formula n(n−1)/2.\n2. 21×20/2.\n3. = 210."}
{"id": "mat_0901", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Kolik řádků má pravdivostní tabulka s 3 proměnnými?", "choices": {"A": "6", "B": "8", "C": "10", "D": "12"}, "answer": "Tabulka má 8 řádků.", "steps": ["2^3 řádků.", "Celkem: 8.", "Vyber možnost B."], "text": "Answer: 8\nSteps:\n1. 2^3 řádků.\n2. Celkem: 8.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0902", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why prove Rolle's theorem?", "choices": null, "answer": "Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem.", "steps": ["Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem.", "So the answer stands: Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem."], "text": "Answer: Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem.\nSteps:\n1. Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem.\n2. So the answer stands: Rolle's theorem finds flat tangents between equal values, anchoring the mean value theorem."}
{"id": "mat_0903", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej det [[6,1],[2,7]] krok za krokem.", "choices": null, "answer": "Determinant: 40.", "steps": ["ad − bc.", "6×7 − 1×2.", "= 40."], "text": "Answer: Determinant: 40.\nSteps:\n1. ad − bc.\n2. 6×7 − 1×2.\n3. = 40."}
{"id": "mat_0904", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why study probability?", "choices": null, "answer": "It quantifies uncertainty everywhere, from games to science to machine learning.", "steps": ["It quantifies uncertainty everywhere, from games to science to machine learning.", "So the answer stands: It quantifies uncertainty everywhere, from games to science to machine learning."], "text": "Answer: It quantifies uncertainty everywhere, from games to science to machine learning.\nSteps:\n1. It quantifies uncertainty everywhere, from games to science to machine learning.\n2. So the answer stands: It quantifies uncertainty everywhere, from games to science to machine learning."}
{"id": "mat_0905", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(20, 8) step by step.", "choices": null, "answer": "lcm = 40.", "steps": ["gcd(20,8) = 4.", "20×8/4.", "= 40."], "text": "Answer: lcm = 40.\nSteps:\n1. gcd(20,8) = 4.\n2. 20×8/4.\n3. = 40."}
{"id": "mat_0906", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(12,3).", "choices": {"A": "208", "B": "220", "C": "232", "D": "244"}, "answer": "C(12,3) is 220.", "steps": ["12×11×10/6.", "= 220.", "Select option B."], "text": "Answer: 220\nSteps:\n1. 12×11×10/6.\n2. = 220.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0907", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why read proofs actively?", "choices": null, "answer": "Active reading fills each step yourself, turning passive lines into owned understanding.", "steps": ["Active reading fills each step yourself, turning passive lines into owned understanding.", "So the answer stands: Active reading fills each step yourself, turning passive lines into owned understanding."], "text": "Answer: Active reading fills each step yourself, turning passive lines into owned understanding.\nSteps:\n1. Active reading fills each step yourself, turning passive lines into owned understanding.\n2. So the answer stands: Active reading fills each step yourself, turning passive lines into owned understanding."}
{"id": "mat_0908", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 10.", "choices": {"A": "990", "B": "1000", "C": "1010", "D": "1020"}, "answer": "The integral is 1000.", "steps": ["Antiderivative x³.", "10³ = 1000.", "Select option B."], "text": "Answer: 1000\nSteps:\n1. Antiderivative x³.\n2. 10³ = 1000.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0909", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Najdi skalární součin (2,3) · (3,4).", "choices": {"A": "14", "B": "22", "C": "26", "D": "18"}, "answer": "Skalární součin je 18.", "steps": ["2×3 + 3×4.", "Součin = 18.", "Vyber možnost D."], "text": "Answer: 18\nSteps:\n1. 2×3 + 3×4.\n2. Součin = 18.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0910", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co aktualizuje s evidencí?", "choices": {"A": "podmíněná pravděpodobnost", "B": "slepé hádání", "C": "tvrdohlavý prior", "D": "divoká naděje"}, "answer": "Odpověď je podmíněná pravděpodobnost.", "steps": ["Prostor se zužuje.", "Tedy: podmíněná pravděpodobnost.", "Vyber možnost A."], "text": "Answer: podmíněná pravděpodobnost\nSteps:\n1. Prostor se zužuje.\n2. Tedy: podmíněná pravděpodobnost.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0911", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is Pell's equation?", "choices": null, "answer": "x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D).", "steps": ["x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D).", "So the answer stands: x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D)."], "text": "Answer: x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D).\nSteps:\n1. x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D).\n2. So the answer stands: x^2 - Dy^2 = 1 has infinite integer solutions from continued fractions of sqrt(D)."}
{"id": "mat_0912", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(12,2).", "choices": {"A": "66", "B": "54", "C": "78", "D": "90"}, "answer": "C(12,2) is 66.", "steps": ["12×11/2.", "= 66.", "Select option A."], "text": "Answer: 66\nSteps:\n1. 12×11/2.\n2. = 66.\n3. Select option A.\nAnswer: (A)"}
{"id": "mat_0913", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Kolik je součet 1 + 2 + ... + 6?", "choices": {"A": "15", "B": "27", "C": "21", "D": "33"}, "answer": "Součet je 21.", "steps": ["6×7/2.", "Součet = 21.", "Vyber možnost C."], "text": "Answer: 21\nSteps:\n1. 6×7/2.\n2. Součet = 21.\n3. Vyber možnost C.\nAnswer: (C)"}
{"id": "mat_0914", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is uniform convergence?", "choices": null, "answer": "Uniform convergence controls error everywhere at once, letting limits pass through sums safely.", "steps": ["Uniform convergence controls error everywhere at once, letting limits pass through sums safely.", "So the answer stands: Uniform convergence controls error everywhere at once, letting limits pass through sums safely."], "text": "Answer: Uniform convergence controls error everywhere at once, letting limits pass through sums safely.\nSteps:\n1. Uniform convergence controls error everywhere at once, letting limits pass through sums safely.\n2. So the answer stands: Uniform convergence controls error everywhere at once, letting limits pass through sums safely."}
{"id": "mat_0915", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why SVD?", "choices": null, "answer": "Singular value decomposition splits any matrix into rotations and scalings, powering data science.", "steps": ["Singular value decomposition splits any matrix into rotations and scalings, powering data science.", "So the answer stands: Singular value decomposition splits any matrix into rotations and scalings, powering data science."], "text": "Answer: Singular value decomposition splits any matrix into rotations and scalings, powering data science.\nSteps:\n1. Singular value decomposition splits any matrix into rotations and scalings, powering data science.\n2. So the answer stands: Singular value decomposition splits any matrix into rotations and scalings, powering data science."}
{"id": "mat_0916", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 22-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/22.", "steps": ["Favourable: 1.", "Possible: 22.", "P = 1/22."], "text": "Answer: P = 1/22.\nSteps:\n1. Favourable: 1.\n2. Possible: 22.\n3. P = 1/22."}
{"id": "mat_0917", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why Gaussian integers?", "choices": null, "answer": "They extend unique factorization to the complex plane, solving sum-of-squares questions.", "steps": ["They extend unique factorization to the complex plane, solving sum-of-squares questions.", "So the answer stands: They extend unique factorization to the complex plane, solving sum-of-squares questions."], "text": "Answer: They extend unique factorization to the complex plane, solving sum-of-squares questions.\nSteps:\n1. They extend unique factorization to the complex plane, solving sum-of-squares questions.\n2. So the answer stands: They extend unique factorization to the complex plane, solving sum-of-squares questions."}
{"id": "mat_0918", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(20,3).", "choices": null, "answer": "P(20,3) = 6840.", "steps": ["n(n−1)(n−2).", "20×19×18.", "= 6840."], "text": "Answer: P(20,3) = 6840.\nSteps:\n1. n(n−1)(n−2).\n2. 20×19×18.\n3. = 6840."}
{"id": "mat_0919", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is a corollary?", "choices": null, "answer": "A corollary follows quickly from a theorem just proved, packaging its immediate consequences.", "steps": ["A corollary follows quickly from a theorem just proved, packaging its immediate consequences.", "So the answer stands: A corollary follows quickly from a theorem just proved, packaging its immediate consequences."], "text": "Answer: A corollary follows quickly from a theorem just proved, packaging its immediate consequences.\nSteps:\n1. A corollary follows quickly from a theorem just proved, packaging its immediate consequences.\n2. So the answer stands: A corollary follows quickly from a theorem just proved, packaging its immediate consequences."}
{"id": "mat_0920", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 19.", "choices": null, "answer": "Integral: 6859.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 19.", "19³ − 0 = 6859."], "text": "Answer: Integral: 6859.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 19.\n3. 19³ − 0 = 6859."}
{"id": "mat_0921", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "worked_example", "question": "Spočítej stopu matice [[7,0],[0,8]].", "choices": null, "answer": "Stopa: 15.", "steps": ["Sečti diagonálu.", "7 + 8.", "= 15."], "text": "Answer: Stopa: 15.\nSteps:\n1. Sečti diagonálu.\n2. 7 + 8.\n3. = 15."}
{"id": "mat_0922", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A fair 15-sided die is rolled. What is the probability of one specific number?", "choices": {"A": "1/14", "B": "1/15", "C": "1/16", "D": "2/15"}, "answer": "The probability is 1/15.", "steps": ["1 face of 15.", "P = 1/15.", "Select option B."], "text": "Answer: 1/15\nSteps:\n1. 1 face of 15.\n2. P = 1/15.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0923", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(72, 108) step by step.", "choices": null, "answer": "gcd = 36.", "steps": ["Factor: 72=4×18, 108=6×18.", "Common: 18×gcd(4,6).", "= 18×2 = 36."], "text": "Answer: gcd = 36.\nSteps:\n1. Factor: 72=4×18, 108=6×18.\n2. Common: 18×gcd(4,6).\n3. = 18×2 = 36."}
{"id": "mat_0924", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(11,2).", "choices": {"A": "99", "B": "110", "C": "121", "D": "132"}, "answer": "P(11,2) is 110.", "steps": ["11×10.", "= 110.", "Select option B."], "text": "Answer: 110\nSteps:\n1. 11×10.\n2. = 110.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0925", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 18 by computing its half.", "choices": null, "answer": "Half: 171, so 342 is even.", "steps": ["18² + 18 = 342.", "342 / 2 = 171.", "Integer, so even."], "text": "Answer: Half: 171, so 342 is even.\nSteps:\n1. 18² + 18 = 342.\n2. 342 / 2 = 171.\n3. Integer, so even."}
{"id": "mat_0926", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: kde se skrývají maxima?", "choices": {"A": "nikde jinde", "B": "i v koncových bodech", "C": "pouze v nule", "D": "daleko"}, "answer": "Odpověď je i v koncových bodech.", "steps": ["Konce kontroluj vždy.", "Tedy: i v koncových bodech.", "Vyber možnost B."], "text": "Answer: i v koncových bodech\nSteps:\n1. Konce kontroluj vždy.\n2. Tedy: i v koncových bodech.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0927", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is a norm?", "choices": null, "answer": "A norm measures vector length consistently, generalizing absolute value to many dimensions.", "steps": ["A norm measures vector length consistently, generalizing absolute value to many dimensions.", "So the answer stands: A norm measures vector length consistently, generalizing absolute value to many dimensions."], "text": "Answer: A norm measures vector length consistently, generalizing absolute value to many dimensions.\nSteps:\n1. A norm measures vector length consistently, generalizing absolute value to many dimensions.\n2. So the answer stands: A norm measures vector length consistently, generalizing absolute value to many dimensions."}
{"id": "mat_0928", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..21}.", "choices": null, "answer": "E = 11.", "steps": ["Mean of 1..21.", "(21+1)/2.", "E = 11."], "text": "Answer: E = 11.\nSteps:\n1. Mean of 1..21.\n2. (21+1)/2.\n3. E = 11."}
{"id": "mat_0929", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(12, 6).", "choices": {"A": "10", "B": "14", "C": "12", "D": "16"}, "answer": "The lcm is 12.", "steps": ["12×6/gcd.", "lcm = 12.", "Select option C."], "text": "Answer: 12\nSteps:\n1. 12×6/gcd.\n2. lcm = 12.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0930", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(13,3).", "choices": {"A": "273", "B": "299", "C": "286", "D": "312"}, "answer": "C(13,3) is 286.", "steps": ["13×12×11/6.", "= 286.", "Select option C."], "text": "Answer: 286\nSteps:\n1. 13×12×11/6.\n2. = 286.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0931", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "How many rows does a truth table with 11 variables have?", "choices": {"A": "2046", "B": "2048", "C": "2050", "D": "2052"}, "answer": "The table has 2048 rows.", "steps": ["2^11 rows.", "Total: 2048.", "Select option B."], "text": "Answer: 2048\nSteps:\n1. 2^11 rows.\n2. Total: 2048.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0932", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why study power series?", "choices": null, "answer": "Power series define functions by coefficients, and radius tells where the definition works.", "steps": ["Power series define functions by coefficients, and radius tells where the definition works.", "So the answer stands: Power series define functions by coefficients, and radius tells where the definition works."], "text": "Answer: Power series define functions by coefficients, and radius tells where the definition works.\nSteps:\n1. Power series define functions by coefficients, and radius tells where the definition works.\n2. So the answer stands: Power series define functions by coefficients, and radius tells where the definition works."}
{"id": "mat_0933", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the trace of [[11,1],[2,14]].", "choices": {"A": "23", "B": "27", "C": "25", "D": "29"}, "answer": "The trace is 25.", "steps": ["11 + 14.", "Trace = 25.", "Select option C."], "text": "Answer: 25\nSteps:\n1. 11 + 14.\n2. Trace = 25.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0934", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Sáček má 6 červených a 4 modré kuličky. Ukaž P(červená).", "choices": null, "answer": "P(červená) = 6/10.", "steps": ["Červené: 6.", "Celkem: 10.", "P = 6/10."], "text": "Answer: P(červená) = 6/10.\nSteps:\n1. Červené: 6.\n2. Celkem: 10.\n3. P = 6/10."}
{"id": "mat_0935", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find gcd(66, 99).", "choices": {"A": "30", "B": "33", "C": "36", "D": "39"}, "answer": "The gcd is 33.", "steps": ["Factor 3 out twice.", "gcd = 33.", "Select option B."], "text": "Answer: 33\nSteps:\n1. Factor 3 out twice.\n2. gcd = 33.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0936", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is Sperner's lemma?", "choices": null, "answer": "It finds fully labelled small triangles, implying Brouwer fixed points combinatorially.", "steps": ["It finds fully labelled small triangles, implying Brouwer fixed points combinatorially.", "So the answer stands: It finds fully labelled small triangles, implying Brouwer fixed points combinatorially."], "text": "Answer: It finds fully labelled small triangles, implying Brouwer fixed points combinatorially.\nSteps:\n1. It finds fully labelled small triangles, implying Brouwer fixed points combinatorially.\n2. So the answer stands: It finds fully labelled small triangles, implying Brouwer fixed points combinatorially."}
{"id": "mat_0937", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "What is the sum 1 + 2 + ... + 14?", "choices": {"A": "91", "B": "119", "C": "105", "D": "133"}, "answer": "The sum is 105.", "steps": ["14×15/2.", "Sum = 105.", "Select option C."], "text": "Answer: 105\nSteps:\n1. 14×15/2.\n2. Sum = 105.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0938", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the limit of (x² − 19²)/(x − 19) as x → 19.", "choices": null, "answer": "Limit: 38.", "steps": ["Factor: (x−19)(x+19).", "Cancel (x−19).", "Plug 19: 19+19 = 38."], "text": "Answer: Limit: 38.\nSteps:\n1. Factor: (x−19)(x+19).\n2. Cancel (x−19).\n3. Plug 19: 19+19 = 38."}
{"id": "mat_0939", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why condition numbers?", "choices": null, "answer": "Condition numbers bound error blow-up, warning when systems are numerically fragile.", "steps": ["Condition numbers bound error blow-up, warning when systems are numerically fragile.", "So the answer stands: Condition numbers bound error blow-up, warning when systems are numerically fragile."], "text": "Answer: Condition numbers bound error blow-up, warning when systems are numerically fragile.\nSteps:\n1. Condition numbers bound error blow-up, warning when systems are numerically fragile.\n2. So the answer stands: Condition numbers bound error blow-up, warning when systems are numerically fragile."}
{"id": "mat_0940", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "Find the expected value of a uniform draw from {1..13}.", "choices": {"A": "8", "B": "6", "C": "7", "D": "9"}, "answer": "The expected value is 7.", "steps": ["(13+1)/2.", "E = 7.", "Select option C."], "text": "Answer: 7\nSteps:\n1. (13+1)/2.\n2. E = 7.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0941", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find (163) mod 9.", "choices": {"A": "2", "B": "3", "C": "1", "D": "4"}, "answer": "The remainder is 1.", "steps": ["163 ÷ 9.", "Remainder 1.", "Select option C."], "text": "Answer: 1\nSteps:\n1. 163 ÷ 9.\n2. Remainder 1.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0942", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(13,2).", "choices": {"A": "65", "B": "78", "C": "91", "D": "104"}, "answer": "C(13,2) is 78.", "steps": ["13×12/2.", "= 78.", "Select option B."], "text": "Answer: 78\nSteps:\n1. 13×12/2.\n2. = 78.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0943", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why number equations?", "choices": null, "answer": "Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable.", "steps": ["Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable.", "So the answer stands: Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable."], "text": "Answer: Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable.\nSteps:\n1. Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable.\n2. So the answer stands: Numbered equations let later steps cite exact lines, keeping long proofs navigable and checkable."}
{"id": "mat_0944", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 10²)/(x − 10) as x → 10.", "choices": {"A": "18", "B": "22", "C": "20", "D": "24"}, "answer": "The limit is 20.", "steps": ["Factor (x−10)(x+10).", "Cancel, get 2×10 = 20.", "Select option C."], "text": "Answer: 20\nSteps:\n1. Factor (x−10)(x+10).\n2. Cancel, get 2×10 = 20.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0945", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is LU decomposition?", "choices": null, "answer": "LU splits a matrix into lower and upper triangles, speeding repeated solves hugely.", "steps": ["LU splits a matrix into lower and upper triangles, speeding repeated solves hugely.", "So the answer stands: LU splits a matrix into lower and upper triangles, speeding repeated solves hugely."], "text": "Answer: LU splits a matrix into lower and upper triangles, speeding repeated solves hugely.\nSteps:\n1. LU splits a matrix into lower and upper triangles, speeding repeated solves hugely.\n2. So the answer stands: LU splits a matrix into lower and upper triangles, speeding repeated solves hugely."}
{"id": "mat_0946", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A bag holds 19 red and 4 blue marbles. Show P(red).", "choices": null, "answer": "P(red) = 19/23.", "steps": ["Red: 19.", "Total: 23.", "P = 19/23."], "text": "Answer: P(red) = 19/23.\nSteps:\n1. Red: 19.\n2. Total: 23.\n3. P = 19/23."}
{"id": "mat_0947", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (226) mod 7.", "choices": null, "answer": "Remainder: 2.", "steps": ["Divide 226 by 7.", "7×32 = 224.", "226 − 224 = 2."], "text": "Answer: Remainder: 2.\nSteps:\n1. Divide 226 by 7.\n2. 7×32 = 224.\n3. 226 − 224 = 2."}
{"id": "mat_0948", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(22,3).", "choices": null, "answer": "C(22,3) = 1540.", "steps": ["n(n−1)(n−2)/6.", "22×21×20/6.", "= 1540."], "text": "Answer: C(22,3) = 1540.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 22×21×20/6.\n3. = 1540."}
{"id": "mat_0949", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "A proof checks all cases of 19 boolean variables. How many rows?", "choices": null, "answer": "Rows: 524288.", "steps": ["2 choices each.", "2^19 = 524288."], "text": "Answer: Rows: 524288.\nSteps:\n1. 2 choices each.\n2. 2^19 = 524288."}
{"id": "mat_0950", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "What is analytic?", "choices": null, "answer": "Analytic functions equal their Taylor series nearby, making them smooth and predictable.", "steps": ["Analytic functions equal their Taylor series nearby, making them smooth and predictable.", "So the answer stands: Analytic functions equal their Taylor series nearby, making them smooth and predictable."], "text": "Answer: Analytic functions equal their Taylor series nearby, making them smooth and predictable.\nSteps:\n1. Analytic functions equal their Taylor series nearby, making them smooth and predictable.\n2. So the answer stands: Analytic functions equal their Taylor series nearby, making them smooth and predictable."}
{"id": "mat_0951", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why iterative methods?", "choices": null, "answer": "Iteration approximates huge systems stepwise, beating direct methods on sparse giants.", "steps": ["Iteration approximates huge systems stepwise, beating direct methods on sparse giants.", "So the answer stands: Iteration approximates huge systems stepwise, beating direct methods on sparse giants."], "text": "Answer: Iteration approximates huge systems stepwise, beating direct methods on sparse giants.\nSteps:\n1. Iteration approximates huge systems stepwise, beating direct methods on sparse giants.\n2. So the answer stands: Iteration approximates huge systems stepwise, beating direct methods on sparse giants."}
{"id": "mat_0952", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "What is risk?", "choices": null, "answer": "Risk combines probability with impact, guiding decisions under uncertainty rationally.", "steps": ["Risk combines probability with impact, guiding decisions under uncertainty rationally.", "So the answer stands: Risk combines probability with impact, guiding decisions under uncertainty rationally."], "text": "Answer: Risk combines probability with impact, guiding decisions under uncertainty rationally.\nSteps:\n1. Risk combines probability with impact, guiding decisions under uncertainty rationally.\n2. So the answer stands: Risk combines probability with impact, guiding decisions under uncertainty rationally."}
{"id": "mat_0953", "subject": "math", "topic": "number_theory", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co hledá celočíselná řešení?", "choices": {"A": "reálné hádání", "B": "diofantická rovnice", "C": "komplexní naděje", "D": "mlhavé přání"}, "answer": "Odpověď je diofantická rovnice.", "steps": ["Podmínky NSD platí.", "Tedy: diofantická rovnice.", "Vyber možnost B."], "text": "Answer: diofantická rovnice\nSteps:\n1. Podmínky NSD platí.\n2. Tedy: diofantická rovnice.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0954", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why poker counting?", "choices": null, "answer": "Hand ranks need exact counts, and nCr ratios give true poker probabilities.", "steps": ["Hand ranks need exact counts, and nCr ratios give true poker probabilities.", "So the answer stands: Hand ranks need exact counts, and nCr ratios give true poker probabilities."], "text": "Answer: Hand ranks need exact counts, and nCr ratios give true poker probabilities.\nSteps:\n1. Hand ranks need exact counts, and nCr ratios give true poker probabilities.\n2. So the answer stands: Hand ranks need exact counts, and nCr ratios give true poker probabilities."}
{"id": "mat_0955", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "What is QED?", "choices": null, "answer": "QED marks a proof's end, signalling the claim now rests on justified steps alone.", "steps": ["QED marks a proof's end, signalling the claim now rests on justified steps alone.", "So the answer stands: QED marks a proof's end, signalling the claim now rests on justified steps alone."], "text": "Answer: QED marks a proof's end, signalling the claim now rests on justified steps alone.\nSteps:\n1. QED marks a proof's end, signalling the claim now rests on justified steps alone.\n2. So the answer stands: QED marks a proof's end, signalling the claim now rests on justified steps alone."}
{"id": "mat_0956", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "For f(x) = 4x² + 18x, compute f'(4) step by step.", "choices": null, "answer": "f'(4) = 50.", "steps": ["f'(x) = 8x + 18.", "Plug x = 4.", "8×4 + 18 = 50."], "text": "Answer: f'(4) = 50.\nSteps:\n1. f'(x) = 8x + 18.\n2. Plug x = 4.\n3. 8×4 + 18 = 50."}
{"id": "mat_0957", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute det [[19,1],[2,20]] step by step.", "choices": null, "answer": "Determinant: 378.", "steps": ["ad − bc.", "19×20 − 1×2.", "= 378."], "text": "Answer: Determinant: 378.\nSteps:\n1. ad − bc.\n2. 19×20 − 1×2.\n3. = 378."}
{"id": "mat_0958", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "A fair 23-sided die is rolled. Show P(one specific number).", "choices": null, "answer": "P = 1/23.", "steps": ["Favourable: 1.", "Possible: 23.", "P = 1/23."], "text": "Answer: P = 1/23.\nSteps:\n1. Favourable: 1.\n2. Possible: 23.\n3. P = 1/23."}
{"id": "mat_0959", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "What is a congruence class?", "choices": null, "answer": "It groups numbers sharing one remainder, turning infinite integers into finite clocks.", "steps": ["It groups numbers sharing one remainder, turning infinite integers into finite clocks.", "So the answer stands: It groups numbers sharing one remainder, turning infinite integers into finite clocks."], "text": "Answer: It groups numbers sharing one remainder, turning infinite integers into finite clocks.\nSteps:\n1. It groups numbers sharing one remainder, turning infinite integers into finite clocks.\n2. So the answer stands: It groups numbers sharing one remainder, turning infinite integers into finite clocks."}
{"id": "mat_0960", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute P(12,2).", "choices": {"A": "120", "B": "144", "C": "132", "D": "156"}, "answer": "P(12,2) is 132.", "steps": ["12×11.", "= 132.", "Select option C."], "text": "Answer: 132\nSteps:\n1. 12×11.\n2. = 132.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0961", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Vypočítej N² + N pro N = 4.", "choices": {"A": "18", "B": "22", "C": "24", "D": "20"}, "answer": "Hodnota je 20.", "steps": ["4² + 4.", "Hodnota = 20.", "Vyber možnost D."], "text": "Answer: 20\nSteps:\n1. 4² + 4.\n2. Hodnota = 20.\n3. Vyber možnost D.\nAnswer: (D)"}
{"id": "mat_0962", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "For f(x) = 1x² + 11x, find f'(2).", "choices": {"A": "13", "B": "15", "C": "17", "D": "19"}, "answer": "The derivative is 15.", "steps": ["f'(x) = 2x + 11.", "At 2: 15.", "Select option B."], "text": "Answer: 15\nSteps:\n1. f'(x) = 2x + 11.\n2. At 2: 15.\n3. Select option B.\nAnswer: (B)"}
{"id": "mat_0963", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co počítá dimenzi obrazu?", "choices": {"A": "hodnost", "B": "velikost", "C": "barva", "D": "stáří"}, "answer": "Odpověď je hodnost.", "steps": ["Nezávislé řádky sečteny.", "Tedy: hodnost.", "Vyber možnost A."], "text": "Answer: hodnost\nSteps:\n1. Nezávislé řádky sečteny.\n2. Tedy: hodnost.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0964", "subject": "math", "topic": "probability", "lang": "en", "kind": "concept", "question": "Why insurance works?", "choices": null, "answer": "Pooling independent risks averages losses predictably, letting premiums cover claims.", "steps": ["Pooling independent risks averages losses predictably, letting premiums cover claims.", "So the answer stands: Pooling independent risks averages losses predictably, letting premiums cover claims."], "text": "Answer: Pooling independent risks averages losses predictably, letting premiums cover claims.\nSteps:\n1. Pooling independent risks averages losses predictably, letting premiums cover claims.\n2. So the answer stands: Pooling independent risks averages losses predictably, letting premiums cover claims."}
{"id": "mat_0965", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "concept", "question": "Why study number theory?", "choices": null, "answer": "It secures the internet and charms minds, blending pure beauty with applied power.", "steps": ["It secures the internet and charms minds, blending pure beauty with applied power.", "So the answer stands: It secures the internet and charms minds, blending pure beauty with applied power."], "text": "Answer: It secures the internet and charms minds, blending pure beauty with applied power.\nSteps:\n1. It secures the internet and charms minds, blending pure beauty with applied power.\n2. So the answer stands: It secures the internet and charms minds, blending pure beauty with applied power."}
{"id": "mat_0966", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "What is a Gray code?", "choices": null, "answer": "It orders binary strings flipping one bit at a time, minimizing switch errors.", "steps": ["It orders binary strings flipping one bit at a time, minimizing switch errors.", "So the answer stands: It orders binary strings flipping one bit at a time, minimizing switch errors."], "text": "Answer: It orders binary strings flipping one bit at a time, minimizing switch errors.\nSteps:\n1. It orders binary strings flipping one bit at a time, minimizing switch errors.\n2. So the answer stands: It orders binary strings flipping one bit at a time, minimizing switch errors."}
{"id": "mat_0967", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Prove by formula: find the sum 1 + 2 + ... + 23.", "choices": null, "answer": "Sum: 276.", "steps": ["Pair 1+23, 2+22, ...", "Use 23×24/2.", "= 276."], "text": "Answer: Sum: 276.\nSteps:\n1. Pair 1+23, 2+22, ...\n2. Use 23×24/2.\n3. = 276."}
{"id": "mat_0968", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Pro f(x) = 3x² + 5x spočítej f'(3) krok za krokem.", "choices": null, "answer": "f'(3) = 23.", "steps": ["f'(x) = 6x + 5.", "Dosaď x = 3.", "6×3 + 5 = 23."], "text": "Answer: f'(3) = 23.\nSteps:\n1. f'(x) = 6x + 5.\n2. Dosaď x = 3.\n3. 6×3 + 5 = 23."}
{"id": "mat_0969", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "What is a linear combination?", "choices": null, "answer": "It mixes vectors with scalar weights, and all mixes form the spanned subspace.", "steps": ["It mixes vectors with scalar weights, and all mixes form the spanned subspace.", "So the answer stands: It mixes vectors with scalar weights, and all mixes form the spanned subspace."], "text": "Answer: It mixes vectors with scalar weights, and all mixes form the spanned subspace.\nSteps:\n1. It mixes vectors with scalar weights, and all mixes form the spanned subspace.\n2. So the answer stands: It mixes vectors with scalar weights, and all mixes form the spanned subspace."}
{"id": "mat_0970", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Hází se férovou 10-stěnnou kostkou. Ukaž P(jedno konkrétní číslo).", "choices": null, "answer": "P = 1/10.", "steps": ["Příznivé: 1.", "Možné: 10.", "P = 1/10."], "text": "Answer: P = 1/10.\nSteps:\n1. Příznivé: 1.\n2. Možné: 10.\n3. P = 1/10."}
{"id": "mat_0971", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(21, 8) step by step.", "choices": null, "answer": "lcm = 168.", "steps": ["gcd(21,8) = 1.", "21×8/1.", "= 168."], "text": "Answer: lcm = 168.\nSteps:\n1. gcd(21,8) = 1.\n2. 21×8/1.\n3. = 168."}
{"id": "mat_0972", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(22,2).", "choices": null, "answer": "C(22,2) = 231.", "steps": ["Formula n(n−1)/2.", "22×21/2.", "= 231."], "text": "Answer: C(22,2) = 231.\nSteps:\n1. Formula n(n−1)/2.\n2. 22×21/2.\n3. = 231."}
{"id": "mat_0973", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co vyvrací obecné tvrzení?", "choices": {"A": "jeden protipříklad", "B": "dva příklady", "C": "hlasité popření", "D": "dlouhý důkaz"}, "answer": "Odpověď je jeden protipříklad.", "steps": ["Jediné selhání je zabíjí.", "Tedy: jeden protipříklad.", "Vyber možnost A."], "text": "Answer: jeden protipříklad\nSteps:\n1. Jediné selhání je zabíjí.\n2. Tedy: jeden protipříklad.\n3. Vyber možnost A.\nAnswer: (A)"}
{"id": "mat_0974", "subject": "math", "topic": "calculus", "lang": "en", "kind": "worked_example", "question": "Evaluate the integral of 3x² from 0 to 20.", "choices": null, "answer": "Integral: 8000.", "steps": ["Antiderivative: x³.", "Evaluate 0 to 20.", "20³ − 0 = 8000."], "text": "Answer: Integral: 8000.\nSteps:\n1. Antiderivative: x³.\n2. Evaluate 0 to 20.\n3. 20³ − 0 = 8000."}
{"id": "mat_0975", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co najde nejbližší bod?", "choices": {"A": "hádání", "B": "projekce", "C": "doufání", "D": "čekání"}, "answer": "Odpověď je projekce.", "steps": ["Chyba kolmá.", "Tedy: projekce.", "Vyber možnost B."], "text": "Answer: projekce\nSteps:\n1. Chyba kolmá.\n2. Tedy: projekce.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0976", "subject": "math", "topic": "probability", "lang": "en", "kind": "worked_example", "question": "Compute E of a uniform draw from {1..22}.", "choices": null, "answer": "E = 11.5.", "steps": ["Mean of 1..22.", "(22+1)/2.", "E = 11.5."], "text": "Answer: E = 11.5.\nSteps:\n1. Mean of 1..22.\n2. (22+1)/2.\n3. E = 11.5."}
{"id": "mat_0977", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "multiple_choice", "question": "Find lcm(13, 6).", "choices": {"A": "76", "B": "80", "C": "82", "D": "78"}, "answer": "The lcm is 78.", "steps": ["13×6/gcd.", "lcm = 78.", "Select option D."], "text": "Answer: 78\nSteps:\n1. 13×6/gcd.\n2. lcm = 78.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0978", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "concept", "question": "Why study combinatorics?", "choices": null, "answer": "It underlies probability, algorithms, and design, making counting a core skill.", "steps": ["It underlies probability, algorithms, and design, making counting a core skill.", "So the answer stands: It underlies probability, algorithms, and design, making counting a core skill."], "text": "Answer: It underlies probability, algorithms, and design, making counting a core skill.\nSteps:\n1. It underlies probability, algorithms, and design, making counting a core skill.\n2. So the answer stands: It underlies probability, algorithms, and design, making counting a core skill."}
{"id": "mat_0979", "subject": "math", "topic": "proofs", "lang": "en", "kind": "multiple_choice", "question": "Compute N² + N for N = 12.", "choices": {"A": "154", "B": "158", "C": "160", "D": "156"}, "answer": "The value is 156.", "steps": ["12² + 12.", "Value = 156.", "Select option D."], "text": "Answer: 156\nSteps:\n1. 12² + 12.\n2. Value = 156.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0980", "subject": "math", "topic": "calculus", "lang": "cs", "kind": "worked_example", "question": "Vyčísli integrál 3x² od 0 do 7.", "choices": null, "answer": "Integrál: 343.", "steps": ["Primitivní: x³.", "Vyčísli 0 až 7.", "7³ − 0 = 343."], "text": "Answer: Integrál: 343.\nSteps:\n1. Primitivní: x³.\n2. Vyčísli 0 až 7.\n3. 7³ − 0 = 343."}
{"id": "mat_0981", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "worked_example", "question": "Compute the trace of [[20,0],[0,21]].", "choices": null, "answer": "Trace: 41.", "steps": ["Sum diagonal.", "20 + 21.", "= 41."], "text": "Answer: Trace: 41.\nSteps:\n1. Sum diagonal.\n2. 20 + 21.\n3. = 41."}
{"id": "mat_0982", "subject": "math", "topic": "probability", "lang": "en", "kind": "multiple_choice", "question": "A bag holds 11 red and 23 blue marbles. What is P(red)?", "choices": {"A": "23/34", "B": "11/35", "C": "12/34", "D": "11/34"}, "answer": "P(red) is 11/34.", "steps": ["11 of 34.", "P = 11/34.", "Select option D."], "text": "Answer: 11/34\nSteps:\n1. 11 of 34.\n2. P = 11/34.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0983", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute gcd(76, 114) step by step.", "choices": null, "answer": "gcd = 38.", "steps": ["Factor: 76=4×19, 114=6×19.", "Common: 19×gcd(4,6).", "= 19×2 = 38."], "text": "Answer: gcd = 38.\nSteps:\n1. Factor: 76=4×19, 114=6×19.\n2. Common: 19×gcd(4,6).\n3. = 19×2 = 38."}
{"id": "mat_0984", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "multiple_choice", "question": "Compute C(14,3).", "choices": {"A": "350", "B": "378", "C": "392", "D": "364"}, "answer": "C(14,3) is 364.", "steps": ["14×13×12/6.", "= 364.", "Select option D."], "text": "Answer: 364\nSteps:\n1. 14×13×12/6.\n2. = 364.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0985", "subject": "math", "topic": "proofs", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co musí případy pokrýt?", "choices": {"A": "snadné", "B": "každou možnost", "C": "první dva", "D": "liché případy"}, "answer": "Odpověď je každou možnost.", "steps": ["Mezery boří důkazy.", "Tedy: každou možnost.", "Vyber možnost B."], "text": "Answer: každou možnost\nSteps:\n1. Mezery boří důkazy.\n2. Tedy: každou možnost.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0986", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Compute the integral of 3x² from 0 to 11.", "choices": {"A": "1320", "B": "1342", "C": "1331", "D": "1353"}, "answer": "The integral is 1331.", "steps": ["Antiderivative x³.", "11³ = 1331.", "Select option C."], "text": "Answer: 1331\nSteps:\n1. Antiderivative x³.\n2. 11³ = 1331.\n3. Select option C.\nAnswer: (C)"}
{"id": "mat_0987", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "multiple_choice", "question": "Find the dot product (10,11) · (3,4).", "choices": {"A": "70", "B": "78", "C": "82", "D": "74"}, "answer": "The dot product is 74.", "steps": ["10×3 + 11×4.", "Product = 74.", "Select option D."], "text": "Answer: 74\nSteps:\n1. 10×3 + 11×4.\n2. Product = 74.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0988", "subject": "math", "topic": "probability", "lang": "cs", "kind": "worked_example", "question": "Spočítej E rovnoměrného tahu z {1..9}.", "choices": null, "answer": "E = 5.", "steps": ["Průměr 1..9.", "(9+1)/2.", "E = 5."], "text": "Answer: E = 5.\nSteps:\n1. Průměr 1..9.\n2. (9+1)/2.\n3. E = 5."}
{"id": "mat_0989", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute (237) mod 7.", "choices": null, "answer": "Remainder: 6.", "steps": ["Divide 237 by 7.", "7×33 = 231.", "237 − 231 = 6."], "text": "Answer: Remainder: 6.\nSteps:\n1. Divide 237 by 7.\n2. 7×33 = 231.\n3. 237 − 231 = 6."}
{"id": "mat_0990", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of P(21,3).", "choices": null, "answer": "P(21,3) = 7980.", "steps": ["n(n−1)(n−2).", "21×20×19.", "= 7980."], "text": "Answer: P(21,3) = 7980.\nSteps:\n1. n(n−1)(n−2).\n2. 21×20×19.\n3. = 7980."}
{"id": "mat_0991", "subject": "math", "topic": "proofs", "lang": "en", "kind": "concept", "question": "Why review proofs?", "choices": null, "answer": "Reviewing proofs catches errors and sharpens arguments, which is why journals use peer referees.", "steps": ["Reviewing proofs catches errors and sharpens arguments, which is why journals use peer referees.", "So the answer stands: Reviewing proofs catches errors and sharpens arguments, which is why journals use peer referees."], "text": "Answer: Reviewing proofs catches errors and sharpens arguments, which is why journals use peer referees.\nSteps:\n1. Reviewing proofs catches errors and sharpens arguments, which is why journals use peer referees.\n2. So the answer stands: Reviewing proofs catches errors and sharpens arguments, which is why journals use peer referees."}
{"id": "mat_0992", "subject": "math", "topic": "calculus", "lang": "en", "kind": "multiple_choice", "question": "Find the limit of (x² − 11²)/(x − 11) as x → 11.", "choices": {"A": "20", "B": "24", "C": "26", "D": "22"}, "answer": "The limit is 22.", "steps": ["Factor (x−11)(x+11).", "Cancel, get 2×11 = 22.", "Select option D."], "text": "Answer: 22\nSteps:\n1. Factor (x−11)(x+11).\n2. Cancel, get 2×11 = 22.\n3. Select option D.\nAnswer: (D)"}
{"id": "mat_0993", "subject": "math", "topic": "linear_algebra", "lang": "cs", "kind": "concept", "question": "Proč skalární součin?", "choices": null, "answer": "Skalární součin měří úhly a délky přes souřadnice.", "steps": ["Skalární součin měří úhly a délky přes souřadnice.", "Odpověď tedy platí: Skalární součin měří úhly a délky přes souřadnice."], "text": "Answer: Skalární součin měří úhly a délky přes souřadnice.\nSteps:\n1. Skalární součin měří úhly a délky přes souřadnice.\n2. Odpověď tedy platí: Skalární součin měří úhly a délky přes souřadnice."}
{"id": "mat_0994", "subject": "math", "topic": "probability", "lang": "cs", "kind": "multiple_choice", "question": "Vyber: co modeluje zvony chyb?", "choices": {"A": "divoké hádání", "B": "normální rozdělení", "C": "rovná čára", "D": "jediný bod"}, "answer": "Odpověď je normální rozdělení.", "steps": ["Průměry jsou normální.", "Tedy: normální rozdělení.", "Vyber možnost B."], "text": "Answer: normální rozdělení\nSteps:\n1. Průměry jsou normální.\n2. Tedy: normální rozdělení.\n3. Vyber možnost B.\nAnswer: (B)"}
{"id": "mat_0995", "subject": "math", "topic": "number_theory", "lang": "en", "kind": "worked_example", "question": "Compute lcm(22, 8) step by step.", "choices": null, "answer": "lcm = 88.", "steps": ["gcd(22,8) = 2.", "22×8/2.", "= 88."], "text": "Answer: lcm = 88.\nSteps:\n1. gcd(22,8) = 2.\n2. 22×8/2.\n3. = 88."}
{"id": "mat_0996", "subject": "math", "topic": "combinatorics", "lang": "en", "kind": "worked_example", "question": "Show the computation of C(23,3).", "choices": null, "answer": "C(23,3) = 1771.", "steps": ["n(n−1)(n−2)/6.", "23×22×21/6.", "= 1771."], "text": "Answer: C(23,3) = 1771.\nSteps:\n1. n(n−1)(n−2)/6.\n2. 23×22×21/6.\n3. = 1771."}
{"id": "mat_0997", "subject": "math", "topic": "proofs", "lang": "en", "kind": "worked_example", "question": "Show N² + N is even for N = 19 by computing its half.", "choices": null, "answer": "Half: 190, so 380 is even.", "steps": ["19² + 19 = 380.", "380 / 2 = 190.", "Integer, so even."], "text": "Answer: Half: 190, so 380 is even.\nSteps:\n1. 19² + 19 = 380.\n2. 380 / 2 = 190.\n3. Integer, so even."}
{"id": "mat_0998", "subject": "math", "topic": "calculus", "lang": "en", "kind": "concept", "question": "Why learn calculus?", "choices": null, "answer": "Calculus models change everywhere, from orbits to markets, making it the core applied language.", "steps": ["Calculus models change everywhere, from orbits to markets, making it the core applied language.", "So the answer stands: Calculus models change everywhere, from orbits to markets, making it the core applied language."], "text": "Answer: Calculus models change everywhere, from orbits to markets, making it the core applied language.\nSteps:\n1. Calculus models change everywhere, from orbits to markets, making it the core applied language.\n2. So the answer stands: Calculus models change everywhere, from orbits to markets, making it the core applied language."}
{"id": "mat_0999", "subject": "math", "topic": "linear_algebra", "lang": "en", "kind": "concept", "question": "Why study linear algebra?", "choices": null, "answer": "It powers graphics, data, and physics, making vectors and matrices the applied core.", "steps": ["It powers graphics, data, and physics, making vectors and matrices the applied core.", "So the answer stands: It powers graphics, data, and physics, making vectors and matrices the applied core."], "text": "Answer: It powers graphics, data, and physics, making vectors and matrices the applied core.\nSteps:\n1. It powers graphics, data, and physics, making vectors and matrices the applied core.\n2. So the answer stands: It powers graphics, data, and physics, making vectors and matrices the applied core."}
{"id": "mat_1000", "subject": "math", "topic": "probability", "lang": "cs", "kind": "concept", "question": "Proč testování hypotéz?", "choices": null, "answer": "Testování váží důkazy proti nulovému tvrzení.", "steps": ["Testování váží důkazy proti nulovému tvrzení.", "Odpověď tedy platí: Testování váží důkazy proti nulovému tvrzení."], "text": "Answer: Testování váží důkazy proti nulovému tvrzení.\nSteps:\n1. Testování váží důkazy proti nulovému tvrzení.\n2. Odpověď tedy platí: Testování váží důkazy proti nulovému tvrzení."}