instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
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|---|---|---|---|---|---|
Let \( f(x) = \cos(\omega x) \) have a minimum positive period of 6, where \( \omega = \frac{\pi}{3} \). Calculate the value of \( \sum_{i=1}^{2018} f(i) \). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Trigonometry -> Functions"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Let the random variable $X \sim N(\mu, \sigma^2)$. Determine the distribution of the variable $\eta = ax+b$. Express your answer as a specific normal distribution with mean and variance. | N(a\mu + b, a^2\sigma^2) | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Let $a \in \mathbb{R}$, and $(a+i)^2i$ is a positive real number. Find the value of $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
What is the condition on the coefficients for the equation $ax^{2}+bx+c=0$ to be considered a quadratic equation in $x$? Express your answer as a condition on $a$. | a \neq 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x^{3}=a_{0}+a_{1}(x-2)+a_{2}(x-2)^{2}+a_{3}(x-2)^{3}$, calculate the value of $a_{2}$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\dfrac {1}{i}+ \dfrac {1}{i^{3}}+ \dfrac {1}{i^{5}}+ \dfrac {1}{i^{7}}+ \dfrac {1}{i^{9}}$. Express your answer as a simplified complex number. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 0.0000084 in scientific notation. Include the coefficient and the exponent in your answer. | 8.4\times 10^{-6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets \( A = \{0.1, 1, 10\} \) and \( B = \{ y \mid y = \lg x, x \in A \} \), calculate the intersection \( A \cap B \). Express your answer as a set. | \{1\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A line $l$, passing through the focus $F$ of the parabola $y^{2}=2px$ ($p > 0$) and forming a $60^{\circ}$ angle with the $x$-axis, intersects the parabola in the first and fourth quadrants at points $A$ and $B$ respectively. Calculate the ratio $\frac {|AF|}{|BF|}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^2 = 5x$, find the distance from its focus to its directrix. Express your answer as a fraction. | \frac{5}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
What is the opposite of -7? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
An isosceles triangle has side lengths of 2 and 4. Calculate the perimeter of the triangle, if it can be formed. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
There are homework assignments from 3 different teachers, and 4 students are all doing homework at the same time. Calculate the number of possible situations where all 4 students are doing homework. Express your answer as a single integer. | 81 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of positive integer points (points with both coordinates being positive integers) through which the graph of the quadratic function y = -x^2 + 20x - 64 passes. Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ (where $a>0$ and $b>0$), one of its foci coincides with the focus of the parabola $y^2=20x$, and the eccentricity of the hyperbola is $\sqrt {5}$. Derive the equation of the hyperbola in the standard form $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = ... | \frac{x^{2}}{5} - \frac{y^{2}}{20} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A circle is inscribed in an equilateral triangle, and a square is inscribed in this circle. Calculate the ratio of the area of the triangle to the area of the square. Express your answer as a simplified ratio, using a colon (:) to separate the parts of the ratio. | 3 \sqrt{3}: 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given a function f(x) defined on ℝ that is symmetric about the point (2, 0) and satisfies f(x) = f(2 - x) for all real numbers x. If f(-5) = -2, find the value of f(2015). Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate \sqrt{(43 imes 17)^{2}+(43 imes 26)^{2}+(17 imes 26)^{2}}. Express your answer as a single integer. | 1407 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the vector $\overset{→}{a}=(1,2)$, with $\overset{→}{a}·\overset{→}{b}=5$ and $|\overset{→}{a}-\overset{→}{b}|=2\sqrt{5}$, calculate the magnitude of vector $\overset{→}{b}$, denoted as $|\overset{→}{b}|$. Express your answer as a numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Let's assume the original number of members in the interest group is x people, and the original average age is y years old. According to the problem, if five 9-year-old members leave, or five 17-year-old members join, the average age of its members will increase by 1 year. Formulate equations based on these scenarios a... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
At a math conference, the following exchange rates are used:
$$
\begin{aligned}
1 \text { calculator } & =100 \text { rulers } \\
10 \text { rulers } & =30 \text { compasses } \\
25 \text { compasses } & =50 \text { protractors }
\end{aligned}
$$
How many protractors are equivalent to 1 calculator? Express your answe... | 600 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If the terminal side of angle α passes through the point (1, -√3), calculate the value of sin(α). Express your answer as a fraction or exact decimal value. | -\dfrac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Sine"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find the difference of the roots of the quadratic equation $x^2 - 7x - 9 = 0$. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{85} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A certain unit has several employees and adopts stratified sampling to select $n$ people for physical examination. The ratio of the number of old, middle-aged, and young people is $4:1:5$. If $10$ middle-aged people were selected, calculate the sample size $n$. Express your answer as a whole number. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
When the complex number $z = (a^2 + 2011) + (a-1)i$ represents a real number, what is the value of $a$? Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence of complex numbers \(\{a_n\}\) with the general term
\[
a_n = (1+i) \left( 1 + \frac{i}{\sqrt{2}} \right) \left( 1 + \frac{i}{\sqrt{3}} \right) \cdots \left( 1 + \frac{i}{\sqrt{n}} \right),
\]
find the value of \(|a_n - a_{n+1}|\). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inverse proportion function y = k/x, and it passes through points A(3,m) and B(m-1,6), calculate the value of k. Provide your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (a^2 - 2)e^x + b, where a ∈ {-2,0,1,2,3} and b ∈ {3,5}, calculate the probability that f(x) is a decreasing function. Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
What is the tens digit of \(3^{2016}\)? Express your answer as a single digit. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function y = g(x) is symmetric to the graph of the function f(x) = a^(x-1) with respect to the line y = x, and it is given that g(4) = 2. Find the value of g(2). Express your answer as a numerical value. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function y = 1/x at x = 1. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Let $m$ and $n$ be natural numbers not greater than 6. How many hyperbolas are represented by the equation $C_6^m x^2 - C_6^n y^2 = 1$? Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are positive real numbers, and the line $(a+1)x+2y-1=0$ is perpendicular to the line $3x+(b-2)y+2=0$, calculate the minimum value of $\dfrac{3}{a} + \dfrac{2}{b}$. Express your answer as a single number. | 25 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Granny Smith has $63. Anjou has one-third as much as Granny Smith. Elberta has $2 more than Anjou. How many dollars does Elberta have? Express your answer as a dollar amount. | 23 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $x^{3}-3x+3- \frac{x}{e^{x}}-a\leqslant 0$ with respect to $x$, where $x\geqslant -2$, find the minimum value of the real number $a$. Express your answer as a single value. | 1- \frac{1}{e} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The function $f(x)$ is an even function with a period of 2, and when $x \in (0, 1)$, $f(x) = x + 1$. Find the expression for $f(x)$ in the interval $(1, 2)$. Express your answer as a function of $x$. | 3-x | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve y = 2x^2 and a point A(2, 8) on it, find the slope of the tangent line at point A. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^\circ$, $|\overrightarrow{a}|=2$, and $|\overrightarrow{b}|=5$, calculate the value of $|2\overrightarrow{a}-\overrightarrow{b}|$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{21} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle $x^2+y^2+2x-2y+2a=0$ intersects the line $x+y+2=0$ and the length of the chord is 4, calculate the value of the real number $a$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the approximate product of $2.46\times 8.163\times (5.17+4.829)$, rounded to the nearest hundred. | 200 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic progression, the sum of the first 10 terms is 100 and the sum of the first 100 terms is 10. Find the sum of the first 110 terms. Express your answer as a single integer. | -110 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function $f(x) = \cos(\dfrac{\pi}{3} - x) + \sin(\dfrac{\pi}{2} + x)$. Express your answer as an interval. | [-\sqrt{3}, \sqrt{3}] | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
If $m, n \in \mathbb{N}^*$ and $m+n \leq 8$, then what is the total number of points $(m, n)$ on the plane? Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
If the terms $-3x^{m+1}y^{4}$ and $2x^{4}y^{n}$ are like terms, find the value of $|m - n|$. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a<sub>n</sub>} is an arithmetic sequence, a<sub>1</sub> + a<sub>3</sub> + a<sub>5</sub> = 18, a<sub>2</sub> + a<sub>4</sub> + a<sub>6</sub> = 24, find the value of a<sub>20</sub>. Express your answer as a single integer. | 40 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x|y=\log_{2}(x-1)\}$ and $B=\{x|x<2\}$, calculate the intersection of sets A and B, denoted as $A\cap B$. Express your answer in interval notation. | (1,2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In 2003, the average monthly rainfall in Mathborough was 41.5 mm. In 2004, the average monthly rainfall in Mathborough was 2 mm more than in 2003. Calculate the total amount of rain that fell in Mathborough in 2004. Express your answer in millimeters as a whole number. | 522 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Amy bought and then sold \(15n\) avocados, for some positive integer \(n\). She made a profit of \$100. She paid \$2 for every 3 avocados. She sold every 5 avocados for \$4. Calculate the value of \(n\). Express your answer as a positive integer. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A rhombus has its diagonals in the ratio of 1:2. If the shorter diagonal is 4 cm long, calculate the length of each side of the rhombus. Express your answer in centimeters (cm), using simplified radical expressions if necessary. | 2\sqrt{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
The total length of the Great Wall is approximately 6,700,010 meters. Represent this using scientific notation with two significant figures. Express your answer in the form $a \times 10^n$ meters, where $1 \leq |a| < 10$ and $n$ is an integer. | 6.7 \times 10^6 \text{ meters} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Nine lines parallel to the base of a triangle divide the other sides each into 10 equal segments and the area into 10 distinct parts. If the area of the largest of these parts is 38, then find the area of the original triangle. Express your answer as a whole number. | 200 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $S_n$ represents the sum of the first $n$ terms of the sequence $\{a_n\}$, and $S_n = n^2$, calculate the value of $a_5 + a_6 + a_7$. Express your answer as a single integer. | 33 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the arithmetic sequence $\{a_{n}\}$ is an increasing sequence, if $a_{1} > 0$ and $2(a_{n+2}-a_{n})=3a_{n+1}$, then find the common ratio $q$ of the sequence $\{a_{n}\}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? Express your answer as a percentage. | 70\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the fraction $\frac{x^2-x}{x-1}$ equals 0, solve for the value of x. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a^{m}=2$, calculate the value of $a^{2m}+a^{3m}$. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In quadrilateral ABCD, AB = √2, BC = √3, CD = 1, ∠ABC = 75°, ∠BCD = 120°. Find the measure of ∠CDA. Express your answer in degrees as a whole number. | 75 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations 5x + 3k = 24 and 5x + 3 = 0, find the value of k such that both equations have the same solution. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of extreme points of the function $f(x)=3x^{2}-\ln x-x$. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_2 = 2 and a_4 + a_5 = 12. Find the value of a_7. Express your answer as a single numerical value. | 10 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola C: $$\frac{y^{2}}{a^{2}}$$ - $$\frac{x^{2}}{b^{2}}$$ = 1 (a > 0, b > 0), one of its asymptotic line equations is y = 2x. Calculate the eccentricity of C. Express your answer as a fraction with a square root. | \frac{\sqrt{5}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the perimeter of an isosceles triangle is 26cm and the base is 11cm, calculate the length of the legs. Express your answer in centimeters. | 7.5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A circular dining table has 6 seats arranged in order: A, B, C, D, E, F. Now, 3 adults and 3 children are to be seated for dining, with the requirement that no two children can sit next to each other. Find the total number of different seating arrangements. Express your answer as a single numerical value. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ in a plane with an angle of 120° between them, $\overrightarrow {a}=(1,0)$, and $|\overrightarrow {b}|=1$, calculate the magnitude of $\overrightarrow {a}+2\overrightarrow {b}$. Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f'(x) = 2$, calculate the limit $\lim_{{\Delta x}\to 0}\frac{f(x)-f(x+2\Delta x)}{\Delta x}$. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
A triangular pyramid has three lateral edges that are mutually perpendicular, with lengths 1, $\sqrt {3}$, and 2 respectively. The four vertices of the pyramid lie on the same spherical surface. Calculate the volume of this sphere. Express your answer in terms of $\pi$. | \frac {8 \sqrt {2}}{3}\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the rectangular coordinate system Oxy, two points A(2, 1) and B(4, 5) are given. Point C satisfies $\overrightarrow{OC} = λ\overrightarrow{OA} + μ\overrightarrow{OB}$, where λ, μ ∈ R, and λ + μ = 1. Derive the equation of the trajectory of point C in the form of y = mx + b or other standard forms. | y = 2x - 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given $i$ is the imaginary unit, and real numbers $x$, $y$ satisfy $(x+i)i+y=1+2i$, calculate the value of $x-y$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} that satisfies a_1 = 1 and a_n - a_{n-1} = 2 (for n >= 2), find the general term a_n. Express your answer as an algebraic expression in terms of n. | a_n = 2n - 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {1+ai}{1-i}(a∈R)$ with an imaginary part of $1$, find the value of $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that 1 mile equals 8 furlongs and 1 furlong equals 40 rods, calculate the number of rods in one mile. Express your answer as a whole number. | 320 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the K^2 statistic has a critical value of 3.841, which corresponds to a p-value of 0.05, what is the confidence level that event A is related to event B when K^2 > 3.841? Express your answer as a percentage. | 95% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If the graph of the function y = 2sin(2x + π/6) is shifted to the right by π/4 units, what is the resulting function? Express your answer as a function of x. | y = 2sin(2x - π/3) | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n}, where a_1 = 1/4 and a_n = 1 - 1/a_{n-1} for n >= 2, find the value of a_{2020}. | \frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $n=\int_{0}^{\frac{\pi}{2}} 4\cos x dx$, find the constant term in the expansion of the binomial $(x-\frac{1}{x})^{n}$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals",
"Mathematics -> Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given a non-zero arithmetic sequence {a_n} that satisfies 2a_3 - a_7^2 + 2a_{11} = 0, and a geometric sequence {b_n} with b_7 = a_7, calculate the value of b_5b_9. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies $a_{n+1}= \begin{cases} 2a_{n}, & 0\leqslant a_{n}\leqslant \dfrac {1}{2}\\ 2a_{n}-1, & \dfrac {1}{2}\leqslant a_{n} < 1\end{cases}$. If $a_{1}= \dfrac {6}{7}$, calculate the value of $a_{2017}$. Express your answer as a fraction. | \dfrac {6}{7} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains 5 balls of the same size, of which 3 are white and 2 are black. If a ball is randomly drawn from the bag, not replaced, and then another ball is randomly drawn from the bag, calculate the probability that the first draw is white and the second draw is black. Express your answer as a simplified fraction i... | \frac{3}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sector with a radius of 1 and an arc length of 4, calculate the area of the sector using the formula $S_{\text{sector}} = \frac{1}{2}lR$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Let g(x) = ax^3 + bx^-1 - 2, where a and b are constants. If g(2) = 2, calculate the exact value of g(-2). Express your answer as a single integer in the format \boxed{answer}. | -6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 - x + sqrt(2x^4 - 6x^2 + 8x + 16), find the minimum value of f(x). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin 2t = - \int_{0}^{t} \cos x \,dx$ where $t \in (0,\pi)$, calculate the exact value of $t$ in radians. | \frac{2\pi}{3} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Translate the parabola $y=-x^{2}$ $3$ units to the right and $2$ units down. Calculate the coordinates of the vertex of the new parabola obtained. Express your answer as an ordered pair $(x, y)$. | (3,-2) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the sum of the first term, three times the eighth term, and the fifteenth term is 120. Using the properties of arithmetic sequences, find the value of the eighth term. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|y=\sqrt{2-{2}^{x}}\}$ and $B=\{x|x^{2}-3x \leqslant 0\}$, find the intersection of sets $A$ and $B$, denoted as $A \cap B$. Express your answer as an interval, using square brackets or parentheses as necessary. | [0, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = xe^{-x} + 2, find the equation of the tangent line at point P(0, 2). Express your answer in the form Ax + By + C = 0. | x - y + 2 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 228 and 1995. Express your answer as a single integer. | 57 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {x| -1 < x < 1} and B = {x| 0 ≤ x ≤ 2}, find the intersection of sets A and B, denoted as A ∩ B. Express your answer in interval notation. | [0,1) | big_math | [
"Mathematics -> Other -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Two identical rectangles are placed obliquely on top of each other. The area X of the overlapping region is one-eighth of the total shaded area. What fraction of the area of one rectangle is X? Express your answer as a fraction. | \frac{2}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The Palace Museum is holding four exhibitions during the May Day holiday: "Opera Culture Exhibition," "Ming Dynasty Imperial Cellar Porcelain Exhibition," "Historical Green Landscape Painting Exhibition," and "Zhao Mengfu Calligraphy and Painting Exhibition." A student decides to visit one of them in the morning and on... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Let \( S = 1! + 2! + 3! + \cdots + 99! \). Calculate the units digit of \( S \). Express your answer as a single digit. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the real number m is the arithmetic mean between 1 and 5, calculate the value of m. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for the complex number z in the equation 2z * i = 1 + 3i, where i is the imaginary unit. Express your answer in the form a + bi, where a and b are real numbers. | \frac{3}{2} - \frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
The China Mobile Data Center (IDC) project recently started construction in the high-tech zone. The planned construction scale is 12.6 square meters. Express this number in scientific notation in the form $a imes 10^{n}$, where $1 \leqslant |a| < 10$ and $n$ is an integer. | 1.26 \times 10^{1} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence \\(\\{a_n\\}\\) where the sum of the first \\(n\\) terms is \\(S_n\\), if \\(a_4 = 18 - a_5\\), calculate the value of \\(S_8\\). Express your answer as a single numerical value. | 72 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Alicia earns 20 dollars per hour, of which 1.45% is deducted to pay local taxes. Calculate the amount of Alicia's wages that is used to pay local taxes in cents per hour. Express your answer as a whole number. | 29 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {0,1,2,3} and the complement of A in U (∁_U A) = {2}, determine the set A. Express your answer as a set of elements enclosed in curly brackets {}, with elements separated by commas. | A = {0,1,3} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $I=\{1,2,3,4,5\}$, $M=\{1,2,3\}$, $N=\{3,4,5\}$, calculate the intersection of the complement of $N$ in $I$ and $M$, denoted as $(\complement _{I}N) \cap M$. Express your answer as a set of numbers using standard set notation. | \{1,2\} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_{n}}$, $a_{1}=-2016$, and the sum of its first $n$ terms is $S_{n}$. If $\frac{S_{2015}}{2015} - \frac{S_{2012}}{2012} = 3$, then find the value of $S_{2016}$. Express your answer as a single integer. | -2016 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $${x^{2}}-3x+2 < 0$$. Express your answer in interval notation. | \left( 1,2 \right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
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