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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|---|---|---|---|---|---|
A person orders 4 pairs of black socks and some pairs of blue socks. The price of each pair of black socks is twice the price of each pair of blue socks. However, the colors were reversed on the order form, causing his expenditure to increase by 50%. Find the original ratio of the number of pairs of black socks to the ... | 1:4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
A regular triangle and a regular hexagon have the same perimeter. If the area of the triangle is 2, what is the area of the hexagon? Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ (with $A$ between $P$ and $B$). Calculate the ratio $PA:AB$. Express your answer as a ratio in the form $a:b$. | 3:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In a sequence, each term after the first two terms is the mean of all the terms which come before that term. Given that the first term is 8 and the tenth term is 26, find the second term of the sequence. | 44 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $P(\sin \frac{3π}{4},\cos \frac{3π}{4})$ lies on the terminal side of angle $θ$, where $θ∈[0,2π)$, find the exact value of $θ$ in radians. | \frac{7π}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a > 0, b > 0$, and $2a+3b=10$, find the maximum value of $\sqrt{3b}+\sqrt{2a}$. Express your answer as a single mathematical expression. | 2\sqrt{5} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $2x - 4 = 0$ for $x$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of x^2 in the expansion of (1+x)(1-2x)^5. | 30 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$ where $a_1=4$ and $a_2=6$, calculate the sum of the first 4 terms ($S_4$). Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
An opaque box contains 20 balls, each of the same size and shape, numbered from 1 to 20. Two balls are randomly chosen from the box. Find the probability that the sum of the numbers on these two balls is divisible by 3. Express your answer as a simplified fraction. | \frac{32}{95} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If the sequence {a_n} is defined by a_1 = 2 and a_{n+1} = a_n + 2n for n >= 1, calculate the value of a_{100}. Express your answer as a single integer. | 9902 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the first term a_1 = 3 and the third term a_3 = 2. Calculate the sum of the first 10 terms S_10. Express your answer as a decimal number. | 7.5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 - x - 2 where x ∈ [-5, 5], find the probability that a randomly chosen point x_0 will satisfy f(x_0) > 0. Express your answer as a decimal value rounded to 1 decimal place. | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors with an angle of $60^{\circ}$ between them, find the magnitude of the vector difference $| \overrightarrow{a} - \overrightarrow{b} |$. Express your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2a=3b$ where $ab\\neq 0$, calculate the ratio $\\frac{a}{b}$. Express your answer as a ratio. | \\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A standard die is rolled consecutively two times. Find the probability that the face-up numbers are adjacent natural numbers. Express your answer as a simplified fraction. | \frac{5}{18} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
If \( 4(\vee)^{2}=144 \), calculate a value of \(\vee\). Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Five numbers in increasing order are 2, 5, x, 10, and y. The median of the numbers is 7 and the mean (average) is 8. Calculate the value of y. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
The hyperbola \(x^2 - \frac{y^2}{m} = 1\) has one of its foci at \((-3,0)\). Find the value of \(m\). Express your answer as a single value. | 8 | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors $\overrightarrow{a} = (x, 1, -2)$ and $\overrightarrow{b} = (3, -x, 4)$, and knowing that $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, find the value of $x$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $R$ is the set of real numbers, let $A=\{x|x^2-2x\geq0\}$, and $B=\{x|x>1\}$. Calculate the intersection of the complement of $A$ and $B$, denoted as $(\complement_R A) \cap B$. Express your answer using standard interval notation. | (1,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given point A(2, -1, 7) and moving along the direction of vector $\overrightarrow{a} = (8, 9, -12)$ to take a segment length $|AB| = 34$, calculate the coordinates of point B. Express your answer as a coordinate (x, y, z). | (18, 17, -17) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the common ratio of the geometric sequence $\{a_n\}$ is a positive number, and $a_4a_8=2a_5^2$, if $a_3=1$, then find the value of $a_7$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $(1+bi)(2+i)$, where $i$ is the imaginary unit and $b$ is a real number, find the range of values for $b$ such that the corresponding point in the complex plane lies in the fourth quadrant. Express your answer as an inequality involving $b$. | b < -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=\frac{{1-i}}{{1+i}}+4-2i$, calculate the magnitude $|z|$. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the annual profit $y$ (in ten thousand yuan) of a certain manufacturer is a function of the annual output $x$ (in ten thousand pieces) as $y=-\frac{1}{3}x^3+81x-234$, find the annual output $x$ that maximizes the manufacturer's annual profit. Express your answer in millions of pieces. | 9 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Xiaolong took an exam with 50 questions, where each correct answer earns 3 points, each incorrect answer deducts 1 point, and unanswered questions neither add nor deduct points. If Xiaolong scored 120 points, what is the maximum number of questions he could have answered correctly? Express your answer as a whole number... | 42 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality |ax + 2| < 4 with a solution set of (-1, 3), find the value of the real number a. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number $0.0000077$ in scientific notation. Use the format $a \times 10^b$, where $a$ is the coefficient and $b$ is the exponent. | 7.7 \times 10^{-6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha$ is an angle in the fourth quadrant, and $\tan \alpha = -\frac{5}{12}$, find the value of $\cos \alpha$. Express your answer as a fraction. | \frac{12}{13} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, find the sum of the first $4n$ positive integers. Express your answer as a single integer. | 300 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets \( A = \left\{ n \left\lvert\, \frac{n}{3} \in \mathbf{N}_{+} \right.\right\} \) and \( B = \left\{ y \mid y = x + 4 + \sqrt{5 - x^2} \right\} \), calculate the number of elements in the intersection \( A \cap B \). Provide your answer as a single integer. | 2 | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an even function defined on $\mathbb{R}$, and for any $x \in \mathbb{R}$, it holds that $f(x+4) = f(x) + f(2)$. If $f(1) = 2$, then find the value of $f(2006) + f(2007)$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The Partridge family pays each of their five children a weekly allowance. The average allowance for each of the three younger children is $8, and the average allowance for each of the two older children is $13. Calculate the total amount of allowance money paid per week to all five children. Express your answer as a si... | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $C=\frac{\pi}{6}$ and $b=2c=2$, calculate the area of $\triangle ABC$. Express your answer as a single value, including the square root symbol if necessary. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the lines \( l_{1}: x + m y + 6 = 0 \) and \( l_{2}: (m-2) x + 3 y + 2m = 0 \) are parallel to each other, find all possible values of the real number \( m \). Express your answer as a numerical value, including all possible values of m. | m = -1, 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)=\ln x- \frac {2}{x}$. Find the value of $\cos\alpha+\sin\alpha$, where $\alpha$ is the slope angle of the tangent line to the curve $y=f(x)$ at $x=1$. Express your answer as a numerical value. | \frac{2\sqrt{10}}{5} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + m and g(x) = 6lnx - 4x defined on (0, +∞), if the tangent lines of the curves y = f(x) and y = g(x) are the same at their common points, find the value of m. Express your answer as an integer in the form \boxed{m}. | -5 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, a_5 = 5, and S_8 = 36, find the sum of the first n terms of the sequence {1/(a_na_{n+1})}. Express your answer as a fraction in simplest form. | \frac{n}{n+1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of the expression $\left(20-\left(2010-201\right)\right)+\left(2010-\left(201-20\right)\right)$? Express your answer as a single integer. | 40 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, $a=3$, $b-c=2$, $\cos B=-\frac{1}{2}$. Find the value of $b$. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, $\overset{.}{z}$ is the conjugate of the complex number $z$. If the complex number $z$ satisfies $(2-5i) \overset{.}{z}=29$, calculate the value of $z$. Express your answer in the form $a + bi$, where $a$ and $b$ are real numbers. | 2-5i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given point P (3, 4) and the line x - y - 1 = 0, find the coordinates (a, b) of point Q that is symmetrical to point P with respect to the line. Express your answer in the format a = x, b = y. | a = 5, b = 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Let $U=\{x \in \mathbb{N} | 0 < x \leq 8\}$, $S=\{1,2,4,5\}$, $T=\{3,5,7\}$. Find the set $S \cap (U \setminus T)$. Express your answer as a set of elements, separated by commas and enclosed in curly brackets. | \{1,2,4\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations 60^a = 3 and 60^b = 5, calculate the value of 12^((1-a-b)/2(1-b)). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 + (a-1)x^2 + ax, where f(x) is an odd function, find the equation of the tangent line to the curve y = f(x) at the point (0, 0). Express your answer as an equation of a line in the form y = mx. | y = x | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given distinct positive real numbers x1, x2 and a real number a in the interval (0,1), define y1 = x1/(1+a) + a*x2/(1+a) and y2 = x2/(1+a) + a*x1/(1+a). Determine the relationship between x1*x2 and y1*y2. Express your answer as a single inequality symbol (>, <, or =). | > | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a_1, a_2, b_1, b_2, b_3$ are real numbers, and $-1, a_1, a_2, -4$ form an arithmetic sequence, $-4, b_1, b_2, b_3, -1$ form a geometric sequence, find the value of $\left(\frac{a_2 - a_1}{b_2}\right)$. Express your answer as a simplified fraction, if possible. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the normal vector of plane \\(\alpha\\) is \\((2,3,-1)\\) and the normal vector of plane \\(\beta\\) is \\((4,\lambda,-2)\\), and knowing that \\(\alpha \perp \beta\\), calculate the value of \\(\lambda\\) that makes the two planes perpendicular. Express your answer as a single number. | -\frac{10}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In the binomial expansion of $(1+x)^n$ ($n\in\mathbb{N}^*$), the coefficient of $x^5$ is the largest. Find the value of $n$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{OA}=(1,2)$ and $\overrightarrow{OB}=(-2,3)$ represent vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ respectively, calculate the magnitude of the sum of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer as a square root if necessary. | \sqrt{26} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \(\left\{a_{n}\right\}\) which satisfies \(a_{1}=2, a_{n+1}=-\frac{1}{a_{n}+1}\), calculate the value of \(a_{2001}\). Express your answer as a fraction. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given set A = {0, 1, 2, 3} and set B defined as {x ∈ N | ln(x) < 1}, find the intersection of sets A and B. Express your answer as a set enclosed in curly brackets. | \{1, 2\} | big_math | [
"Mathematics -> Other -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given $i$ as the imaginary unit and $z=1+i$, calculate $z^{2}-|z|^{2}$. Express your answer as a complex number in the form $a + bi$, where $a$ and $b$ are real numbers. | -2 + 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In a space experiment conducted by astronauts, six procedures need to be implemented in sequence. Among them, Procedure A can only appear in the first or last step, and Procedures B and C must be adjacent. Calculate the total number of possible arrangements for the experiment sequence. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Let $x_0$ be a zero of the function $f(x) = \sin \pi x$, and it satisfies $|x_{0}| + f(x_{0} + \frac {1}{2}) < 11$. Calculate the number of such zeros. | 21 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(φ) = 3/5 and φ ∈ (π/2, π), the distance between two adjacent symmetry axes of the graph of the function f(x) = sin(ωx + φ) (ω > 0) is equal to π/2. Calculate the value of f(π/4). Express your answer as a fraction, including a negative sign if applicable. | -\frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given $$a=2 \int_{0}^{\pi} (\cos(x+ \frac{\pi}{6}))dx$$, find the coefficient of $x$ in the expansion of the binomial $$(x^{2}+ \frac{a}{x})^{5}$$. | -80 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals",
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 25,000,000 in scientific notation in the form $a imes 10^{n}$, where $1 \leqslant |a| < 10$, and $n$ is an integer. | 2.5 \times 10^{7} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Among 10 lottery tickets, only 3 are winning tickets. If 5 people each buy one ticket, calculate the probability that at least one person wins a prize. Express your answer as a simplified fraction. | \frac{11}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the problem from "Nine Chapters on the Mathematical Art," the ratios of consumption among cows, horses, sheep, and pigs are given. A pig eats half as much as a sheep, a sheep eats half as much as a horse, and a horse eats half as much as a cow. If the total compensation for the crops is 9 dou of grain, and they want... | \frac{9}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of (2+x)^4, find the coefficient of x^2. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Theorem"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence ${{a_n}}$, let ${S_n}$ denote the sum of its first $n$ terms. If ${a_2 = 1}$ and ${a_3 = 3}$, find the value of ${S_4}$. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_4 + a_8 = 16. Find the value of a_2 + a_6 + a_10. Express your answer as a single numerical value. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Let $z=\\frac{2+i}{i}$ (where $i$ is the imaginary unit). Calculate the conjugate $\\overline{z}$ and express your answer in the form $\\boxed{a+bi}$. | 1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^{3}$ in the expansion of $(1-2x)(1-x)^{5}$. Express your answer as a single integer. | -30 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
To rent a kayak and a paddle, there is a fixed fee to use the paddle, plus a charge of $5 per hour to use the kayak. For a three hour rental, the total cost is $30. What is the total cost for a six hour rental? Express your answer in dollars as a whole number. | 45 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\sqrt{0.09}$. Express your answer as a single number. | 0.3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given a Pascal distribution with parameters r = 3 and p, where 0 < p < 1, find the maximum value of p that satisfies the condition P(X=6) >= P(X=5). Express your answer as a single numerical value. | 2/5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $\left(\frac{1+i}{1-i}\right)^{2023}$. Express your answer in the form $\boxed{a+bi}$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. | -i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} and a fixed line l: (m+3)x - (2m+4)y - m - 9 = 0, if the point (n, a_n) lies on the line l for each term of the sequence, then what is the sum of the first 13 terms of the sequence {a_n}? Express your answer as a single numerical value. | 39 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $(x^2+2)\left(\frac{1}{x}-1\right)^6$. Express your answer as a single integer. | 17 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line passing through point P(1,2) that is equidistant from points A(2,2) and B(4,-6). Express your answer as a standard equation of a line (e.g., Ax + By + C = 0). | 2x + y - 4 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x||x| \lt 3\}$ and $B=\{x|x^{2}-4x+3 \lt 0\}$, find the intersection $A\cap B$. Express your answer in interval notation. | (1,3) | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the probability for each seed to germinate is $\frac {3}{5}$, calculate the probability that exactly 2 out of 3 planted seeds will germinate. Express your answer as a simplified fraction. | \frac {54}{125} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $b=2a$ and $b\sin A=c\sin C$. Calculate the value of $\cos C$. | \frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_{n}\}$ defined by the recursive relationship $a_{n+1}=a_{n}-n$ for $n \in \mathbb{N}_{+}$ and the initial condition $a_{1}=1$, calculate the exact value of $a_{5}$. Express your answer as an integer. | -9 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
What is the sum of the exterior angles of a quadrilateral? Express your answer in degrees (°) as a whole number. | 360 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
A rhombus is formed by two radii and two chords of a circle whose radius is 16 feet. Calculate the area of the rhombus in square feet. Express your answer as a numerical value with or without a square root. | 128\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that circles $C_1: x^2+y^2+2x+8y-8=0$ and $C_2: x^2+y^2-4x-4y-2=0$ intersect, find the equation of the straight line that contains the common chord of circles $C_1$ and $C_2$. Express your answer in the form Ax + By + C = 0. | x + 2y - 1 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( f(x) = \frac{x-1}{x+1} \), and define \( f_{n}(x) = \underbrace{f\{f[f \cdots f}_{n \uparrow f}(x)]\} \). Calculate \( f_{1999}(x) \). Express your answer as a function of x, including all necessary mathematical notation. | \frac{x-1}{x+1} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(α) = 3/5 and α is an angle in the second quadrant, find the value of tan(2α + π/4). Express your answer as a simplified fraction. | -\frac{17}{31} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(x-1)^2 + |y+2| = 0$, find the value of $x+y$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the definite integral: $\int_{1}^{e} \frac{1}{x}dx$. Express your answer as a single value. | 1 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $\frac{1}{i-2}$. Express your answer as a fraction or decimal, including a negative sign if applicable. | -\frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, if the complex number $(1+ai)(2+i)$ is a pure imaginary number, find the real number $a$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A merchant buys goods at a 20% discount on the list price and wants to make a 20% profit on the selling price after applying a 20% discount on the marked price. Calculate the percentage of the list price at which the merchant should mark the goods to achieve this. Express your answer as a percentage. | 125\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given set $A=\{x|x^{2}-3x-4\leqslant 0\}$ and set $B=\{x\in N\left|\right.2x-3 \gt 0\}$, calculate the intersection $A\cap B$. Express your answer as a set of numbers, including all numbers in the intersection. | \{2, 3, 4\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, and the complex number $z = \frac{a+i}{1-i}$, where $a$ is a real number, and $|z| = 1$, calculate the value of $a$. Express your answer as a real number, including the ± symbol if necessary. | ±1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If the complex number $z=\frac{i}{1+i}$, then find the conjugate of $z$. Express your answer as a complex number in the form $a + bi$, where $a$ and $b$ are fractions if necessary. | \frac{1}{2} - \frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
The preservation time y (in hours) of a certain food is related to the storage temperature x (in degrees Celsius) by the function y = e^(kx + b), where e = 2.718... is the base of the natural logarithm, and k and b are constants. If the food is designed to last for 192 hours at 0°C and 48 hours at 22°C, calculate the p... | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular box has a sum of the lengths of its twelve edges equal to 140, and the distance from one corner of the box to the furthest opposite corner is 21. Find the total surface area of the box. Express your answer as a whole number. | 784 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
There is a square whose perimeter is the same as the perimeter of a given triangle. The triangle is a right-angled triangle with side lengths 6 cm and 8 cm. Find the area of the square. Express your answer in square centimeters (cm²). | 36 \text{ cm}^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of linear equations in x and y:
$$
\begin{cases}
& x+y=5k \\
& x-y=9k \\
\end{cases}
$$
If the solution to this system is also a solution to the linear equation 2x+3y=6, find the value of k. Express your answer as a numerical value in the form $\boxed{[answer]}$. | \dfrac{3}{4} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Among the following algebraic expressions, how many are polynomials?
$2a$; $\frac{m-n}{6}$; $\frac{3}{π}+a$; $\frac{5}{a-b}$; $2(x^{2}-4)$.
Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of subsets of the set $\{x | x^2 - 1 = 0\}$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Combinatorics -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate 50% of 2002. Express your answer as a whole number. | 1001 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
If $n \in \mathbb{N}^*$, and $3C_{n-1}^{n-5} = 5A_{n-2}^2$, calculate the value of $n$. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations",
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
How many positive integers k are there such that the equation a^2 + b^2 = k * ab has a positive integer solution (a, b)? Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number equation $(3+4i)z=5$, calculate the imaginary part of $z$. Express your answer as a fraction or decimal. | - \frac {4}{5} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
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