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Find the roots of the quadratic equation: 3x^2 + 1x + -5 = 0 | 1. Identify coefficients: a=3, b=1, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -60 = 61.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √61) / 6.
5. Calculate roots: x1 = 1.135041612651109, x2 = -1.4683749459844424.
Final Answe... |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Find the roots of the quadratic equation: 10x^2 + 10x + 9 = 0 | 1. Identify coefficients: a=10, b=10, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 10^2 - 4(10)(9) = 100 - 360 = -260.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-10 ± √260i) / 20.
Final Answer: x = -0.5 ± 0.8062257748298549i |
Multiply the following 2x2 matrices:
A =
[2, 2]
[-1, -1]
B =
[4, 3]
[1, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*4) + (2*1) = 10
C12 = (A11 * B12) + (A12 * B22) = (2*3) + (2*-2) = 2
C21 = (A21 * B11) + (A22 * B21) = (-1*4) + (-1*1) = -5
C22 = (A21 * B12) + (A22 * B22) = (-1*3) + (-1*-2) = -1
Result M... |
In a right-angled triangle, the lengths of the two legs are a = 20 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 20^2 + 10^2 = c^2
2. Calculate the squares: 400 + 100 = c^2
3. Sum the squares: 500 = c^2
4. Take the square root of both sides: c = √500
Final A... |
Multiply the following 2x2 matrices:
A =
[2, -2]
[-1, 4]
B =
[4, 5]
[-5, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*4) + (-2*-5) = 18
C12 = (A11 * B12) + (A12 * B22) = (2*5) + (-2*-3) = 16
C21 = (A21 * B11) + (A22 * B21) = (-1*4) + (4*-5) = -24
C22 = (A21 * B12) + (A22 * B22) = (-1*5) + (4*-3) = -17
Res... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Multiply the following 2x2 matrices:
A =
[4, 2]
[1, 2]
B =
[-2, 1]
[-4, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*-2) + (2*-4) = -16
C12 = (A11 * B12) + (A12 * B22) = (4*1) + (2*3) = 10
C21 = (A21 * B11) + (A22 * B21) = (1*-2) + (2*-4) = -10
C22 = (A21 * B12) + (A22 * B22) = (1*1) + (2*3) = 7
Result M... |
Evaluate the logarithm: log_3(27) | To evaluate log_3(27), we ask the question: '3 raised to what power equals 27?'
Let x be the unknown power: 3^x = 27
Since 3^3 = 27, it follows that x = 3.
Final Answer: 3 |
Compute the indefinite integral of f(x) = 8x^3. | To find the indefinite integral ∫ 8x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 8 / 4 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^3 dx = 2x^4 + C |
Find the roots of the quadratic equation: 6x^2 + 6x + 6 = 0 | 1. Identify coefficients: a=6, b=6, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(6)(6) = 36 - 144 = -108.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √108i) / 12.
Final Answer: x = -0.5 ± 0.8660254037844387i |
Compute the quotient and remainder of 10049 divided by 41. Show the long division steps. | To divide 10049 by 41:
1. Determine how many times 41 goes into the leading digits of 10049.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 10049 / 41 = 245 with a remainder of 4.
Proof: (245 * 41) +... |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Find the derivative of f(x) = (2x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Find the roots of the quadratic equation: 9x^2 + -10x + 4 = 0 | 1. Identify coefficients: a=9, b=-10, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = -10^2 - 4(9)(4) = 100 - 144 = -44.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (10 ± √44i) / 18.
Final Answer: x = 0.5555555555555556 ± 0.3685138655950444i |
Add the fractions 1/3 and 10/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 10, which is 30.
2. Convert 1/3 by multiplying top and bottom by 10: 10/30.
3. Convert 10/10 by multiplying top and bottom by 3: 30/30.
4. Add the numerators: 10 + 30 = 40.
5. The resulting fraction is 40/30.
6. Simplify by... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 17, 19, 10, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 17 + 19 + 10 + 11) / 5 = 76 / 5 = 15.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.8, 1.8, 3.8, -5.2, -4.2]
Squared Differences: [14.44, 3.24, 14.44, 27.04, 17.64... |
Find the roots of the quadratic equation: 9x^2 + -10x + 7 = 0 | 1. Identify coefficients: a=9, b=-10, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -10^2 - 4(9)(7) = 100 - 252 = -152.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (10 ± √152i) / 18.
Final Answer: x = 0.5555555555555556 ± 0.6849348892187751i |
Compute the quotient and remainder of 29520 divided by 43. Show the long division steps. | To divide 29520 by 43:
1. Determine how many times 43 goes into the leading digits of 29520.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 29520 / 43 = 686 with a remainder of 22.
Proof: (686 * 43) ... |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Find the roots of the quadratic equation: 5x^2 + -2x + -8 = 0 | 1. Identify coefficients: a=5, b=-2, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -160 = 164.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √164) / 10.
5. Calculate roots: x1 = 1.4806248474865698, x2 = -1.0806248474865696.
Final ... |
Find the roots of the quadratic equation: 4x^2 + -7x + -6 = 0 | 1. Identify coefficients: a=4, b=-7, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -96 = 145.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √145) / 8.
5. Calculate roots: x1 = 2.380199322349037, x2 = -0.630199322349037.
Final Ans... |
Multiply the following 2x2 matrices:
A =
[1, 3]
[-2, 4]
B =
[5, 1]
[-1, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*5) + (3*-1) = 2
C12 = (A11 * B12) + (A12 * B22) = (1*1) + (3*0) = 1
C21 = (A21 * B11) + (A22 * B21) = (-2*5) + (4*-1) = -14
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (4*0) = -2
Result Mat... |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Evaluate tan(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 16331239353195370.0000 (exact form depends on root simplification if applicable) |
Multiply the following 2x2 matrices:
A =
[-4, -5]
[-4, -5]
B =
[-3, -3]
[-3, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-3) + (-5*-3) = 27
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (-5*-2) = 22
C21 = (A21 * B11) + (A22 * B21) = (-4*-3) + (-5*-3) = 27
C22 = (A21 * B12) + (A22 * B22) = (-4*-3) + (-5*-2) = 2... |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 12^2 + 20^2 = c^2
2. Calculate the squares: 144 + 400 = c^2
3. Sum the squares: 544 = c^2
4. Take the square root of both sides: c = √544
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 11, 6, 5, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 11 + 6 + 5 + 17) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.8, 0.8, -4.2, -5.2, 6.8]
Squared Differences: [3.24, 0.64, 17.64, 27.04, 46.24]
... |
Add the fractions 12/14 and 12/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 3, which is 42.
2. Convert 12/14 by multiplying top and bottom by 3: 36/42.
3. Convert 12/3 by multiplying top and bottom by 14: 168/42.
4. Add the numerators: 36 + 168 = 204.
5. The resulting fraction is 204/42.
6. Simpli... |
Compute the indefinite integral of f(x) = 9x^4. | To find the indefinite integral ∫ 9x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 9 / 5 = 9/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^4 dx = 9/5x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 18. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 18^2 = c^2
2. Calculate the squares: 169 + 324 = c^2
3. Sum the squares: 493 = c^2
4. Take the square root of both sides: c = √493
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 1, 3, 19, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 1 + 3 + 19 + 20) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.0, -11.0, -9.0, 7.0, 8.0]
Squared Differences: [25.0, 121.0, 81.0, 49.0, 64.0]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 3, 1, 8, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 3 + 1 + 8 + 20) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.0, -6.0, -8.0, -1.0, 11.0]
Squared Differences: [16.0, 36.0, 64.0, 1.0, 121.0]
S... |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Find the roots of the quadratic equation: 6x^2 + 2x + -5 = 0 | 1. Identify coefficients: a=6, b=2, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -120 = 124.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √124) / 12.
5. Calculate roots: x1 = 0.7612940604716703, x2 = -1.0946273938050035.
Final ... |
In a right-angled triangle, the lengths of the two legs are a = 3 and b = 18. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 3^2 + 18^2 = c^2
2. Calculate the squares: 9 + 324 = c^2
3. Sum the squares: 333 = c^2
4. Take the square root of both sides: c = √333
Final Answ... |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Compute the quotient and remainder of 84034 divided by 71. Show the long division steps. | To divide 84034 by 71:
1. Determine how many times 71 goes into the leading digits of 84034.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 84034 / 71 = 1183 with a remainder of 41.
Proof: (1183 * 71... |
Multiply the following 2x2 matrices:
A =
[-4, -3]
[-5, -1]
B =
[-1, 4]
[4, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-1) + (-3*4) = -8
C12 = (A11 * B12) + (A12 * B22) = (-4*4) + (-3*0) = -16
C21 = (A21 * B11) + (A22 * B21) = (-5*-1) + (-1*4) = 1
C22 = (A21 * B12) + (A22 * B22) = (-5*4) + (-1*0) = -20
Re... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 1, 9, 10, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 1 + 9 + 10 + 6) / 5 = 35 / 5 = 7.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.0, -6.0, 2.0, 3.0, -1.0]
Squared Differences: [4.0, 36.0, 4.0, 9.0, 1.0]
Sum of S... |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 14. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 14^2 = c^2
2. Calculate the squares: 81 + 196 = c^2
3. Sum the squares: 277 = c^2
4. Take the square root of both sides: c = √277
Final Ans... |
Compute the indefinite integral of f(x) = 6x^4. | To find the indefinite integral ∫ 6x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 6 / 5 = 6/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^4 dx = 6/5x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 15. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 5^2 + 15^2 = c^2
2. Calculate the squares: 25 + 225 = c^2
3. Sum the squares: 250 = c^2
4. Take the square root of both sides: c = √250
Final Ans... |
Compute the indefinite integral of f(x) = 9x^2. | To find the indefinite integral ∫ 9x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 9 / 3 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^2 dx = 3x^3 + C |
Compute the indefinite integral of f(x) = 2x^4. | To find the indefinite integral ∫ 2x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 2 / 5 = 2/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^4 dx = 2/5x^5 + C |
Evaluate the logarithm: log_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
Find the roots of the quadratic equation: 10x^2 + -3x + -6 = 0 | 1. Identify coefficients: a=10, b=-3, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -240 = 249.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √249) / 20.
5. Calculate roots: x1 = 0.938986691902975, x2 = -0.638986691902975.
Final A... |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 7, 15, 14, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 7 + 15 + 14 + 8) / 5 = 54 / 5 = 10.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.8, -3.8, 4.2, 3.2, -2.8]
Squared Differences: [0.64, 14.44, 17.64, 10.24, 7.84]
... |
Evaluate cos(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5000 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 9x^2. | To find the indefinite integral ∫ 9x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 9 / 3 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^2 dx = 3x^3 + C |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
Multiply the following 2x2 matrices:
A =
[-4, 0]
[-5, -2]
B =
[-1, 2]
[2, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-1) + (0*2) = 4
C12 = (A11 * B12) + (A12 * B22) = (-4*2) + (0*3) = -8
C21 = (A21 * B11) + (A22 * B21) = (-5*-1) + (-2*2) = 1
C22 = (A21 * B12) + (A22 * B22) = (-5*2) + (-2*3) = -16
Result... |
Multiply the following 2x2 matrices:
A =
[2, -4]
[-2, 2]
B =
[0, -3]
[-1, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*0) + (-4*-1) = 4
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (-4*1) = -10
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (2*-1) = -2
C22 = (A21 * B12) + (A22 * B22) = (-2*-3) + (2*1) = 8
Result... |
In a right-angled triangle, the lengths of the two legs are a = 18 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 18^2 + 10^2 = c^2
2. Calculate the squares: 324 + 100 = c^2
3. Sum the squares: 424 = c^2
4. Take the square root of both sides: c = √424
Final A... |
Add the fractions 10/9 and 5/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 10, which is 90.
2. Convert 10/9 by multiplying top and bottom by 10: 100/90.
3. Convert 5/10 by multiplying top and bottom by 9: 45/90.
4. Add the numerators: 100 + 45 = 145.
5. The resulting fraction is 145/90.
6. Simplif... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 2, 12, 19, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 2 + 12 + 19 + 13) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.4, -8.4, 1.6, 8.6, 2.6]
Squared Differences: [19.36, 70.56, 2.56, 73.96, 6.76]
... |
In a right-angled triangle, the lengths of the two legs are a = 11 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 11^2 + 8^2 = c^2
2. Calculate the squares: 121 + 64 = c^2
3. Sum the squares: 185 = c^2
4. Take the square root of both sides: c = √185
Final Ans... |
Multiply the following 2x2 matrices:
A =
[4, -3]
[-3, 5]
B =
[3, -4]
[-1, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*3) + (-3*-1) = 15
C12 = (A11 * B12) + (A12 * B22) = (4*-4) + (-3*4) = -28
C21 = (A21 * B11) + (A22 * B21) = (-3*3) + (5*-1) = -14
C22 = (A21 * B12) + (A22 * B22) = (-3*-4) + (5*4) = 32
Res... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 20, 20, 8, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 20 + 20 + 8 + 9) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-10.6, 8.4, 8.4, -3.6, -2.6]
Squared Differences: [112.36, 70.56, 70.56, 12.96, 6.76... |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 14, 18, 11, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 14 + 18 + 11 + 7) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.4, 0.4, 4.4, -2.6, -6.6]
Squared Differences: [19.36, 0.16, 19.36, 6.76, 43.56]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 1, 4, 9, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 1 + 4 + 9 + 19) / 5 = 36 / 5 = 7.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.2, -6.2, -3.2, 1.8, 11.8]
Squared Differences: [17.64, 38.44, 10.24, 3.24, 139.24]
... |
Evaluate cos(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 9x^2 + 8x + 5 = 0 | 1. Identify coefficients: a=9, b=8, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(9)(5) = 64 - 180 = -116.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √116i) / 18.
Final Answer: x = -0.4444444444444444 ± 0.5983516452371671i |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 10, 15, 3, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 10 + 15 + 3 + 20) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.6, -2.4, 2.6, -9.4, 7.6]
Squared Differences: [2.56, 5.76, 6.76, 88.36, 57.76]
... |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 57718 divided by 46. Show the long division steps. | To divide 57718 by 46:
1. Determine how many times 46 goes into the leading digits of 57718.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 57718 / 46 = 1254 with a remainder of 34.
Proof: (1254 * 46... |
Compute the indefinite integral of f(x) = 5x^3. | To find the indefinite integral ∫ 5x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 5 / 4 = 5/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^3 dx = 5/4x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 4. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 4^2 = c^2
2. Calculate the squares: 169 + 16 = c^2
3. Sum the squares: 185 = c^2
4. Take the square root of both sides: c = √185
Final Ans... |
Find the roots of the quadratic equation: 5x^2 + 6x + -3 = 0 | 1. Identify coefficients: a=5, b=6, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -60 = 96.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √96) / 10.
5. Calculate roots: x1 = 0.37979589711327116, x2 = -1.579795897113271.
Final An... |
Find the roots of the quadratic equation: 6x^2 + 8x + 7 = 0 | 1. Identify coefficients: a=6, b=8, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(6)(7) = 64 - 168 = -104.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √104i) / 12.
Final Answer: x = -0.6666666666666666 ± 0.8498365855987974i |
Multiply the following 2x2 matrices:
A =
[0, 2]
[0, 0]
B =
[-4, 5]
[0, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*-4) + (2*0) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (2*-1) = -2
C21 = (A21 * B11) + (A22 * B21) = (0*-4) + (0*0) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*5) + (0*-1) = 0
Result Matri... |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Compute the indefinite integral of f(x) = 3x^1. | To find the indefinite integral ∫ 3x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 3 / 2 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^1 dx = 3/2x^2 + C |
Find the roots of the quadratic equation: 1x^2 + 0x + -5 = 0 | 1. Identify coefficients: a=1, b=0, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -20 = 20.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √20) / 2.
5. Calculate roots: x1 = 2.23606797749979, x2 = -2.23606797749979.
Final Answer: x... |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 10^2 + 8^2 = c^2
2. Calculate the squares: 100 + 64 = c^2
3. Sum the squares: 164 = c^2
4. Take the square root of both sides: c = √164
Final Ans... |
Find the derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Multiply the following 2x2 matrices:
A =
[3, 3]
[-4, 4]
B =
[5, 2]
[5, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*5) + (3*5) = 30
C12 = (A11 * B12) + (A12 * B22) = (3*2) + (3*1) = 9
C21 = (A21 * B11) + (A22 * B21) = (-4*5) + (4*5) = 0
C22 = (A21 * B12) + (A22 * B22) = (-4*2) + (4*1) = -4
Result Matrix... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 16, 5, 15, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 16 + 5 + 15 + 19) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.0, 4.0, -7.0, 3.0, 7.0]
Squared Differences: [49.0, 16.0, 49.0, 9.0, 49.0]
Su... |
Find the roots of the quadratic equation: 8x^2 + -2x + 9 = 0 | 1. Identify coefficients: a=8, b=-2, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(8)(9) = 4 - 288 = -284.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √284i) / 16.
Final Answer: x = 0.125 ± 1.0532687216470449i |
Find the derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Add the fractions 15/11 and 14/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 7, which is 77.
2. Convert 15/11 by multiplying top and bottom by 7: 105/77.
3. Convert 14/7 by multiplying top and bottom by 11: 154/77.
4. Add the numerators: 105 + 154 = 259.
5. The resulting fraction is 259/77.
6. Simp... |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 17, 16, 2, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 17 + 16 + 2 + 15) / 5 = 64 / 5 = 12.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.2, 4.2, 3.2, -10.8, 2.2]
Squared Differences: [1.44, 17.64, 10.24, 116.64, 4.84]... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 14, 7, 19, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 14 + 7 + 19 + 15) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.4, 0.6, -6.4, 5.6, 1.6]
Squared Differences: [1.96, 0.36, 40.96, 31.36, 2.56]
... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Compute the quotient and remainder of 44996 divided by 52. Show the long division steps. | To divide 44996 by 52:
1. Determine how many times 52 goes into the leading digits of 44996.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 44996 / 52 = 865 with a remainder of 16.
Proof: (865 * 52) ... |
Compute the quotient and remainder of 48729 divided by 70. Show the long division steps. | To divide 48729 by 70:
1. Determine how many times 70 goes into the leading digits of 48729.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 48729 / 70 = 696 with a remainder of 9.
Proof: (696 * 70) +... |
End of preview. Expand in Data Studio
Massive Step-by-Step Mathematics Instruction Dataset
Dataset Description
This is a 60GB, highly knowledge-dense dataset designed to teach Large Language Models (LLMs) rigorous mathematical reasoning.
Unlike standard math datasets that only provide the final answer, this dataset emphasizes Chain-of-Thought (CoT) reasoning. Every single row contains a detailed, step-by-step breakdown of how to arrive at the solution, making it ideal for supervised fine-tuning (SFT) to improve an LLM's logical reasoning capabilities.
Dataset Structure
The dataset is stored as a single CSV file (math_dataset.csv) and contains two columns:
prompt: The mathematical question or instruction.completion: The step-by-step reasoning and final answer.
Mathematical Domains Covered
The dataset contains hundreds of millions of examples spanning 10 core mathematical disciplines:
- Arithmetic: Long division with rigorous verification (quotient and remainder proofs).
- Algebra: Quadratic equations utilizing the discriminant (handling both real and complex roots).
- Calculus: Indefinite integration using the power rule, and derivatives using the chain rule.
- Linear Algebra: 2x2 matrix multiplication with explicit dot-product calculations.
- Geometry: Pythagorean theorem calculations for right-angled triangles.
- Fractions: Addition of fractions requiring Least Common Multiple (LCM) calculation and simplification via Greatest Common Divisor (GCD).
- Statistics: Calculation of mean, population variance, and standard deviation.
- Trigonometry: Exact trigonometric values derived from the unit circle and special triangles.
- Logarithms: Evluation of logarithms by converting between logarithmic and exponential forms.
License
Open Data Attribution Training Disclosure License (ODATL‑1.0)
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