id stringlengths 36 36 | source stringclasses 6
values | question stringlengths 12 855 | steps listlengths 3 36 | answer stringlengths 102 1.68k |
|---|---|---|---|---|
dc5e03aa-4d4a-5763-a3b0-365a38620b5e | open-r1/OpenR1-Math-220k/cn_k12 | Given $i$ is the imaginary unit, the imaginary part of the complex number $1-2i$ is ( )
A: 2
B: 1
C: -1
D: -2 | [
"Okay, so I need to figure out the imaginary part of the complex number 1 - 2i. Let me start by recalling what a complex number is. A complex number is usually written in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. The 'i' here represents the imaginary unit, which is the square root o... | To determine the imaginary part of the complex number \(1 - 2i\), we recall that a complex number is generally written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
For the given complex number \(1 - 2i\), we can rewrite it as \(1 + (-2)i\). Here, the real part is \(1\) and the... |
08f08abe-2f70-5353-a5f7-f5489cbd8eda | open-r1/OpenR1-Math-220k/cn_k12 | The ratio of the number of students in grades 10, 11, and 12 at a certain school is 4:3:3. If a stratified sampling method is used to draw a sample of 50 students from these three grades, then the number of students drawn from the 11th grade is ( )
A: 15
B: 20
C: 25
D: 30 | [
"Okay, so I need to figure out how many 11th grade students will be in the stratified sample of 50 students. The ratio of students in grades 10, 11, and 12 is 4:3:3. Hmm, stratified sampling means that the sample is proportionally representative of each group, right? So each grade's representation in the sample sho... | The ratio of the number of students in grades 10, 11, and 12 is given as 4:3:3. To determine the number of students drawn from the 11th grade in a stratified sample of 50 students, we first calculate the total number of parts in the ratio:
\[ 4 + 3 + 3 = 10 \]
The proportion of students in grade 11 is \(\frac{3}{10}... |
dbb6e346-17d8-5107-804c-31ef3a2b27e6 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following operations is correct?
A: $a^{5}\div a^{2}=a^{3}$
B: $a^{3}+a^{3}=a^{6}$
C: $(a^{3})^{2}=a^{5}$
D: $\sqrt{{a^2}}=a$ | [
"Okay, let's see. The question is asking which of the operations A through D is correct. I need to check each one step by step. Let me start by recalling some exponent rules.",
"Starting with option A: a⁵ ÷ a² = a³. Hmm, when you divide exponents with the same base, you subtract the exponents, right? So, a^(5-2) ... | To determine which operation is correct, we analyze each option using exponent rules:
**A: \(a^{5} \div a^{2} = a^{3}\)**
When dividing exponents with the same base, subtract the exponents:
\(a^{5-2} = a^{3}\). **Correct.**
**B: \(a^{3} + a^{3} = a^{6}\)**
Adding terms with the same base and exponent combines c... |
73a5fc6f-76bd-5dfc-8127-63a53d41820b | open-r1/OpenR1-Math-220k/cn_k12 | The opposite of $23$ is ( )
A: $-23$
B: $23$
C: $-\frac{1}{23}$
D: $\frac{1}{23}$ | [
"Okay, let's see. The problem is asking for the opposite of 23, and the options are A: -23, B: 23, C: -1/23, D: 1/23. Hmm, opposite. So, what does opposite mean in math? I think it could mean the additive inverse or maybe the multiplicative inverse. Let me recall. The additive inverse of a number is what you add to... | The opposite of a number in mathematics typically refers to its additive inverse, which is the number that, when added to the original, results in zero. For 23, the additive inverse is \(-23\) because \(23 + (-23) = 0\). Options C and D involve reciprocals (multiplicative inverses), which are not relevant here. Option ... |
b7055e74-7435-55fc-a20a-d2dc8a0d961c | open-r1/OpenR1-Math-220k/cn_k12 | The value of $\cos^2 15^\circ - \sin^2 15^\circ$ is ( )
A: $- \frac { \sqrt {3}}{2}$
B: $\frac { \sqrt {3}}{2}$
C: $- \frac {1}{2}$
D: $\frac {1}{2}$ | [
"Okay, let me see. The problem is asking for the value of cos²15° minus sin²15°, and there are four options to choose from. Hmm, I need to remember my trigonometric identities. Wait, isn't there a double-angle formula related to cos²θ minus sin²θ? Let me recall. Oh, right! The double-angle formula for cosine is cos... | The problem asks for the value of \(\cos^2 15^\circ - \sin^2 15^\circ\).
We recall the double-angle formula for cosine:
\[
\cos(2\theta) = \cos^2\theta - \sin^2\theta
\]
Applying this identity with \(\theta = 15^\circ\), we get:
\[
\cos^2 15^\circ - \sin^2 15^\circ = \cos(2 \times 15^\circ) = \cos(30^\circ)
\]
The ... |
7d5d4b9f-3af5-50f6-b0d4-23b3c3f62112 | open-r1/OpenR1-Math-220k/cn_k12 | Given that the line $ax + y - 1 = 0$ is parallel to the line $2x - y + 2 = 0$, find the value of $a$. | [
"Okay, let's see. I need to find the value of 'a' such that the line ax + y - 1 = 0 is parallel to the line 2x - y + 2 = 0. Hmm, right. Parallel lines have the same slope, so first, I should find the slope of each line and set them equal.",
"Let me start with the second line, 2x - y + 2 = 0. Maybe I can rearrange... | To find the value of \( a \) such that the line \( ax + y - 1 = 0 \) is parallel to the line \( 2x - y + 2 = 0 \), we need to ensure that both lines have the same slope.
First, we convert the given lines into slope-intercept form \( y = mx + b \):
1. For the line \( 2x - y + 2 = 0 \):
\[
-y = -2x - 2 \implies y... |
7cb0fcb4-fa11-553c-ba8e-52e8ff95eb1b | open-r1/OpenR1-Math-220k/cn_k12 | Calculate the expression that equals $2022$.
A: $1-\left(-2021\right)$
B: $1^{2022}$
C: $0-2022$
D: $2022^{-1}$ | [
"Okay, let's see. I need to figure out which of these options A, B, C, or D equals 2022. Let me go through each one step by step.",
"Starting with option A: 1 - (-2021). Hmm, subtracting a negative number is like adding its positive counterpart, right? So 1 - (-2021) would be 1 + 2021. Let me do that addition: 1 ... | To determine which expression equals 2022, we evaluate each option step by step:
- **Option A**: \(1 - (-2021)\). Subtracting a negative number is equivalent to adding its positive counterpart:
\[
1 - (-2021) = 1 + 2021 = 2022
\]
This equals 2022.
- **Option B**: \(1^{2022}\). Any number raised to any power r... |
d6e6b8de-90d8-5a13-98e7-10ca6fa66765 | open-r1/OpenR1-Math-220k/cn_k12 | The slope of the line $x+y=0$ is ( ).
A: $45^{\circ}$
B: $60^{\circ}$
C: $90^{\circ}$
D: $135^{\circ}$ | [
"Okay, let's see. The problem is asking for the slope of the line given by the equation x + y = 0, and the options are angles: 45°, 60°, 90°, and 135°. Hmm, wait, actually, the question says \"the slope of the line,\" but the options are angles. That's a bit confusing because slope is usually a number, like a gradi... | To determine the angle of inclination of the line given by the equation \(x + y = 0\), we first convert the equation to slope-intercept form. Subtracting \(x\) from both sides, we get:
\[ y = -x \]
This shows that the slope (\(m\)) of the line is \(-1\). The angle of inclination \(\theta\) of a line with slope \(m\) ... |
4be9d3e6-e079-5cce-9754-5780613981b7 | open-r1/OpenR1-Math-220k/cn_k12 | A student's test scores for a certain sports event were as follows (unit: points): $9,7,10,8,10,9,10$. The mode of this data set is
A: $10$
B: $9$
C: $8$
D: $7$ | [
"Okay, let me see. The problem is asking for the mode of the student's test scores in a sports event. The scores given are 9, 7, 10, 8, 10, 9, 10. The options are A: 10, B: 9, C: 8, D: 7. Hmm, mode. Right, the mode is the number that appears most frequently in a data set. So I need to count how many times each scor... | To determine the mode of the data set, we need to identify the number that appears most frequently. The given test scores are: 9, 7, 10, 8, 10, 9, 10.
1. Count the occurrences of each score:
- 9 appears twice.
- 7 appears once.
- 10 appears three times.
- 8 appears once.
2. The score 10 appears the most f... |
74c69c47-06f1-5f1e-92d5-5f62234da3f0 | open-r1/OpenR1-Math-220k/cn_k12 | If the arithmetic mean of 1 and $a$ is 2, then the value of $a$ is ( )
A: 4
B: 3
C: 1
D: -4 | [
"Okay, let me try to figure out this problem. The question says that the arithmetic mean of 1 and a is 2, and we need to find the value of a from the given options. Alright, arithmetic mean, right? That's just the average of two numbers. So if I remember correctly, the arithmetic mean of two numbers is calculated b... | The arithmetic mean of 1 and \( a \) is given as 2. The arithmetic mean is calculated by adding the numbers together and dividing by the number of terms. Therefore, we set up the equation:
\[
\frac{1 + a}{2} = 2
\]
To solve for \( a \), we first multiply both sides of the equation by 2:
\[
1 + a = 4
\]
Next, we sub... |
33b061da-1bde-5a88-bbd3-9ba166cc543e | open-r1/OpenR1-Math-220k/cn_k12 | Given vectors $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(x,1)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, find $|\overrightarrow{b}|$. | [
"Okay, so I need to find the magnitude of vector b, which is given as (x, 1), but there's a condition that vector a is parallel to vector b. Vector a is (1, 2). Hmm, let me remember what it means for two vectors to be parallel. I think if two vectors are parallel, one must be a scalar multiple of the other. So, tha... | Given vectors \(\overrightarrow{a} = (1, 2)\) and \(\overrightarrow{b} = (x, 1)\), and the condition that \(\overrightarrow{a} \parallel \overrightarrow{b}\), we need to find \(|\overrightarrow{b}|\).
Since \(\overrightarrow{a}\) is parallel to \(\overrightarrow{b}\), there exists a scalar \(k\) such that:
\[
\overrig... |
b73d6111-ca81-57bd-9850-7daa81be130f | open-r1/OpenR1-Math-220k/cn_k12 | The domain of the function $y=\log_{2}(x-3)$ is ( )
A: $[3,+\infty)$
B: $(3,+\infty)$
C: $(-\infty,-3)$
D: $\mathbb{R}$ | [
"Okay, so I need to find the domain of the function y equals log base 2 of (x minus 3). Let me remember what the domain of a logarithmic function is.",
"Hmm, log functions are only defined when their argument is positive. That is, for log_b(z), z has to be greater than zero. So in this case, the argument is (x - ... | To determine the domain of the function \( y = \log_{2}(x - 3) \), we need to ensure the argument of the logarithm is positive. The logarithmic function is defined only when its argument is greater than zero.
For the given function, the argument is \( x - 3 \). We set up the inequality:
\[
x - 3 > 0
\]
Solving this i... |
8d38f93d-0162-5209-9eeb-2518d8377151 | open-r1/OpenR1-Math-220k/olympiads | Task 2 - 200512 To transport a certain amount of gravel, a truck with a 5 t loading capacity would have had to make exactly 105 fully loaded trips. After 35 of these trips, it was replaced by another truck with a 7 t loading capacity.
Determine how many fully loaded trips this second truck still has to make to transpo... | [
"Okay, let's see. The problem is about transporting gravel with two different trucks. The first truck has a 5-ton capacity and was supposed to make 105 fully loaded trips. But after 35 trips, they switch to a second truck with a 7-ton capacity. We need to figure out how many fully loaded trips the second truck stil... | To determine how many fully loaded trips the second truck still has to make to transport the remaining amount of gravel, we start by calculating the total amount of gravel. The first truck, with a 5-ton capacity, was supposed to make 105 trips:
\[
\text{Total gravel} = 105 \, \text{trips} \times 5 \, \text{t/trip} = 5... |
bbe81a48-0158-55fe-8422-0012630deb9e | open-r1/OpenR1-Math-220k/olympiads | 2. Jenny had to add 26 to a certain number. Instead she subtracted 26 and obtained -14 . What number should she have obtained?
A 28
B 32
C 36
D 38
E 42 | [
"Okay, so the problem says that Jenny had to add 26 to a certain number, but instead she subtracted 26 and got -14. We need to find out what number she should have obtained if she did it correctly. The answer choices are A 28, B 32, C 36, D 38, E 42. Let me try to figure this out step by step.",
"First, let's par... | Jenny was supposed to add 26 to a certain number \( x \), but instead she subtracted 26 and obtained -14. To find the correct result, we first determine the value of \( x \).
Given the incorrect operation:
\[
x - 26 = -14
\]
Solving for \( x \):
\[
x = -14 + 26
\]
\[
x = 12
\]
Now, we calculate the correct result by... |
c01f0263-e9db-549c-98b5-2f7b5ba191d7 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following sequences is in the correct order according to the "syllogism" model?
① $y=\cos x$ ($x\in R$) is a trigonometric function;
② Trigonometric functions are periodic functions;
③ $y=\cos x$ ($x\in R$) is a periodic function.
A: ①②③
B: ②①③
C: ②③①
D: ③②① | [
"Okay, let's see. The question is about syllogism and the order of the statements. Hmm, syllogism... I remember that from logic class. It's a form of deductive reasoning where a conclusion is drawn from two given premises. The classic example is like:",
"All men are mortal. Socrates is a man. Therefore, Socrates ... | To determine the correct order of the statements according to the syllogism model, we need to identify the major premise, minor premise, and conclusion.
- **Statement ②** ("Trigonometric functions are periodic functions") is a general statement and serves as the major premise.
- **Statement ①** ("y = cos x (x ∈ R) is... |
c1521f61-54b9-59ab-a19f-6004c18bfb07 | open-r1/OpenR1-Math-220k/olympiads | ## Task 8
The area of a new playground is square and measures $1600 \mathrm{~m}^{2}$.
How long is one side of the playground?
How many meters of fence are needed for three sides? | [
"Okay, so I need to figure out the length of one side of a square playground that has an area of 1600 square meters. Then, I also need to determine how many meters of fence are required for three sides of this playground. Let me start with the first question.",
"Since the playground is square, all four sides are ... | The area of the square playground is given as \(1600 \, \mathrm{m}^2\). To find the length of one side, we use the formula for the area of a square, which is \(s^2\), where \(s\) is the side length.
\[
s^2 = 1600
\]
Taking the square root of both sides, we get:
\[
s = \sqrt{1600} = 40 \, \mathrm{m}
\]
Next, we nee... |
ba68d43a-f85f-58cf-aa94-ab9ffff9f8dd | open-r1/OpenR1-Math-220k/olympiads | If $8+6=n+8$, then $n$ equals
(A) 14
(B) 22
(C) 6
(D) -2
(E) 9 | [
"Okay, so I need to solve this equation: 8 + 6 = n + 8. And then find out what n equals from the given options. Let me think step by step.",
"First, let me write down the equation again to make sure I have it right. It says 8 plus 6 equals n plus 8. So, 8 + 6 on the left side and n + 8 on the right side. The ques... | First, we start with the given equation:
\[ 8 + 6 = n + 8 \]
We simplify the left side by adding 8 and 6:
\[ 14 = n + 8 \]
Next, we need to isolate \( n \). We do this by subtracting 8 from both sides of the equation:
\[ 14 - 8 = n + 8 - 8 \]
This simplifies to:
\[ 6 = n \]
To verify, we substitute \( n = 6 \) ... |
bbacee97-3885-5113-a7f7-4a0061d06d3a | open-r1/OpenR1-Math-220k/cn_k12 | The positional relationship between two non-overlapping lines in the same plane is ( ).
A: There are two types: perpendicular or intersecting
B: There are three types: parallel, perpendicular, or intersecting
C: There are two types: parallel or intersecting
D: There are two types: parallel or perpendicular | [
"Okay, let's see. The question is about the positional relationship between two non-overlapping lines in the same plane. The options are A, B, C, D. I need to figure out which one is correct.",
"First, let me recall what the possible relationships between two lines in a plane are. In a plane, two lines can either... | The positional relationship between two non-overlapping lines in the same plane can be categorized as either **parallel** or **intersecting**.
- **Parallel lines** do not meet, regardless of their extension.
- **Intersecting lines** cross at a single point, which includes the special case of **perpendicular lines**... |
43a90665-ab98-5f89-9c5d-cf344d2b1239 | open-r1/OpenR1-Math-220k/cn_k12 | Given that the sequence of positive numbers $\{a\_n\}$ satisfies $a_{n+1}=2a_{n}$, determine the type of sequence $\{a\_n\}$.
A: An increasing sequence
B: A decreasing sequence
C: A constant sequence
D: The monotonicity of the sequence cannot be determined | [
"Okay, so I need to figure out what type of sequence {a_n} is, given that each term is twice the previous one. The options are increasing, decreasing, constant, or undetermined. Let me start by recalling what these terms mean.",
"An increasing sequence is one where each term is larger than the one before it, righ... | Given the sequence of positive numbers $\{a_n\}$ that satisfies the recurrence relation $a_{n+1} = 2a_n$, we need to determine the type of sequence.
1. **Understanding the Recurrence Relation**: The relation $a_{n+1} = 2a_n$ indicates that each term is twice the previous term. This is characteristic of a geometric seq... |
e1cc1f33-ea60-585b-9ef2-1cfda7707233 | open-r1/OpenR1-Math-220k/olympiads | ## Task 5
Mr. Müller has manufactured 16 chair legs.
How many chairs can he build? | [
"Okay, so Mr. Müller made 16 chair legs. The question is asking how many chairs he can build with those legs. Hmm, let's see. I need to figure out how many legs each chair requires.",
"From what I know, most chairs have four legs each. Like, a standard chair, right? You know, one at each corner. But wait, maybe s... | Mr. Müller has manufactured 16 chair legs. Since each standard chair requires 4 legs, the number of chairs he can build is calculated by dividing the total number of legs by the number needed per chair:
\[
\frac{16 \text{ legs}}{4 \text{ legs per chair}} = 4 \text{ chairs}
\]
**Answer:** \boxed{4} |
1cdd0373-376d-55b7-83ac-6d9836c19b61 | open-r1/OpenR1-Math-220k/cn_k12 | "x=1" is a ( ) condition for "x^2=1".
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Both sufficient and necessary
D: Neither sufficient nor necessary | [
"Okay, so I need to figure out whether \"x=1\" is a sufficient, necessary, both, or neither condition for \"x²=1\". Let me start by recalling what these terms mean.",
"A sufficient condition means that if the condition is met, then the statement must be true. So, if x=1 is a sufficient condition for x²=1, then wh... | The statement "x=1" is a **sufficient** condition for "x²=1" because if \( x = 1 \), then squaring it necessarily gives \( x^2 = 1 \). However, it is **not necessary** because \( x^2 = 1 \) could also hold if \( x = -1 \). Thus, "x=1" guarantees the truth of "x²=1" but is not required for it.
**Answer:**
\boxed{A} |
c0635d90-03eb-506a-ad7f-b218890ab707 | open-r1/OpenR1-Math-220k/cn_k12 | If $|m-3|+(n+2)^2=0$, find the value of $m+2n$. | [
"Okay, let's see. The problem is: If |m - 3| + (n + 2)² = 0, find the value of m + 2n. Hmm, so we have an equation involving absolute value and a square, and their sum is zero. I remember that absolute values are always non-negative, and squares are also non-negative. So, if the sum of two non-negative things is ze... | Given the equation \(|m - 3| + (n + 2)^2 = 0\), we need to find the value of \(m + 2n\).
Since both the absolute value and the square term are non-negative, their sum can only be zero if each term is individually zero. Therefore, we set each term to zero:
1. \(|m - 3| = 0\) implies \(m - 3 = 0\), so \(m = 3\).
2. \((... |
b07ded54-08a9-5307-ab42-4c893ed2020e | open-r1/OpenR1-Math-220k/cn_k12 | Given that the lengths of two adjacent sides of a parallelogram are $3$ and $5$, then its perimeter is ( )
A: $8$
B: $15$
C: $16$
D: $20$ | [
"Okay, so I have this problem here about a parallelogram. The question says that the lengths of two adjacent sides are 3 and 5, and we need to find the perimeter. The options are A: 8, B: 15, C: 16, D: 20. Hmm, let me think about how to approach this.",
"First, I remember that a parallelogram has opposite sides t... | Given that the lengths of two adjacent sides of a parallelogram are 3 and 5, we need to find its perimeter.
A parallelogram has opposite sides that are equal in length. Therefore, if one pair of adjacent sides are 3 and 5, the opposite sides will also be 3 and 5. The perimeter of a parallelogram is the sum of all its... |
2675c898-1964-5f06-8ddd-1cf6f4f6ca40 | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following options can form a set?
A: Students with a higher level of basketball skills in the school
B: Tall trees in the campus
C: All the countries in the European Union in 2012
D: Economically developed cities in China | [
"Okay, so I have this question here asking which of the given options can form a set. The options are A, B, C, D. Let me try to remember what a set is in mathematics. A set is a well-defined collection of distinct objects. The key term here is \"well-defined,\" which means that it must be clear whether any particul... | To determine which options can form a set, we evaluate whether the collection is **well-defined** (i.e., unambiguous criteria for inclusion/exclusion):
- **A**: "Students with a higher level of basketball skills" is subjective. What constitutes "higher level" is not objectively measurable. ❌
- **B**: "Tall trees in ... |
ad939184-7954-5e0a-a58f-7353a99a0bfd | open-r1/OpenR1-Math-220k/cn_k12 | Calculate the value of $|-3| \times 2$.
A: 6
B: -6
C: $\pm$6
D: -1 | [
"Okay, so I have this math problem here: Calculate the value of |-3| × 2. The options are A: 6, B: -6, C: ±6, D: -1. Hmm, let me think about how to solve this.",
"First, I need to remember what the absolute value symbols mean. Absolute value, right? The absolute value of a number is its distance from zero on the ... | To calculate the value of \(|-3| \times 2\):
1. First, determine the absolute value of \(-3\). The absolute value of a number is its distance from zero on the number line, which is always non-negative. Therefore, \(|-3| = 3\).
2. Next, multiply the result by 2: \(3 \times 2 = 6\).
The correct answer is \(\boxed{A}\). |
7772fa90-da19-53dd-9871-b2cf00efaf12 | open-r1/OpenR1-Math-220k/cn_k12 | Simplify: $\frac{{x}^{2}+1}{x-1}-\frac{2x}{x-1}$. | [
"Okay, let's see. I need to simplify this expression: (x squared plus 1) over (x minus 1) minus (2x) over (x minus 1). Hmm, both fractions have the same denominator, right? So maybe I can combine them into a single fraction by subtracting the numerators. Let me write that down.",
"First, the expression is \\(\\fr... | To simplify the given expression \(\frac{x^2 + 1}{x - 1} - \frac{2x}{x - 1}\):
1. Combine the fractions since they have the same denominator:
\[
\frac{(x^2 + 1) - 2x}{x - 1}
\]
2. Simplify the numerator:
\[
x^2 + 1 - 2x = x^2 - 2x + 1
\]
3. Factor the quadratic expression in the numerator:
\[
... |
9cead0a3-54c0-5bc5-b3de-9aa7ac0842f0 | open-r1/OpenR1-Math-220k/cn_k12 | The point $(-1, a^{2}+1)$ must lie in which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | [
"Alright, let's see. The problem is asking which quadrant the point (-1, a² + 1) must lie in. The options are first, second, third, or fourth quadrant. Hmm, okay. Let me recall how the quadrants are divided in a coordinate plane.",
"So, the coordinate plane is divided into four quadrants. The first quadrant is wh... | The point given is \((-1, a^2 + 1)\).
1. **Analyzing the x-coordinate**: The x-coordinate is \(-1\), which is negative.
2. **Analyzing the y-coordinate**: The y-coordinate is \(a^2 + 1\). Since \(a^2\) is always non-negative for any real number \(a\), adding 1 ensures that \(a^2 + 1\) is always positive (at least 1).... |
c89d9bf4-c938-59ff-8094-e299f1837e47 | open-r1/OpenR1-Math-220k/cn_k12 | At the market, tomatoes cost a yuan per kilogram and cabbage costs b yuan per kilogram. The school cafeteria buys 30 kg of tomatoes and 50 kg of cabbage, in total costing _____ yuan. | [
"Okay, so I have this problem here about the school cafeteria buying tomatoes and cabbage, and I need to figure out the total cost. Let me start by reading the problem again to make sure I understand all the details.",
"Alright, tomatoes cost 'a' yuan per kilogram, and cabbage costs 'b' yuan per kilogram. The caf... | The problem requires calculating the total cost for the school cafeteria's purchase of tomatoes and cabbage.
- Tomatoes cost \(a\) yuan per kilogram, and the cafeteria buys 30 kg. Therefore, the cost for tomatoes is \(30a\) yuan.
- Cabbage costs \(b\) yuan per kilogram, and the cafeteria buys 50 kg. Therefore, the co... |
20d46f66-8581-54e1-b6dd-868a9eebd039 | open-r1/OpenR1-Math-220k/olympiads | 29. The numbers $1, 2, 3, 4, 5, 6, 7, 8, 9, 0$ that we use for counting are called Arabic numerals, and they have become the universal symbols for numbers used around the world. Arabic numerals were actually invented by the ancient $\qquad$.
A. Indians
B. China
C. Babylon
D. Arabs | [
"Okay, let's see. The question is about the origin of Arabic numerals. The options are Indians, China, Babylon, and Arabs. Hmm, I remember that Arabic numerals are the numbers we use today, like 1, 2, 3, etc. But the name might be misleading because sometimes things are named after the people who popularized them r... | The numbers known as Arabic numerals were actually invented by ancient Indian mathematicians. Although they are named after the Arabs who later introduced these numerals to the Western world, the origin lies in India, particularly with the development of the decimal system and the concept of zero.
**Answer:**
\boxe... |
51055f8d-9e1a-5ba7-80f1-7d147ed6b767 | open-r1/OpenR1-Math-220k/amc_aime | The arithmetic mean of a set of $50$ numbers is $38$. If two numbers of the set, namely $45$ and $55$, are discarded,
the arithmetic mean of the remaining set of numbers is:
$\textbf{(A)}\ 38.5 \qquad \textbf{(B)}\ 37.5 \qquad \textbf{(C)}\ 37 \qquad \textbf{(D)}\ 36.5 \qquad \textbf{(E)}\ 36$ | [
"Okay, let me try to solve this problem. So, the question is about finding the new arithmetic mean after removing two numbers from a set of 50 numbers. The original mean is 38, and the two numbers being discarded are 45 and 55. The options are given from A to E, with the answers ranging from 36 to 38.5. Hmm, let's ... | The arithmetic mean of a set of 50 numbers is 38. Therefore, the total sum of all 50 numbers is calculated as follows:
\[
38 \times 50 = 1900
\]
When two numbers, 45 and 55, are discarded, their sum is:
\[
45 + 55 = 100
\]
Subtracting this sum from the original total sum gives:
\[
1900 - 100 = 1800
\]
The number of ... |
4e4f22a2-d659-59f1-8f82-2047f4ffe99f | open-r1/OpenR1-Math-220k/olympiads | ## Task 3
$1 \mathrm{~m}=\ldots \mathrm{cm} ; \quad 40 \mathrm{~mm}=\ldots \mathrm{cm} ; \quad 10 \mathrm{~cm}=\ldots \mathrm{dm}$ | [
"Okay, so I need to solve these three unit conversion problems. Let me take them one by one.",
"First, 1 meter equals how many centimeters. Hmm, I remember that the metric system is based on powers of ten, which should make this easier. The prefix \"centi-\" means one hundredth, right? So, 1 centimeter is 1/100 o... | $1 \mathrm{~m}=\boxed{100} \mathrm{~cm}$;
$40 \mathrm{~mm}=\boxed{4} \mathrm{~cm}$;
$10 \mathrm{~cm}=\boxed{1} \mathrm{~dm}$.
**Step-by-Step Explanations:**
1. **1 meter to centimeters:**
Since "centi-" means $1/100$, there are $100$ centimeters in a meter.
$1 \, \text{m} \times 100 \, \text{cm/m} = 1... |
b73517a9-5779-5745-8526-6fcc7bebc784 | open-r1/OpenR1-Math-220k/cn_k12 | Given sets $M=\{0, 1, 2, 3\}$, $N=\{-1, 1\}$, which of the following statements is true? ( )
A: $N \subseteq M$
B: $M \cup N = M$
C: $M \cap N = N$
D: $M \cap N = \{1\}$ | [
"Okay, let's see. I have this problem here with two sets, M and N. M is {0, 1, 2, 3} and N is {-1, 1}. The question is asking which of the statements A through D is true. Let me go through each option one by one.",
"First, let's recall what each symbol means. The subset symbol ⊆ means that every element of the le... | Given sets \( M = \{0, 1, 2, 3\} \) and \( N = \{-1, 1\} \), we need to determine which of the given statements is true.
- **Statement A**: \( N \subseteq M \).
-1 is an element of \( N \) but not of \( M \). Therefore, \( N \) is not a subset of \( M \). This statement is false.
- **Statement B**: \( M \cup N = ... |
9bb56dad-3644-5781-80f3-b1ccd6e8bbaf | open-r1/OpenR1-Math-220k/cn_k12 | Given that for all $x \in \mathbb{R}$, it holds that $f(x) = f(2-x)$, and the equation $f(x) = 0$ has five distinct real roots, find the sum of these five roots. | [
"Okay, so I need to find the sum of the five distinct real roots of the equation f(x) = 0, given that for all real numbers x, f(x) = f(2 - x). Hmm, let's see. The function f is symmetric about x = 1, right? Because replacing x with 2 - x gives the same function value. So the graph of f(x) is symmetric with respect ... | Given that \( f(x) = f(2 - x) \) for all \( x \in \mathbb{R} \), the function \( f \) is symmetric about the line \( x = 1 \). This symmetry implies that if \( a \) is a root of \( f(x) = 0 \), then \( 2 - a \) must also be a root.
Since the equation \( f(x) = 0 \) has five distinct real roots, one of these roots mus... |
8163c966-58fb-5abe-98e6-493ded8e43fa | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following options does not reflect the same type of correlation between two variables as the other three?
( )
A: A great teacher produces outstanding students
B: A rising tide lifts all boats
C: The brighter the moon, the fewer the stars
D: Climbing high to see far | [
"Okay, let's see. I need to figure out which of these options doesn't match the others in terms of correlation. The question is asking about the type of correlation between two variables. So, first, I should recall that correlation can be positive or negative. Positive correlation means as one variable increases, t... | The question asks which option does **not** reflect the same type of correlation (positive/negative) as the others.
- **A**, **B**, and **D** describe **positive correlations**:
- A: Better teachers correlate with better student outcomes.
- B: A rising tide (increase) lifts all boats (increase).
- D: Cli... |
9503b700-334e-5379-a645-f4dc5285db6a | open-r1/OpenR1-Math-220k/cn_k12 | Calculate: $(-2m^{3})^{2}$ is equal to ( ).
A: $-2m^{6}$
B: $4m^{6}$
C: $4m^{5}$
D: $-4m^{6}$ | [
"Okay, so I need to calculate (-2m³) squared and choose the correct answer from the options given. Let me think about how to approach this.",
"First, when you square a term that has both a coefficient and a variable raised to a power, you need to square each part separately. That is, both the coefficient (-2) and... | To calculate \((-2m^3)^2\), we need to square both the coefficient and the variable part separately.
1. **Coefficient part**: \((-2)^2 = 4\) because squaring a negative number results in a positive number.
2. **Variable part**: \((m^3)^2 = m^{3 \cdot 2} = m^6\) because when raising a power to another power, we multipl... |
19a65cec-f79b-5fd9-abe8-bda7cd96a23d | open-r1/OpenR1-Math-220k/cn_k12 | The coefficient of the linear term in the quadratic equation $x^{2}-x=0$ is ____. | [
"Okay, so I need to find the coefficient of the linear term in the quadratic equation x squared minus x equals zero. Let me start by recalling what a quadratic equation looks like in general form. I think the standard form is ax² + bx + c = 0, where a, b, and c are coefficients, right? The coefficient of the x² ter... | To find the coefficient of the linear term in the quadratic equation \(x^2 - x = 0\), we start by recalling the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are the coefficients of the quadratic term, linear term, and constant term, respectively.
Comparing the gi... |
0d3d603c-8ffe-5a1b-aef2-fc2fa55c74bd | open-r1/OpenR1-Math-220k/cn_k12 | The domain of the function $f\left(x\right)=\sqrt{x+2}+\dfrac{1}{x-1}$ is ( ).
A: $\left(-\infty ,1\right)\cup \left(1,+\infty \right)$
B: $ \left[-2,+\infty \right)$
C: $ \left[-2,1\right)\cup \left(1,+\infty \right)$
D: $ \left(1,+\infty \right)$ | [
"Okay, let's see. I need to find the domain of the function f(x) = sqrt(x + 2) + 1/(x - 1). Hmm, the domain is all the real numbers x for which the function is defined. Since this function has two parts: a square root and a rational expression, I need to consider the domains of each part separately and then find th... | To find the domain of the function \( f(x) = \sqrt{x+2} + \frac{1}{x-1} \), we need to consider the domains of each part separately and then find their intersection.
1. **Square Root Part**: The expression inside the square root, \( x + 2 \), must be non-negative:
\[
x + 2 \geq 0 \implies x \geq -2
\]
This... |
8168a9ca-ef06-5a73-8685-160e8fab1e44 | open-r1/OpenR1-Math-220k/cn_k12 | Given points $A(3,2,3)$ and $B(1,1,4)$, the coordinates of the midpoint of $A$ and $B$ are:
A: $(1, \frac{1}{2}, -\frac{1}{2})$
B: $(2, \frac{3}{2}, \frac{7}{2})$
C: $(4,3,7)$
D: $(-1, -\frac{1}{2}, \frac{1}{2})$ | [
"Okay, let's see. The problem is asking for the midpoint between points A(3,2,3) and B(1,1,4). The options are given as A, B, C, D with different coordinates. Hmm, midpoints. I remember that the midpoint formula in coordinate geometry is the average of the corresponding coordinates of the two points. So, for each c... | To find the midpoint of points \( A(3,2,3) \) and \( B(1,1,4) \), we use the midpoint formula, which averages the corresponding coordinates of the two points.
For the \( x \)-coordinate:
\[
\frac{3 + 1}{2} = \frac{4}{2} = 2
\]
For the \( y \)-coordinate:
\[
\frac{2 + 1}{2} = \frac{3}{2} = 1.5
\]
For the \( z \)-coo... |
9bddbb4f-5ea6-5fc9-88be-002067dfd7bd | open-r1/OpenR1-Math-220k/cn_k12 | In the Cartesian coordinate system, the coordinates of the point symmetric to point $P(-2,3)$ relative to the $x$-axis are ( ).
A: $(-2,-3)$
B: $(2,-3)$
C: $(-3,-2)$
D: $(3,-2)$ | [
"Okay, so I need to find the coordinates of the point that's symmetric to P(-2, 3) relative to the x-axis. Let me think. Symmetric relative to the x-axis... Hmm, when you reflect a point over the x-axis, what changes? I remember that reflecting over the x-axis affects the y-coordinate. Let me visualize it. If the o... | To find the coordinates of the point symmetric to \( P(-2, 3) \) relative to the \( x \)-axis, we need to reflect the point over the \( x \)-axis. Reflecting a point over the \( x \)-axis changes the sign of the \( y \)-coordinate while keeping the \( x \)-coordinate the same.
The original point is \( P(-2, 3) \). Ref... |
f6230031-bd7e-5c43-adf6-16fd7621c16a | open-r1/OpenR1-Math-220k/cn_k12 | Xiaochao is 1 meter and 36 centimeters tall, which, when written as a decimal, is ____ meters. | [
"Okay, let's see. The problem says Xiaochao is 1 meter and 36 centimeters tall, and we need to write that as a decimal in meters. Hmm. Alright, so first, I know that centimeters and meters are units of length in the metric system. Since the metric system is decimal-based, converting between them should be straightf... | Xiaochao is 1 meter and 36 centimeters tall. To convert this height into a decimal in meters, we need to convert the centimeters part to meters.
Since 1 meter is equal to 100 centimeters, we know that 1 centimeter is 0.01 meters. Therefore, 36 centimeters can be converted to meters by multiplying 36 by 0.01:
\[
36 \... |
fd26fccd-e9b4-59a5-a1ab-ad21c6c4c1bb | open-r1/OpenR1-Math-220k/cn_k12 | The medicine is on sale everywhere. You can get it at \_\_\_ chemist’s.
A. each B. some C. certain D. any | [
"Okay, let's tackle this question. The sentence is: \"The medicine is on sale everywhere. You can get it at _____ chemist’s.\" The options are A. each, B. some, C. certain, D. any.",
"First, I need to figure out which determiner fits best in the blank. The first sentence says the medicine is available everywhere,... | The sentence states that the medicine is available everywhere, implying it can be obtained at **any** chemist’s without restriction.
- **A. each** emphasizes individual instances but feels overly specific for this context.
- **B. some** and **C. certain** contradict "everywhere" by implying limited availability.
... |
33bb773e-7a75-53ae-be42-ae09d1eb72ae | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following statements is incorrect?
A: The diagonals of a parallelogram bisect each other.
B: The diagonals of a rectangle are equal and bisect each other.
C: The diagonals of a rhombus are equal and bisect each other.
D: The diagonals of a square are equal and perpendicular bisectors of each other. | [
"Okay, so I need to figure out which of these statements about the diagonals of different quadrilaterals is incorrect. Let's go through each option one by one.",
"Starting with option A: \"The diagonals of a parallelogram bisect each other.\" Hmm, I remember that in a parallelogram, which has opposite sides equal... | To determine which statement is incorrect, we analyze each option:
- **Statement A**: The diagonals of a parallelogram bisect each other.
This is correct because in a parallelogram, the diagonals intersect each other at their midpoints.
- **Statement B**: The diagonals of a rectangle are equal and bisect each oth... |
e6f0869a-29ad-546a-a41b-51833a93b4fa | open-r1/OpenR1-Math-220k/cn_k12 | In the process of deriving a contradiction using the method of proof by contradiction, which of the following should be used as conditions? ( )
① The judgment opposite to the conclusion, i.e., the assumption
② The conditions of the original proposition
③ Axioms, theorems, definitions, etc.
④ The original concl... | [
"Okay, let's see. The question is about proof by contradiction and which conditions should be used in deriving a contradiction. The options are combinations of ①, ②, ③, ④. Let me recall how proof by contradiction works.",
"So, in a proof by contradiction, you start by assuming the opposite of what you want to pro... | In a proof by contradiction, the process involves assuming the opposite of the conclusion (①), using the original conditions of the proposition (②), and applying axioms, theorems, and definitions (③) to derive a contradiction. The original conclusion (④) is not used as a condition but is what we aim to prove by contrad... |
5df736ab-72c8-5be6-9a8b-87031c1aebf6 | open-r1/OpenR1-Math-220k/cn_k12 | Convert radians to degrees: $$\frac {2}{3}\pi$$ = \_\_\_\_\_\_. | [
"Okay, so I need to convert 2/3 pi radians to degrees. Hmm, I remember that radians and degrees are both units for measuring angles. But how do you actually convert from one to the other? Let me think.",
"Wait, I think the key formula here is that pi radians is equal to 180 degrees. So, if I can use that as a con... | To convert \(\frac{2}{3}\pi\) radians to degrees, we use the conversion factor that \(\pi\) radians is equivalent to 180 degrees. The formula for converting radians to degrees is:
\[
\text{degrees} = \text{radians} \times \left(\frac{180^\circ}{\pi}\right)
\]
Applying this to \(\frac{2}{3}\pi\) radians:
\[
\frac{2}{... |
54fb7e63-c5dc-5087-b6cf-9134829f830c | open-r1/OpenR1-Math-220k/cn_k12 | The complex number $(1+i)(1- i) =\_\_\_\_\_\_\_\_\_$ | [
"Okay, so I need to figure out what (1 + i)(1 - i) is. Hmm, complex numbers can be tricky at first, but let me remember how multiplication works here.",
"First, I recall that when multiplying two binomials, we use the distributive property, also known as the FOIL method (First, Outer, Inner, Last). Let me try app... | To find the product of the complex numbers \((1 + i)(1 - i)\), we can use the distributive property (FOIL method):
1. Multiply the first terms: \(1 \cdot 1 = 1\).
2. Multiply the outer terms: \(1 \cdot (-i) = -i\).
3. Multiply the inner terms: \(i \cdot 1 = i\).
4. Multiply the last terms: \(i \cdot (-i) = -i^2\).
Ad... |
e9f21e54-cba3-5d24-891c-49eb98775c04 | open-r1/OpenR1-Math-220k/cn_k12 | Among the four numbers $0$, $-1$, $1$, $-5$, the smallest number is
A: $-5$
B: $-1$
C: $1$
D: $0$ | [
"Okay, so I need to figure out which of the numbers 0, -1, 1, or -5 is the smallest. Let me start by recalling how numbers are ordered on the number line. Negative numbers are to the left of zero, and positive numbers are to the right. The further left a number is, the smaller it is. Similarly, the further right a ... | To determine the smallest number among \(0\), \(-1\), \(1\), and \(-5\), we start by recalling the order of numbers on the number line. Negative numbers are to the left of zero, and the further left a number is, the smaller it is.
The numbers given are \(0\), \(-1\), \(1\), and \(-5\). Separating them into positive, ... |
084c25dc-5741-5202-9748-c173ae84342e | open-r1/OpenR1-Math-220k/cn_k12 | If $x \gt y \gt 0$, which of the following inequalities is correct?
A: $|x| \lt |y|$
B: $x^{2} \lt y^{2}$
C: $\frac{1}{x}＜\frac{1}{y}$
D: $\frac{x+y}{2}≤\sqrt{xy}$ | [
"Okay, let's see. The problem is asking which of the given inequalities is correct if x is greater than y and both are greater than 0. The options are A, B, C, D. Let me go through each one step by step.",
"First, since both x and y are positive numbers, their absolute values are just themselves. So |x| is x and ... | Given \( x > y > 0 \), we need to determine which of the given inequalities is correct.
**Option A: \( |x| < |y| \)**
Since \( x \) and \( y \) are positive, \( |x| = x \) and \( |y| = y \). Given \( x > y \), it follows that \( |x| > |y| \). Therefore, A is incorrect.
**Option B: \( x^2 < y^2 \)**
Squaring both ... |
4aaf0934-dc82-5ef5-aa12-03fd9a9598ac | open-r1/OpenR1-Math-220k/cn_k12 | Let $U=\{1,2,4,6,8\}$, $A=\{1,2,4\}$, $B=\{2,4,6\}$. Which of the following conclusions is correct?
A: $A\subseteq B$
B: $B\subseteq A$
C: $A\cap B=\{2\}$
D: $A\cap (\complement _{U}B)=\{1\}$ | [
"Okay, let's see. I need to figure out which of the options A, B, C, or D is correct based on the given sets. Let me start by recalling the definitions of the set operations mentioned here: subset, intersection, and complement.",
"First, the universal set U is {1,2,4,6,8}. Set A is {1,2,4}, and set B is {2,4,6}. ... | Given the sets \( U = \{1, 2, 4, 6, 8\} \), \( A = \{1, 2, 4\} \), and \( B = \{2, 4, 6\} \), we need to determine which of the given options is correct.
- **Option A**: \( A \subseteq B \). This is incorrect because the element 1 in set \( A \) is not in set \( B \).
- **Option B**: \( B \subseteq A \). This is incor... |
62a2d7f9-4094-5161-bcf2-026e799de7c6 | open-r1/OpenR1-Math-220k/cn_k12 | Given that $p$ is a true statement, and $q$ is a false statement, then ( )
A: The statement $p \land q$ is true.
B: The statement $p \lor q$ is false.
C: The statement $\lnot p$ is true.
D: The statement $\lnot q$ is true. | [
"Okay, let's see. I need to figure out which of the options A, B, C, or D is correct given that p is true and q is false. Hmm, okay, let me start by recalling what each logical connective means.",
"First, p is true, q is false. So, let's go through each option one by one.",
"Option A: The statement p ∧ q is tru... | Given that \( p \) is a true statement and \( q \) is a false statement, we need to determine which of the given options is correct.
- **Option A**: The statement \( p \land q \) is true.
The logical "and" (∧) is true only if both statements are true. Since \( p \) is true and \( q \) is false, \( p \land q \) is ... |
7fde4650-61ee-57eb-9ef4-9ad1b4123e08 | open-r1/OpenR1-Math-220k/cn_k12 | Currently, mobile WeChat payment has become a common payment method, greatly loved by consumers. It can be said that "with a mobile phone in hand, travel worry-free." If WeChat receives $22$ yuan, it is recorded as $+22$ yuan. Then, if WeChat spends $10.6$ yuan, it is recorded as: ______ yuan. | [
"Alright, let me see. The problem here is about how WeChat payment records transactions. So, when you receive money, they note it as a positive number, like +22 yuan. Now, the question is, if you spend 10.6 yuan, how would they record that?",
"Okay, first, I need to make sure I understand the system. So receiving... | If WeChat receives $22$ yuan, it is recorded as $+22$ yuan. This indicates that receiving money is recorded as a positive value. Conversely, spending money would be the opposite transaction, so it should be recorded as a negative value. Therefore, if WeChat spends $10.6$ yuan, it is recorded as $-10.6$ yuan.
\[
\boxed... |
0fe604f7-6b48-5bed-a46b-00bfefdbc065 | open-r1/OpenR1-Math-220k/olympiads | You are writing a multiple choice test and on one question you guess and pick an answer at random. If there are five possible choices $(A, B, C, D, E)$, what is the probability that you guessed correctly?
(A) $\frac{1}{5}$
(B) $\frac{5}{5}$
(C) $\frac{4}{5}$
(D) $\frac{2}{5}$
(E) $\frac{3}{5}$ | [
"Okay, so I need to figure out the probability of guessing the correct answer on a multiple-choice question that has five options: A, B, C, D, and E. Hmm, let me start by recalling what probability means. Probability is the chance that a certain event will happen, right? It's calculated by dividing the number of fa... | To determine the probability of guessing the correct answer on a multiple-choice question with five possible choices (A, B, C, D, E), we start by noting that probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes.
Here, the favorable outcome is selecting the cor... |
a4c3c7ab-1413-5fe7-9e13-ce3937680cb0 | open-r1/OpenR1-Math-220k/cn_k12 | Is an 8 kg block of wood heavier than an 8000 g brick? (Judge true or false) | [
"Okay, so the question is whether an 8 kg block of wood is heavier than an 8000 g brick. Hmm, let me think. I know that kilograms and grams are units of mass, and heavier refers to weight, but in everyday language, people often use them interchangeably. Wait, but if we're just comparing masses, then it's about whic... | The block of wood is 8 kg, and the brick is 8000 g. Converting 8 kg to grams:
\[ 8\,\text{kg} \times 1000\,\frac{\text{g}}{\text{kg}} = 8000\,\text{g} \]
Both masses are equal (8000 g = 8000 g). Since weight depends on mass and gravity (assumed identical here), neither is heavier.
**Answer:** \boxed{False} |
715a26a6-4b8b-5a6a-b60b-68d743806463 | open-r1/OpenR1-Math-220k/olympiads | 1. Given the set $M$ satisfies $M \subseteq\{0,1,2,3,4\}$ and $M \subseteq\{0,2,4,8\}$, the maximum number of elements in set $M$ is ( ).
A. 1
B. 2
C. 3
D. 6 | [
"Okay, so I need to figure out the maximum number of elements that set M can have. The problem says that M is a subset of both {0,1,2,3,4} and {0,2,4,8}. Hmm, subsets... right, so M has to be contained within both of these sets. That means M can only have elements that are common to both sets, right? Because if it'... | Given the set \( M \) must satisfy \( M \subseteq \{0,1,2,3,4\} \) and \( M \subseteq \{0,2,4,8\} \), we need to find the maximum number of elements in \( M \).
1. **Identify the intersection of the two sets**:
- The first set is \(\{0,1,2,3,4\}\).
- The second set is \(\{0,2,4,8\}\).
- The intersection of th... |
3de1787c-c690-5388-9cb8-082138b9ad81 | open-r1/OpenR1-Math-220k/cn_k12 | A number $x$ is randomly selected from the interval $[-1, 2]$. Determine the probability that $x$ belongs to the interval $[0, 1]$. | [
"Okay, so I need to find the probability that a randomly selected number x from the interval [-1, 2] falls into the interval [0, 1]. Hmm, let's think about how probability works with continuous intervals. I remember that when dealing with uniform distributions over intervals, the probability of a number falling int... | To determine the probability that a number \( x \) randomly selected from the interval \([-1, 2]\) belongs to the interval \([0, 1]\), we need to consider the lengths of these intervals.
1. The total length of the interval \([-1, 2]\) is calculated as:
\[
2 - (-1) = 3
\]
2. The length of the subinterval \([0... |
704b5f6a-7b5f-5a74-908f-803f5dd1d0d8 | open-r1/OpenR1-Math-220k/olympiads | ## Task 1
In every 100 m preliminary race, 8 athletes start. The two best runners will participate in the final. How many athletes participated in the preliminaries if 8 of them competed in the final. | [
"Okay, let's see. So the problem is about figuring out how many athletes participated in the preliminaries if 8 of them ended up competing in the final. Hmm. Let me read the question again to make sure I understand it correctly.",
"\"In every 100 m preliminary race, 8 athletes start. The two best runners will par... | To determine the number of athletes who participated in the preliminaries, we start by noting that each preliminary race has 8 athletes and the top 2 from each race qualify for the final. Given that 8 athletes competed in the final, we need to find out how many preliminary races were held.
1. Each preliminary race con... |
57313792-beae-5590-9b80-d21b6f61810a | open-r1/OpenR1-Math-220k/olympiads | 61. To pave a 112-meter road, Team A started and paved for several days, paving 12 meters per day. Team A had to leave due to some matters, and Team B took over, paving 20 meters per day. When the 112-meter road was completed, it was found that the average daily paving by Teams A and B was 14 meters. How many days did ... | [
"Okay, let's see here. So there's this problem where Team A starts paving a 112-meter road, doing 12 meters each day. Then Team B takes over and does 20 meters per day. The total road is 112 meters, and the average daily paving by both teams together is 14 meters. We need to find out how many days Team A worked.",
... | Let \( x \) be the number of days Team A paved the road, and \( y \) be the number of days Team B paved the road. We know the following:
1. The total length of the road is 112 meters:
\[
12x + 20y = 112
\]
2. The average daily paving by both teams was 14 meters, which means the total number of days is:
\[
... |
51a59986-96ee-5f37-9773-259e97f35592 | open-r1/OpenR1-Math-220k/cn_k12 | Given the sets $A = \{2, 5, 6\}$ and $B = \{3, 5\}$, then the union of sets $A \cup B = \_\_\_\_$. | [
"Okay, so I need to find the union of sets A and B. Let me recall what a union of sets means. If I remember correctly, the union of two sets A and B is a set that contains all the elements that are in A, in B, or in both. So basically, I need to combine the elements from both sets but without repeating any elements... | To find the union of sets \( A \) and \( B \), we need to combine all elements from both sets without repeating any duplicates.
Set \( A = \{2, 5, 6\} \) contains the elements 2, 5, and 6. Set \( B = \{3, 5\} \) contains the elements 3 and 5.
Combining these elements, we list all unique elements from both sets:
- F... |
aafc275e-346f-58b4-80c9-1605e7207e6e | open-r1/OpenR1-Math-220k/cn_k12 | $i^{2010} =$ （ ）
A: $2i$
B: $-i$
C: $-1$
D: $1$ | [
"Okay, let's see. The problem is to find the value of i raised to the power of 2010, and then choose the correct option among A, B, C, D. The options are A: 2i, B: -i, C: -1, D: 1. Hmm, I need to figure out what i^2010 is.",
"First, I remember that the powers of i cycle every four exponents. Let me recall the cyc... | To determine the value of \(i^{2010}\), we use the cyclicity of the powers of \(i\), which cycle every four exponents: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).
We need to find the remainder when 2010 is divided by 4. Performing the division, we get:
\[ 2010 \div 4 = 502 \text{ remainder } 2 \]
This m... |
a282d45c-e795-53a2-a01e-1c8d352e5bfb | open-r1/OpenR1-Math-220k/cn_k12 | Which of the following equations is a linear equation in two variables?
A: $xy=2$
B: $\frac{1}{x}+y=3$
C: $3x+y^{2}=1$
D: $2x+y=5$ | [
"Okay, so I need to figure out which of these equations is a linear equation in two variables. Let me remember what a linear equation in two variables is. From what I recall, a linear equation is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not bot... | To determine which equation is a linear equation in two variables, we recall that a linear equation can be written in the form \(ax + by + c = 0\) where \(a\), \(b\), and \(c\) are real numbers, and \(a\) and \(b\) are not both zero. The variables must be to the first power and not multiplied together or part of any ot... |
b1d647a1-56e6-5fa2-a7e6-480a76478431 | open-r1/OpenR1-Math-220k/cn_k12 | If the coordinates of point $A(m,n)$ with respect to the origin are $\left(-3,2\right)$, then $m+n=$____. | [
"Okay, so I have this problem here: If the coordinates of point A(m, n) with respect to the origin are (-3, 2), then m + n equals what? Hmm, let me think.",
"First, I need to recall what coordinates mean. In a Cartesian plane, any point is represented by (x, y), where x is the horizontal distance from the origin ... | The coordinates of point \( A(m, n) \) with respect to the origin are given as \((-3, 2)\). This means that \( m \) (the x-coordinate) is \(-3\) and \( n \) (the y-coordinate) is \(2\).
To find \( m + n \):
\[
m + n = -3 + 2 = -1
\]
Thus, the value of \( m + n \) is \(\boxed{-1}\). |
094fc99f-4063-5642-ae4d-5e8c3eec06b6 | open-r1/OpenR1-Math-220k/cn_k12 | Solve the quadratic equation $x^{2}-8x+6=0$ using the completing the square method. The correct completion is:
A: $(x-4)^{2}=22$
B: $(x-4)^{2}=10$
C: $(x+4)^{2}=22$
D: $(x+4)^{2}=10$ | [
"Okay, let me try to solve this quadratic equation using the completing the square method. The equation is x² - 8x + 6 = 0. Hmm, I remember that completing the square involves turning the quadratic part into a perfect square trinomial. Let me recall the steps.",
"First, I need to make sure the equation is in the ... | To solve the quadratic equation \(x^2 - 8x + 6 = 0\) using the completing the square method, follow these steps:
1. Start with the given equation:
\[
x^2 - 8x + 6 = 0
\]
2. Move the constant term to the other side of the equation:
\[
x^2 - 8x = -6
\]
3. Find the coefficient of \(x\), which is \(-8\... |
49d0cde8-eb9b-5d0d-b416-b0ffd42d8b74 | open-r1/OpenR1-Math-220k/olympiads | Janne buys a camera which costs $\$ 200.00$ without tax. If she pays $15 \%$ tax on this purchase, how much tax does she pay?
(A) $\$ 30.00$
(B) $\$ 18.00$
(C) $\$ 20.00$
(D) $\$ 15.00$
(E) $\$ 45.00$ | [
"Okay, so Janne buys a camera that costs $200 before tax, and there's a 15% tax on it. I need to figure out how much tax she pays. Hmm, let's see. The question is asking for the amount of tax, not the total price, right? So it's just 15% of $200.",
"First, I remember that percentages can be converted to decimals ... | Janne buys a camera that costs $200 before tax, and she needs to pay 15% tax on this purchase. To find the amount of tax, we calculate 15% of $200.
First, convert 15% to a decimal:
\[ 15\% = 0.15 \]
Next, multiply the original price by the tax rate in decimal form:
\[ 200 \times 0.15 = 30 \]
To verify, we can brea... |
2119e027-c7bc-5862-8f17-aa5fd635e418 | open-r1/OpenR1-Math-220k/cn_k12 | The imaginary part of the complex number $z=i(1-i)$ is. | [
"Okay, so I need to find the imaginary part of the complex number z, which is given by z = i times (1 minus i). Hmm, let's start by recalling what a complex number is. A complex number is usually written as a + bi, where 'a' is the real part and 'b' is the imaginary part. So, the imaginary part is the coefficient o... | To find the imaginary part of the complex number \( z = i(1 - i) \), we start by expanding the multiplication:
\[
z = i(1 - i)
\]
Distribute \( i \) across the parentheses:
\[
z = i \cdot 1 - i \cdot i
\]
This simplifies to:
\[
z = i - i^2
\]
Since \( i^2 = -1 \), we substitute:
\[
z = i - (-1)
\]
Which further... |
9cbd7c85-504f-55a1-85d2-ddb1fe83107a | open-r1/OpenR1-Math-220k/cn_k12 | If the slope of the tangent line to the curve $y=2x^2+1$ at point $M$ is $-4$, then the coordinates of point $M$ are _____. | [
"Okay, so I need to find the coordinates of point M on the curve y = 2x² + 1 where the slope of the tangent line is -4. Hmm, let me remember how to approach this.",
"First, I know that the slope of the tangent line to a curve at a particular point is given by the derivative of the function at that point. So, mayb... | To find the coordinates of point \( M \) on the curve \( y = 2x^2 + 1 \) where the slope of the tangent line is \(-4\), we start by computing the derivative of the function. The derivative \( \frac{dy}{dx} \) gives the slope of the tangent line at any point \( x \).
The derivative of \( y = 2x^2 + 1 \) is:
\[
\frac{dy... |
f17cd77e-e875-5c18-8045-7def37937e12 | open-r1/OpenR1-Math-220k/cn_k12 | Determine the value of $a$ in the equation $\int_{0}^{a}{2dx=2}$. | [
"Okay, so I need to find the value of 'a' in the equation ∫ from 0 to a of 2 dx equals 2. Hmm, let's start by recalling some basic integral rules. The integral of a constant function, like 2, with respect to x should be straightforward.",
"First, the integral of 2 dx is 2x, right? Because when you integrate a con... | To determine the value of \( a \) in the equation \(\int_{0}^{a} 2 \, dx = 2\), we start by evaluating the integral. The integral of the constant function 2 with respect to \( x \) is:
\[
\int_{0}^{a} 2 \, dx = 2x \Big|_{0}^{a} = 2a - 2 \cdot 0 = 2a
\]
We are given that this integral equals 2, so we set up the equati... |
94f30313-35bc-54c9-98e2-df6da7ac158f | open-r1/OpenR1-Math-220k/cn_k12 | Xiaoqiang's average score in Chinese, Mathematics, and English is 97 points, with a score of 100 in Mathematics. Then, the total score for Chinese and English is ___ points. | [
"Okay, so Xiaoqiang has an average score of 97 in three subjects: Chinese, Mathematics, and English. His math score is 100. We need to find the total score for Chinese and English. Hmm, let me think.",
"First, average score is total points divided by the number of subjects, right? So if the average is 97 over thr... | Xiaoqiang's average score in Chinese, Mathematics, and English is 97 points. To find the total score for all three subjects, we multiply the average by the number of subjects:
\[
\text{Total score} = 97 \times 3 = 291
\]
We know his Mathematics score is 100. To find the combined score for Chinese and English, we subt... |
e4290238-27c3-53b4-9d73-43938f608caf | open-r1/OpenR1-Math-220k/cn_k12 | Given sets $A=\{1, 2, 5\}$ and $B=\{1, 3, 5\}$, then $A \cap B=$ ? | [
"Alright, so I need to find the intersection of sets A and B. Let me remember, the intersection of two sets is the set of elements that are common to both sets. So, basically, I need to look at each element in set A and check if it's also in set B. Then, collect all those common elements into a new set, which will ... | To find the intersection of sets \(A\) and \(B\), we need to identify the elements that are common to both sets.
Set \(A\) is \(\{1, 2, 5\}\) and set \(B\) is \(\{1, 3, 5\}\).
1. Check each element of set \(A\) to see if it is also in set \(B\):
- The element \(1\) is in both \(A\) and \(B\).
- The element \(2... |
eec737a6-12ff-55f1-832d-4e10878c29b3 | open-r1/OpenR1-Math-220k/cn_k12 | The relationship between the electricity bill $y$ (in yuan) and the electricity consumption $x$ (in kilowatt-hours) is given by the equation $y = 0.54x$. Identify the constant in this equation. | [
"Okay, let's see. The problem says that the electricity bill y in yuan is related to the electricity consumption x in kilowatt-hours by the equation y = 0.54x. And I need to identify the constant in this equation. Hmm, constants in equations are the numbers that don't change, right? So variables like x and y can va... | The constant in the equation \( y = 0.54x \) is the numerical coefficient that does not change regardless of the values of \( x \) or \( y \). Here, \( 0.54 \) is the constant because it represents the fixed rate per kilowatt-hour of electricity consumed.
**Answer:** The constant is \boxed{0.54}. |
eefcb7ab-27fa-58c0-8f7d-597083347559 | open-r1/OpenR1-Math-220k/cn_k12 | According to many years of meteorological statistics, the probability of rain on June 1st in a certain area is 0.45, and the probability of cloudy weather is 0.20. Therefore, the probability of sunny weather on that day is ( )
A: 0.65
B: 0.55
C: 0.35
D: 0.75 | [
"Okay, let's see here. The problem is about calculating the probability of sunny weather on June 1st in a certain area. They've given the probability of rain as 0.45 and the probability of cloudy weather as 0.20. The options are A: 0.65, B: 0.55, C: 0.35, D: 0.75. Hmm, I need to figure out which one is correct.",
... | The problem requires calculating the probability of sunny weather on June 1st given the probabilities of rain and cloudy weather.
1. The probability of rain is 0.45.
2. The probability of cloudy weather is 0.20.
Assuming the weather conditions (rain, cloudy, and sunny) are mutually exclusive and collectively exhaust... |
3ffe4832-e0f2-59b7-ae9f-de4c70c2327c | open-r1/OpenR1-Math-220k/cn_k12 | Given the equation of line $l$ as $y= \frac {1}{2}x+1$, the slope of line $l$ is $(\ \ \ \ )$
A: $ \frac {1}{2}$
B: $-2$
C: $2$
D: $- \frac {1}{2}$ | [
"Okay, so I need to figure out the slope of the line l given by the equation y = (1/2)x + 1. Hmm, let me remember how to find the slope from an equation. I think the slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. Is that right? Yeah, I've seen that before in my math cla... | To determine the slope of the line \( l \) given by the equation \( y = \frac{1}{2}x + 1 \), we recall the slope-intercept form of a line, which is \( y = mx + b \). In this form, \( m \) represents the slope and \( b \) represents the y-intercept.
The given equation \( y = \frac{1}{2}x + 1 \) is already in slope-inte... |
59641455-a231-571e-88a6-198e82f1a49f | open-r1/OpenR1-Math-220k/cn_k12 | Given the general term formula of a sequence $a_n$ as follows:
$$
a_n = \begin{cases}
3n + 1, & \text{if } n \text{ is odd} \\
2n - 2, & \text{if } n \text{ is even}
\end{cases}
$$
then the value of $a_2a_3$ is ( )
A: 70
B: 28
C: 20
D: 8 | [
"Okay, let's see. I need to find the value of a_2 times a_3. The sequence a_n has different formulas depending on whether n is odd or even. Let me break this down step by step.",
"First, let's understand the general term formula. For any term a_n in the sequence, if the position n is odd, then the term is calcula... | To find the value of \(a_2 \cdot a_3\), we need to determine the values of \(a_2\) and \(a_3\) using the given formulas for the sequence.
1. **Calculating \(a_2\)**:
- Since \(n = 2\) is even, we use the formula for even \(n\): \(2n - 2\).
- Substituting \(n = 2\):
\[
a_2 = 2 \cdot 2 - 2 = 4 - 2 = 2
... |
8e015852-97cc-598f-81f1-91ae43e8a53f | open-r1/OpenR1-Math-220k/cn_k12 | Given $\frac{a}{b}=2$, find $\frac{a}{a-b}$. | [
"Okay, so I need to find the value of \\(\\frac{a}{a - b}\\) given that \\(\\frac{a}{b} = 2\\). Hmm, let's see. First, maybe I can express one variable in terms of the other using the given ratio. Since \\(\\frac{a}{b} = 2\\), that means \\(a = 2b\\), right? Because if I multiply both sides by \\(b\\), I get \\(a =... | Given \(\frac{a}{b} = 2\), we can express \(a\) in terms of \(b\) as \(a = 2b\). Substituting \(a = 2b\) into the expression \(\frac{a}{a - b}\):
\[
\frac{a}{a - b} = \frac{2b}{2b - b}
\]
Simplifying the denominator:
\[
2b - b = b
\]
Thus, the expression becomes:
\[
\frac{2b}{b} = 2
\]
Therefore, the value of \(\... |
fad768a4-ba62-5b23-a177-2bcb11b8c4b8 | open-r1/OpenR1-Math-220k/cn_k12 | If ¬(p∧q) is a false proposition, then ( )
A: p is true, q is false
B: p is false, q is false
C: p is true, q is true
D: p is false, q is true | [
"Okay, let's see. The problem says that ¬(p ∧ q) is a false proposition. Then we need to figure out which of the options A to D is correct. Hmm, let me recall some logic basics. So, ¬ is the negation symbol, and ∧ is the conjunction, meaning \"and\". So ¬(p ∧ q) being false means that the negation of \"p and q\" is... | Given that ¬(p ∧ q) is a false proposition, we need to determine the truth values of p and q.
1. Since ¬(p ∧ q) is false, its negation (p ∧ q) must be true.
2. The conjunction p ∧ q is true only when both p and q are true.
3. Evaluating the options:
- Option A: p is true, q is false. This would make p ∧ q false, so... |
42ab7c59-f5d0-5429-bf2f-8e0b96ce2447 | open-r1/OpenR1-Math-220k/cn_k12 | $\sqrt[3]{{(\pi-2)}^{3}}=\left(\ \ \right)$
A: $\pi -4$
B: $\pi -3$
C: $\pi -2$
D: $\pi -1$ | [
"Okay, let's see here. The problem is asking for the cube root of (π - 2) cubed. So, written out, that's the cube root of [(π - 2)^3]. And the options are A through D, with C being π - 2. Hmm, seems straightforward, but maybe there's a trick?",
"Wait, so normally, if you have the cube root of something cubed, lik... | The problem asks for the cube root of \((\pi - 2)^3\).
To solve this, we recognize that the cube root and the cube are inverse operations. Therefore, the cube root of \((x)^3\) is simply \(x\). Applying this to the given problem:
\[
\sqrt[3]{(\pi - 2)^3} = \pi - 2
\]
We verify that \(\pi - 2\) is a positive number ... |
735f9bb0-75b1-5e9c-b8a5-039dbb54f7e5 | open-r1/OpenR1-Math-220k/olympiads | 1. (3 points) $20.16 \times 32 + 2.016 \times 680=$ | [
"Okay, let's see. I need to calculate 20.16 multiplied by 32 plus 2.016 multiplied by 680. Hmm, both terms have numbers that look similar, maybe there's a way to factor something out or make it simpler.",
"First, let me write down the problem again to make sure I have it right: 20.16 × 32 + 2.016 × 680. Yeah, tha... | To solve the problem \(20.16 \times 32 + 2.016 \times 680\), we can factor out the common term \(2.016\):
1. Notice that \(20.16 = 2.016 \times 10\).
2. Rewrite the original expression using this observation:
\[
20.16 \times 32 + 2.016 \times 680 = (2.016 \times 10) \times 32 + 2.016 \times 680
\]
3. Factor o... |
2be0ae73-c168-5118-b11b-bda34e7296ad | open-r1/OpenR1-Math-220k/olympiads | 2. 15 students join a summer course. Every day, 3 students are on duty after school to clean the classroom. After the course, it was found that every pair of students have been on duty together exactly once. How many days does the course last for?
(1 mark)
2. 15 students join a summer course. Every day, 3 students are ... | [
"Okay, let's see. The problem is about 15 students in a summer course. Each day, 3 students are on duty to clean the classroom. After the course, it's found that every pair of students has been on duty together exactly once. We need to find out how many days the course lasted.",
"Hmm, so first, let me parse the p... | The problem involves 15 students where each day 3 students are on duty, and every pair of students must have been on duty together exactly once.
To solve this, we first calculate the total number of unique pairs of students, which is given by the combination formula \( \binom{15}{2} \):
\[
\binom{15}{2} = \frac{15 \t... |
f1e517fb-8bc1-5461-9106-80cd3ea0d06c | open-r1/OpenR1-Math-220k/cn_k12 | Given the set $M=\{0,1\}$, and the set $N=\{x|x^{2}+x=0\}$, then the set $M\cup N$ equals to ( ).
A: $0$
B: $\{0\}$
C: $\varnothing$
D: $\{-1,0,1\}$ | [
"Okay, let's try to figure out this problem step by step. So, the question is about sets. We have set M defined as {0, 1}, and set N defined as all x such that x squared plus x equals zero. Then we need to find the union of M and N, and choose the correct option from the given choices.",
"First, let's recall what... | Given the set \( M = \{0, 1\} \), we need to find the set \( N \) defined by the equation \( x^2 + x = 0 \). Solving this equation, we factor it as \( x(x + 1) = 0 \), which gives the solutions \( x = 0 \) or \( x = -1 \). Therefore, the set \( N \) is \( \{-1, 0\} \).
Next, we find the union of sets \( M \) and \( N ... |
185c641b-c732-59d2-b7a1-1b6cbcbea878 | open-r1/OpenR1-Math-220k/olympiads | 3. In the Caribbean, loggerhead turtles lay three million eggs in twenty thousand nests. On average, how many eggs are in each nest?
A 15
B 150
C 1500
D 15000
E 150000 | [
"Okay, so I need to figure out the average number of eggs per nest for these loggerhead turtles in the Caribbean. The problem says there are three million eggs laid in twenty thousand nests. Hmm, averages... right, average is total divided by the number of items. So, if there are 3,000,000 eggs and 20,000 nests, I ... | To find the average number of eggs per nest, divide the total number of eggs by the number of nests:
\[
\frac{3{,}000{,}000 \text{ eggs}}{20{,}000 \text{ nests}} = \frac{3{,}000{,}000 \div 10{,}000}{20{,}000 \div 10{,}000} = \frac{300}{2} = 150
\]
Verification: \(20{,}000 \times 150 = 3{,}000{,}000\), confirming the ... |
d66b5d06-0300-56f9-95a4-fca6c2cb9f0d | open-r1/OpenR1-Math-220k/cn_k12 | To make the fraction $\frac{3}{{x-2}}$ meaningful, the value of $x$ should satisfy ______. | [
"Okay, so I need to figure out what value of x makes the fraction 3 divided by (x minus 2) meaningful. Hmm, let's start by recalling what makes a fraction undefined or meaningless. From what I remember, a fraction becomes undefined when its denominator is zero because you can't divide by zero. So, in this case, the... | To determine the value of \( x \) that makes the fraction \(\frac{3}{x-2}\) meaningful, we need to ensure the denominator is not zero.
1. Set the denominator equal to zero:
\[
x - 2 = 0
\]
2. Solve for \( x \):
\[
x = 2
\]
When \( x = 2 \), the denominator becomes zero, which makes the fraction und... |
62ff0e26-19a0-5637-bb87-9664231ef3f0 | open-r1/OpenR1-Math-220k/cn_k12 | The reciprocal of $-3$ is
A: $-\frac{1}{3}$
B: $\frac{1}{3}$
C: $-3$
D: $3$ | [
"Okay, so the question is asking for the reciprocal of -3, and there are four options to choose from. Let me think about what a reciprocal means. Hmm, reciprocal... right, the reciprocal of a number is 1 divided by that number. So, if you have a number x, its reciprocal should be 1/x. Let me make sure I remember th... | The reciprocal of a number \( x \) is \( \frac{1}{x} \). For \(-3\), this reciprocal is \(\frac{1}{-3} = -\frac{1}{3}\). Multiplying \(-3\) by \(-\frac{1}{3}\) yields \(1\), confirming the correct reciprocal. Among the options, only \(-\frac{1}{3}\) (Option A) satisfies this condition.
\(\boxed{A}\) |
a6e3530e-55f7-5d6c-a259-3ec0fd38ff61 | open-r1/OpenR1-Math-220k/cn_k12 | The number of subsets of the set $\{x | x^2 - 1 = 0\}$ is ( )
A: 1
B: 2
C: 3
D: 4 | [
"Okay, let's see here. The problem is asking for the number of subsets of the set {x | x² - 1 = 0}, and the options are A:1, B:2, C:3, D:4. Hmm. Alright, so first, I need to figure out what the set {x | x² - 1 = 0} actually is. The notation here is set-builder notation, right? So this is the set of all x such that ... | To determine the number of subsets of the set \(\{x | x^2 - 1 = 0\}\), we first need to identify the elements of the set. Solving the equation \(x^2 - 1 = 0\):
\[
x^2 - 1 = 0 \implies x^2 = 1 \implies x = \pm 1
\]
Thus, the set is \(\{1, -1\}\), which has 2 elements. The number of subsets of a set with \(n\) elements... |
0c8a6e6c-8138-5ec7-a796-611ac57d9146 | open-r1/OpenR1-Math-220k/cn_k12 | The real part of the complex number $z=(1+i)(1+2i)$ ($i$ is the imaginary unit) is. | [
"Okay, let me try to figure out the real part of the complex number z = (1 + i)(1 + 2i). Hmm, complex numbers can sometimes be tricky, but I remember that multiplying them involves using the distributive property, like the FOIL method for binomials. Let me recall: FOIL stands for First, Outer, Inner, Last, right? S... | To find the real part of the complex number \( z = (1 + i)(1 + 2i) \), we use the distributive property (FOIL method):
1. Multiply the First terms: \( 1 \cdot 1 = 1 \).
2. Multiply the Outer terms: \( 1 \cdot 2i = 2i \).
3. Multiply the Inner terms: \( i \cdot 1 = i \).
4. Multiply the Last terms: \( i \cdot 2i = 2i^2... |
72a2a3bb-c65d-5d83-b596-442747a04dc8 | open-r1/OpenR1-Math-220k/cn_k12 | Divide a set of data into 6 groups, where the sum of the frequencies of the first 3 groups is 0.65, and the sum of the frequencies of the last two groups is 0.32. Then, the frequency of the fourth group is ( )
A: 0.02
B: 0.01
C: 0.03
D: 0.04 | [
"Okay, so I have this problem here where I need to divide a set of data into 6 groups. The conditions given are that the sum of the frequencies of the first 3 groups is 0.65, and the sum of the frequencies of the last two groups is 0.32. Then, I need to find the frequency of the fourth group, with the options given... | Given a set of data divided into 6 groups, we need to find the frequency of the fourth group. The conditions provided are:
1. The sum of the frequencies of the first 3 groups is 0.65.
2. The sum of the frequencies of the last two groups is 0.32.
We denote the frequencies of the six groups as \( f_1, f_2, f_3, f_4, f_... |
10b6d5a6-4d1e-504c-ab78-f691e9d46e29 | open-r1/OpenR1-Math-220k/olympiads | 77. $\int \frac{\sin 2 x}{\cos x} d x$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
77. $\int \frac{\sin 2 x}{\cos x} d x$. | [
"Okay, so I have to solve the integral of sin(2x) divided by cos(x) with respect to x. Let me think about how to approach this.",
"First, I remember that sin(2x) is a trigonometric identity. Maybe I can use that to simplify the expression. The double angle formula for sine is sin(2x) = 2 sinx cosx. Let me write t... | 77. $\int \frac{\sin 2 x}{\cos x} d x$.
First, use the double-angle identity $\sin 2x = 2 \sin x \cos x$ to simplify the integrand:
$$
\int \frac{\sin 2x}{\cos x} \, dx = \int \frac{2 \sin x \cos x}{\cos x} \, dx = \int 2 \sin x \, dx.
$$
Integrate $2 \sin x$:
$$
\int 2 \sin x \, dx = -2 \cos x + C.
$$
**Ans... |
92453736-6122-52c9-b6aa-2ff3c48959da | open-r1/OpenR1-Math-220k/cn_k12 | Point $A(5,-4)$ lies in which quadrant?
A: Quadrant I
B: Quadrant II
C: Quadrant III
D: Quadrant IV | [
"Okay, so I need to figure out in which quadrant the point A(5, -4) lies. Let me start by recalling what the quadrants are in a coordinate plane. From what I remember, the coordinate plane is divided into four quadrants by the x-axis and y-axis. The quadrants are labeled I, II, III, and IV, starting from the upper ... | Point \( A(5, -4) \) has an x-coordinate of 5 (positive) and a y-coordinate of -4 (negative). In the coordinate plane, Quadrant IV is where the x-coordinate is positive and the y-coordinate is negative.
**Answer:** D: Quadrant IV
\(\boxed{D}\) |
b2a4b0b6-0fb0-5426-b9d1-a926e29348c2 | open-r1/OpenR1-Math-220k/cn_k12 | Given the set M={x|-2≤x＜2} and the set N={x|x≥-2}, find M∩N=（ ）
A: {x|-2≤x＜2}
B: {x|x≥-2}
C: {x|x＜2}
D: {x|1＜x＜2} | [
"Okay, so I have this problem here where I need to find the intersection of two sets, M and N. Let me start by understanding what each set is.",
"Set M is defined as {x | -2 ≤ x < 2}. That means all real numbers x such that x is greater than or equal to -2 and less than 2. So, it includes -2 but doesn't include 2... | Given the set \( M = \{x \mid -2 \leq x < 2\} \) and the set \( N = \{x \mid x \geq -2\} \), we need to find their intersection \( M \cap N \).
- Set \( M \) includes all real numbers from -2 (inclusive) to 2 (exclusive).
- Set \( N \) includes all real numbers from -2 (inclusive) to infinity.
The intersection \( M \... |
a5be959f-f25d-5275-aec5-5985319246b4 | open-r1/OpenR1-Math-220k/olympiads | Which one of the following statements is not true?
(A) 25 is a perfect square.
(B) 31 is a prime number.
(C) 3 is the smallest prime number.
(D) 8 is a perfect cube.
(E) 15 is the product of two prime numbers. | [
"Okay, let's see. The question is asking which of the given statements is not true. There are five options: A to E. I need to check each statement one by one and find out which one is false. Let me start with option A.",
"Statement A: 25 is a perfect square. Hmm, a perfect square means a number that can be expres... | To determine which statement is not true, we analyze each option:
- **Statement A**: 25 is a perfect square.
\(25 = 5^2\), so it is a perfect square. This statement is true.
- **Statement B**: 31 is a prime number.
31 has no divisors other than 1 and itself. Testing divisibility by primes less than 31 (2, 3, ... |
1e50fb9d-f10f-5ee7-958f-342c23b090a5 | open-r1/OpenR1-Math-220k/olympiads | 2. How many of these four integers are prime?
$\begin{array}{llll}1 & 11 & 111 & 1111\end{array}$
A 0
B 1
C 2
D 3
E 4 | [
"Okay, let me try to figure out how many of these numbers are prime: 1, 11, 111, and 1111. The options are from A to E, with E being all four. I need to check each one step by step.",
"First, starting with the number 1. Hmm, I remember that prime numbers are defined as numbers greater than 1 that have no positive... | To determine how many of the given integers are prime, we analyze each number:
1. **1**: By definition, prime numbers must be greater than 1. Therefore, 1 is not a prime.
2. **11**: 11 is a prime number as it has no divisors other than 1 and itself.
3. **111**: The sum of the digits (1 + 1 + 1 = 3) is divisible by 3, ... |
a3023fcb-fd49-523c-8ba3-dcb6e5c805d9 | open-r1/OpenR1-Math-220k/cn_k12 | Given a sample with a total capacity of 100 that is divided into several groups, if the frequency of a certain group is 0.3, then the frequency number of that group is ( ).
A: 3
B: 30
C: 10
D: 300 | [
"Okay, so I have this problem here. It says that there's a sample with a total capacity of 100 divided into several groups. The frequency of a certain group is 0.3, and we need to find the frequency number of that group from the options given. The options are A: 3, B: 30, C: 10, D: 300. Hmm, let me think about how ... | The problem states a sample with a total capacity of 100 divided into groups. The frequency given (0.3) is a relative frequency, representing the proportion of the group relative to the total. To find the absolute frequency (frequency number), multiply the relative frequency by the total capacity:
\[
\text{Absolute Fr... |
c267e7c4-b248-5cd8-a05e-48724ac87d65 | open-r1/OpenR1-Math-220k/cn_k12 | In the Cartesian coordinate system, there is a point $P\left(x,y\right)$. If point $P$ is located in the second quadrant, and the distance from point $P$ to the $x$-axis and $y$-axis are $5$ and $2$ respectively, then the coordinates of point $P$ are ( ).
A: $(5,2)$
B: $(2,5)$
C: $(2,-5)$
D: $(-2,5)$ | [
"Okay, so I need to find the coordinates of point P which is in the second quadrant. The problem says that the distance from P to the x-axis is 5 and to the y-axis is 2. Hmm, let me think.",
"First, I remember that in the Cartesian coordinate system, the quadrants are divided based on the signs of the x and y coo... | Point \( P(x, y) \) is located in the second quadrant, meaning \( x \) is negative and \( y \) is positive. The distance from \( P \) to the \( x \)-axis is 5, which implies \( |y| = 5 \). Since \( y \) must be positive, \( y = 5 \). The distance from \( P \) to the \( y \)-axis is 2, which implies \( |x| = 2 \). Since... |
fc7f2211-996d-5a5b-a2ee-1ca358efcab1 | open-r1/OpenR1-Math-220k/cn_k12 | The coordinates of point P (-2, -3) symmetric about the origin are ( )
A: (-2, 3)
B: (2, -3)
C: (2, 3)
D: (-3, -2) | [
"Okay, so I need to find the coordinates of point P (-2, -3) when it's symmetric about the origin. Hmm, symmetry about the origin... I think that means reflecting the point over the origin. But wait, reflection over the origin is different from reflecting over the x-axis or y-axis, right? Let me recall.",
"When y... | To find the coordinates of point \( P(-2, -3) \) symmetric about the origin, we need to reflect the point over the origin. Reflecting a point \((a, b)\) about the origin results in the point \((-a, -b)\).
Applying this to point \( P(-2, -3) \):
- The x-coordinate \(-2\) becomes \(2\).
- The y-coordinate \(-3\) become... |
18d7b74c-645c-51f4-9af4-2e0e68bd617b | open-r1/OpenR1-Math-220k/olympiads | Sarah multiplied an integer by itself. Which of the following could be the result?
(A) 32
(B) 33
(C) 34
(D) 35
(E) 36 | [
"Okay, so Sarah multiplied an integer by itself, and we need to figure out which of the options A to E could be the result. The options are 32, 33, 34, 35, and 36. Let me think about how to approach this.",
"First, when an integer is multiplied by itself, the result is a perfect square. So, the question is really... | Sarah multiplied an integer by itself, which means the result must be a perfect square. We need to determine which of the given options is a perfect square.
- **32**: The square root of 32 is approximately 5.656, which is not an integer.
- **33**: The square root of 33 is approximately 5.744, which is not an integer.
... |
debd9c5f-0e94-501b-b211-21465e79d5af | open-r1/OpenR1-Math-220k/cn_k12 | Given that the central angle of a sector of a circle is 2 and its radius is 1, determine the area of the sector. | [
"Okay, so I need to find the area of a sector where the central angle is 2 radians and the radius is 1. Hmm, let me remember how sector area works. I think the formula involves the radius squared and the angle, but I should probably double-check that.",
"Wait, right, the area of a sector is given by (1/2) * r² * ... | To determine the area of a sector with a central angle of 2 radians and a radius of 1, we use the formula for the area of a sector, which is \(\frac{1}{2} r^2 \theta\). Here, \(r = 1\) and \(\theta = 2\).
1. Substitute the given values into the formula:
\[
\text{Area} = \frac{1}{2} \times (1)^2 \times 2
\]
2.... |
7dec570b-eafa-5ebe-a537-22395608bf9a | open-r1/OpenR1-Math-220k/cn_k12 | A rectangle with a length of 9 meters and a width of 6 meters is cut into the largest possible square. The area of the square is ____. | [
"Okay, let's see. There's a rectangle that's 9 meters long and 6 meters wide. The problem is asking me to cut it into the largest possible square and find the area of that square. Hmm, so first, I need to figure out what's the biggest square that can fit into this rectangle.",
"Alright, rectangles have length and... | To determine the largest possible square that can be cut from a rectangle with a length of 9 meters and a width of 6 meters, we need to consider the shorter side of the rectangle. The shorter side is 6 meters, which constrains the maximum side length of the square.
A square with a side length of 6 meters can fit perf... |
17b8f5e4-2a4a-5530-a42e-a891713bdcc6 | open-r1/OpenR1-Math-220k/cn_k12 | The y-intercept of the line $y = -\sqrt{3}x + 4$ on the $y$-axis is ________. | [
"Okay, so I need to find the y-intercept of the line given by the equation y equals negative square root of three x plus four. Hmm, let's see. I remember from algebra class that the y-intercept is the point where the line crosses the y-axis. And at that point, the x-coordinate should be zero because the y-axis is w... | To find the y-intercept of the line given by the equation \( y = -\sqrt{3}x + 4 \), we note that the equation is in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
By comparing the given equation with the slope-intercept form, we see that the constant term \( b \) is ... |
aa1f27ae-d16b-56dc-b182-ec7e14907a3d | open-r1/OpenR1-Math-220k/olympiads | The value of $999+999$ is
(A) 2999
(B) 181818
(C) 1998
(D) 999999
(E) 198 | [
"Okay, so I have this problem here: What is the value of 999 plus 999? And then there are five options to choose from: (A) 2999, (B) 181818, (C) 1998, (D) 999999, and (E) 198. Hmm, let me think about how to approach this.",
"First, I need to add 999 and 999. Let me recall how addition works with three-digit numbe... | To find the value of \(999 + 999\), we can break down the addition step by step:
1. Add the units place: \(9 + 9 = 18\). Write down 8 and carry over 1.
2. Add the tens place: \(9 + 9 = 18\), plus the carried over 1 gives 19. Write down 9 and carry over 1.
3. Add the hundreds place: \(9 + 9 = 18\), plus the carried ove... |
ad0fa471-e751-594e-bc52-00a7ecd2eee3 | open-r1/OpenR1-Math-220k/cn_k12 | Given the set $P = \{x \,|\, x^2 - x - 2 = 0\}$, and the set $T = \{x \,|\, -1 < x \leq 2\}$, find $P \cap T$. | [
"Okay, so I need to find the intersection of sets P and T. Let me start by understanding each set individually.",
"First, set P is defined as all x such that x squared minus x minus 2 equals zero. That's a quadratic equation. Maybe I should solve for x to find the elements of P. Let me write that down: x² - x - 2... | To find the intersection of sets \( P \) and \( T \), we start by determining the elements of each set.
Set \( P \) is defined by the quadratic equation \( x^2 - x - 2 = 0 \). Factoring the quadratic equation, we get:
\[
(x - 2)(x + 1) = 0
\]
The solutions are \( x = 2 \) and \( x = -1 \). Therefore, \( P = \{-1, 2\} ... |
6dc78bd9-1052-586d-9b07-e6a1c8631b81 | open-r1/OpenR1-Math-220k/cn_contest | 8. Using 3 ones, 2 twos, and 1 three, the number of different 6-digit numbers that can be formed is ( ).
(A) 45
(B) 50
(C) 60
(D) 70 | [
"Okay, so I need to figure out how many different 6-digit numbers can be formed using 3 ones, 2 twos, and 1 three. The options are given as (A) 45, (B) 50, (C) 60, (D) 70. Hmm, let's see.",
"First, I know that when we're dealing with permutations of multiset, the formula is usually the factorial of the total numb... | To determine the number of different 6-digit numbers that can be formed using 3 ones, 2 twos, and 1 three, we use the formula for permutations of a multiset. The formula is given by:
\[
\frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!}
\]
where \( n \) is the total number of items, and \( n_1, n_2, \ldots, n_k \) ar... |
33db7ec2-b91f-59cb-9701-c28cb6058062 | open-r1/OpenR1-Math-220k/cn_k12 | The program obtained by using the bisection method to solve the equation $x^2-2=0$ is ( )
A: Organizational structure diagram
B: Process flow diagram
C: Knowledge structure diagram
D: Program flowchart | [
"Okay, let's see. The question is about which diagram represents the program that uses the bisection method to solve the equation x² - 2 = 0. The options are organizational structure, process flow, knowledge structure, or program flowchart.",
"Hmm, first, I need to remember what each of these diagrams represents.... | The bisection method algorithm involves iterative steps, decisions, and loops to find a root of the equation. To represent the structure of such a program, a **program flowchart** (Option D) is the appropriate choice, as it visually outlines the control flow, sequences, and logical decisions within the algorithm. Other... |
fa1c9fe3-659c-5988-bfc2-896440794235 | open-r1/OpenR1-Math-220k/cn_k12 | The domain of the power function $f(x) = x^{-2}$ is ( )
A: $\mathbb{R}$
B: $\{x | x \in \mathbb{R} \text{ and } x \neq 0\}$
C: $[0, +\infty)$
D: $(0, +\infty)$ | [
"Okay, so I need to figure out the domain of the power function f(x) = x^{-2}. The options are A: all real numbers, B: all real numbers except 0, C: from 0 to infinity including 0, and D: from 0 to infinity not including 0. Hmm, let me think.",
"First, what is a power function? A power function is generally defin... | To determine the domain of the power function \( f(x) = x^{-2} \):
1. Recognize that \( x^{-2} \) is equivalent to \( \frac{1}{x^2} \).
2. The function \( \frac{1}{x^2} \) is defined for all real numbers \( x \) except where the denominator is zero.
3. The denominator \( x^2 \) is zero when \( x = 0 \), which makes th... |
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