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
value |
|---|---|---|---|---|---|
Given that there are 2 defective products in 6 items, calculate the probability of selecting exactly 1 defective product when taking 2 items at random. Express your answer as a simplified fraction. | \dfrac{8}{15} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
There are 2012 backgammon checkers with one side black and the other side white, arranged in a line such that no two consecutive checkers are the same color. At each move, two checkers are chosen, and they and all checkers between them are flipped. What is the minimum number of moves required to make all checkers the s... | 1006 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,4)$ and $\overrightarrow{b}=(x,6)$ are collinear, use the necessary and sufficient condition for vector collinearity to find the real number $x$. Express your answer as a single real number. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A cookie jar contains a total of 100 raisins. All but one of the cookies are the same size and contain the same number of raisins. One cookie is larger and contains one more raisin than each of the others. The number of cookies in the jar is between 5 and 10, inclusive. Calculate the number of raisins in the larger coo... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
A regular hexagon is divided into 6 equilateral triangles. How many parallelograms can be formed within the hexagon? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Mary thought of a positive two-digit number. She multiplied it by 3 and added 11. Then she switched the digits of the result, obtaining a number between 71 and 75, inclusive. What was Mary's original number? Express your answer as a two-digit integer. | 12 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The vertex of the new parabola obtained by first shifting the parabola y = -(x+1)^2 + 4 one unit to the right and then two units down can be found by applying the transformation steps to the original equation. Calculate the vertex coordinates of the new parabola. Express your answer as an ordered pair (x, y). | (0, 2) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the general equation of a circle $C$ as ${x^2}+{y^2}-2x+y+\frac{1}{4}=0$, find the coordinates of the center of $C$. Express your answer as an ordered pair (x, y). | (1, -\frac{1}{2}) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2x / (x - 1), find the maximum value of the real number a such that f(x) >= a holds true in the interval [3, 5]. Express your answer as a single real number. | \frac{5}{2} | big_math | [
"Mathematics -> Calculus -> Applications of Derivatives"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,-1)$ and $\overrightarrow{b}=(1,3)$, and $\overrightarrow{a}\perp (\overrightarrow{a}+m\overrightarrow{b})$, find the value of $m$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with the first term a_1 = 2 and common difference d = 1, calculate the value of the fifth term, a_5. | 6 | big_math | [
"Mathematics -> Algebra -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations:
1. \( a(c+d) + b(c+d) = 42 \)
2. \( c + d = 3 \)
Calculate the value of \( a + b + c + d \). Express your answer as a single integer. | 17 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimal positive period of the function $f(x) = \frac{\tan x}{1 + \tan^2 x}$. Express your answer in radians. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function y = 1/4*sin(x - π/6) - cos(x - 2π/3). Express your answer as a single numerical value. | \frac{3}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A high school has a total of 1600 students, with 400 being third-year students. If a sample of 160 students is to be drawn from the entire high school population using stratified sampling, calculate the number of third-year students that should be drawn into the sample. Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In a bag, there are a total of 15 balls, some black and some white. All the balls are identical except for their colors. The balls in the bag are thoroughly mixed. One ball is randomly drawn from the bag, its color is noted, and then it is put back into the bag. This process is repeated 100 times. It is found that 60 t... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
There are 2 girls and 6 boys playing a game. How many additional girls must join the game so that \(\frac{5}{8}\) of the players are girls? Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given two real numbers x and y randomly chosen from the interval [0,1], find the probability that √(x²+y²) ≤ 1. Express your answer as a fraction or decimal value. | \frac{\pi}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
In the positive geometric sequence {a_n}, a_2 = 3 and a_8 = 27. Calculate the value of the fifth term a_5. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A and B independently shoot at the same target once. The probability of A hitting the target is 0.5, and the probability of B hitting the target is 0.6. Calculate the probability that the target is hit. Express your answer as a decimal value between 0 and 1. | 0.8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Let $D$ be a point in the plane of $\triangle ABC$, and let $\overrightarrow{BC}=3\overrightarrow{CD}$. Express the vector $\overrightarrow{AD}$ in terms of vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$. Provide your answer in the format $\boxed{\overrightarrow{AD} = a\overrightarrow{AB} + b\overrightarrow{AC... | \overrightarrow{AD} = -\dfrac{1}{3}\overrightarrow{AB} + \dfrac{4}{3}\overrightarrow{AC} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\alpha = \frac{4}{5}$ and $\alpha$ is an acute angle, calculate the exact value of $\cos \frac{\alpha}{2}$. Express your answer as a simplified fraction, including square roots if necessary. | \frac{2\sqrt{5}}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given a two-digit number with tens digit $x$ and units digit $y$, the original number can be represented as $10x+y$. When we swap the tens and units digits, the new number becomes $10y+x$. Find the algebraic expression for the difference between the new number and the original number, expressed in terms of $x$ and $y$. | -9x+9y | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the numbers $85_{(9)}$, $200_{(6)}$, $68_{(11)}$, and $70$, convert each number to decimal and determine the largest one. Express your answer as a single decimal number. | 77 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overrightarrow{a}=(1, \sqrt {3})$ and $\overrightarrow{b}=(3,m)$, find the value of $m$ such that the projection of vector $\overrightarrow{b}$ in the direction of vector $\overrightarrow{a}$ is $3$. Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the axis of symmetry of the parabola given by the equation $y=-\frac{1}{2}x^2+x-\frac{5}{2}$. Express your answer as an equation of a line in the format $x = c$. | x = 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = log2(x), where x ∈ [1/2, 2], find the probability of randomly selecting a point x0 in the interval [1/2, 2] such that f(x0) >= 0. Express your answer as a simplified fraction. | 2/3 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given an even function $f(x)$ and its derivative $f'(x)$, it is known that $\dfrac{x}{2}f'(x) + f(-x) \lt 0$ for $x \geqslant 0$. If $g(x) = x^{2}f(x)$, find the solution set of the inequality $g(x) \lt g(1-2x)$. Express your answer as an interval or a combination of intervals. | (\dfrac{1}{3}, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A certain city generates an average of 700 tons of domestic waste per day, which is processed by Factory A and Factory B. It is known that Factory A can process 55 tons of waste per hour at a cost of 550 yuan, while Factory B can process 45 tons of waste per hour at a cost of 495 yuan. If it is stipulated that the dail... | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the direction vector (Δx, Δy) of the line given by the equation y = -1/2x + 1. Express your answer as an ordered pair (Δx, Δy), where Δx and Δy are integers. | (2, -1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence \(\{a_n\}\) with a common ratio \(q = -\frac{1}{3}\), calculate the value of \(\frac{a_1 + a_3 + a_5 + a_7}{a_2 + a_4 + a_6 + a_8}\). Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A sector of a circle has a perimeter of 10 and an area of 4. Determine the central angle of the sector in radians. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, point P lies on the y-axis, point Q has coordinates (4,0), and PQ passes through the point R(2,4). Calculate the area of triangle OPQ. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given \( k \in \mathbf{Z}_{+} \), express the fraction \(\frac{7}{51}\) in base \(k\) as \(\overline{0.23}_{k} = (0.2323)_{k}\) and solve for \(k\). Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions that satisfy the inequality x - 1 < (x - 1)^2 < 3x + 7. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function $y=\cos \left( \frac {\pi}{3}- \frac {2}{5}x\right)$. Express your answer in terms of π. | 5\pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
If $f\left(x\right)=x^{2}+2\sin x$, find the value of ${f'}\left(1\right)$. Express your answer as a mathematical expression involving trigonometric functions. | 2+2\cos 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a right-angled triangle, one of the legs is 24cm long and the hypotenuse is 25cm long. Find the length of the other leg. Express your answer in centimeters (cm) as a whole number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of values for $x$ that make the fraction $\frac{3}{{2-x}}$ valid. Express your answer as an inequality or equation. | x \neq 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Four students go to a shop to buy either a brush pen or a pencil. Each student buys exactly one item, and at least one person buys a pencil. Calculate the total number of different buying combinations possible. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2(5-a)(6+a) = 100$, calculate the value of $a^2 + a + 1$. Express your answer as a single integer. | -19 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given \(x, y, z \in \mathbb{R}^{+}\) and \(\\frac{1}{x} + \\frac{2}{y} + \\frac{3}{z} = 1\), find the minimum value of \(x + \\frac{y}{2} + \\frac{z}{3}\). Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number which is 6 less than twice the square of 4. Provide your answer as a single number. | 26 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} xe^{x}, & x \geqslant 0 \\ -\frac{x}{e^{x}}, & x < 0 \end{cases}$, find the solution set for the inequality $f(x-2) < e$. Express your answer in interval notation. | (1,3) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=2+\frac{1}{i}$ (where $i$ is the imaginary unit), find the conjugate of the complex number $z$. Express your answer in the form $a+bi$, where $a$ and $b$ are real numbers. | 2+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A soccer ball is made with 32 pieces of leather. Of these pieces, 12 are identical regular pentagons, and the other 20 are identical regular hexagons. The sides of the pentagons are equal to the sides of the hexagons. To join two sides of these pieces, a seam is required. Calculate the total number of seams needed to m... | 90 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given that the perimeter of a sector of a circle is 5 cm and its area is 1 cm^2, determine the measure of the central angle of the sector in radians. Express your answer as a single value in radians. | 1/2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Which of the following operations is correct according to the rules of exponents: $x^{5}+x^{5}$, $(x^{3}y^{2})^{2}$, $x^{6}\div x^{2}$, or $x^{2}\cdot x^{3}$? Identify the correct operation and provide the result of that operation.
However, this reformulation still has multiple parts to consider. A better reformulatio... | $x^{5}$ | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)= \dfrac {3x^{2}}{ \sqrt {1-x}}+\log (-3x^{2}+5x+2)$. Express your answer in interval notation. | (- \dfrac {1}{3},1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is an odd function defined on the domain R, and is monotonically increasing on the interval [0, +inf), find the range of x that satisfies the inequality |f(ln x) - f(ln 1/x)| / 2 < f(1). Express your answer as an interval. | (1/e, e) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of distinct mappings from set \(M\) (\(=\{0,1,2\}\)) to set \(N\) (\(=\{1,2,3,4\}\)). Express your answer as a whole number. | 64 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
The sum of two natural numbers is 17402. One of these numbers is a multiple of 10, and when the last digit of this number is removed, the result is the other number. Calculate the difference between these two numbers. Express your answer as a single integer. | 14238 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $z=\frac{1-2i}{2+i}+\frac{2}{1+i}$, where $i$ is the imaginary unit. Express your answer in the form a+bi. | 1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
If the focal length, the length of the minor axis, and the length of the major axis of an ellipse form an arithmetic sequence, calculate the eccentricity of the ellipse. Express your answer as a fraction in simplest form. | \frac{3}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Convert the base-4 number $1010_{(4)}$ to a decimal number. Express your answer as a single decimal number. | 68 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\frac{1}{3}{{x}^{3}}+a{{x}^{2}}+{{b}^{2}}x+1$, where $a$ is a number chosen from $1$, $2$, $3$, and $b$ is a number chosen from $0$, $1$, $2$, calculate the probability that this function has two extreme points. Express your answer as a simplified fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $(x+3)(x-1)<0$. Express your answer as an interval in interval notation. | (-3,1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
An item is originally priced at 200 yuan per unit. After two consecutive discounts of m%, the selling price becomes 162 yuan. Calculate the value of m. Express your answer as a percentage. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function f(x) is \mathbb{R} and it satisfies f(x+4) = f(x), and f(0.5) = 9, calculate the value of f(8.5). | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Each number from 1 to 6 replaces one of the letters \(P, Q, R, S, T\), and \(U\). The sum of \(P\) and \(Q\) is 5, and the difference between \(R\) and \(S\) is 5. If \(T\) is greater than \(U\), what number replaces the letter \(T\)? Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A specific type of computer virus that invades memory space starts by occupying 2KB of memory when the computer is turned on. Then, every 3 minutes, it replicates itself, and the memory it occupies doubles. If the virus occupies 64MB of memory (1MB = $2^{10}$KB), calculate the time taken in minutes for the virus to rea... | 45 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Five circles are each tangent to lines \( L_{1} \) and \( L_{2} \) and are sequentially tangent to each other. If the radius of the largest circle is 18 and the radius of the smallest circle is 8, what is the radius of the middle circle? Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the fixed point (x, y) that satisfies the equation y=k(x-2)+3 for any value of k. Express your answer as an ordered pair (x, y). | (2, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
An ordered list of four numbers is called a quadruple.
A quadruple (p, q, r, s) of integers with p, q, r, s >= 0 is chosen at random such that
2p + q + r + s = 4
What is the probability that p + q + r + s = 3? Express your answer as a simplified fraction. | \frac{3}{11} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given two cones, cone A and cone B, with equal base radii and side areas $S_{A}$ and $S_{B}$, and volumes $V_{A}$ and $V_{B}$, respectively. If the lateral surface of cone A can be unfolded into a semicircle, and $\frac{{S}_{A}}{{S}_{B}}=\frac{2}{3}$, then find the ratio of their volumes $\frac{{V}_{A}}{{V}_{B}}$. Expr... | \frac{\sqrt{6}}{4} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x-1)=x^2$, find the expression for $f(x)$. Express your answer as a function of $x$. | x^2 + 2x + 1 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C_{1}$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 \ (a > 0, b > 0)$ passes through the focus of the parabola $C_{2}$: $y^{2} = 2px \ (p > 0)$, and the asymptotes of the hyperbola form an equilateral triangle with the directrix of the parabola, calculate the eccentricity of the hyperbola $C_{... | \dfrac{2\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n}, where a_3 and a_{15} are the two roots of the equation x^2 - 6x + 1 = 0, find the product a_7a_8a_9a_{10}a_{11}. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\mathrm{ABCD}$ is a right trapezoid with $\angle \mathrm{DAB} = \angle \mathrm{ABC} = 90^\circ$. A rectangle $\mathrm{ADEF}$ is constructed externally along $\mathrm{AD}$, with an area of 6.36 square centimeters. Line $\mathrm{BE}$ intersects $\mathrm{AD}$ at point $\mathrm{P}$, and line $\mathrm{PC}$ ... | 3.18 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Alain and Louise are driving on a circular track with a radius of 25 km. Alain leaves the starting line first, going clockwise around the track at a speed of 80 km/h. Fifteen minutes after Alain starts, Louise leaves the same starting line, going counterclockwise around the track at a speed of 100 km/h. Find the time L... | \frac{10\pi - 1}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A computer user plans to spend no more than 500 yuan to purchase software and boxed disks, with unit prices of 60 yuan and 70 yuan, respectively. According to the requirements, at least 3 pieces of software and at least 2 boxes of disks must be purchased. Calculate the number of different purchasing methods that satisf... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The equation $$\frac{x^2}{m-2} + \frac{y^2}{m+3} = 1$$ represents a hyperbola. Find the range of values for the real number $m$. Express your answer in interval notation. | (-3, 2) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A group of girls stands in a circle. Florence is the fourth on the left from Jess and the seventh on the right from Jess. Find the total number of girls in the group. Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (\cos\alpha, \sin\alpha)$ and $\overrightarrow{b} = (\cos^2\alpha, \sin\alpha-1)$, with $\alpha \in \left(\frac{\pi}{2}, \pi\right)$. If $\overrightarrow{a} \cdot \overrightarrow{b} = \frac{1}{5}$, calculate $\tan\left(\alpha + \frac{\pi}{4}\right)$. Express your answer as a single f... | -\frac{1}{7} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A test consists of 6 multiple-choice questions. The grading rule is as follows: each person initially scores 6 points, each correct answer adds 4 points, each incorrect answer deducts 1 point, and no answer gets 0 points. If 51 students take the test, what is the minimum number of students who must have the same score?... | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $\sin \frac{4\pi }{3}\cos \frac{5\pi }{6}\tan \left(- \frac{4\pi}{3}\right)$. Provide your answer as a single number. | -\frac{3\sqrt{3}}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {1+bi}{2+i}$ where $b \in \mathbb{R}$, if $z$ is a purely imaginary number, then find the value of $z$. Express your answer as a complex number in the form $a+bi$ and simplify to its simplest form. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Point P starts from (1,0) and moves counterclockwise along the unit circle for an arc length of 2π/3 to reach point Q. Determine the coordinates of point Q. Express your answer as an ordered pair (x, y) with exact values. | \left( -\frac{1}{2},\frac{\sqrt{3}}{2} \right) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of terms in the arithmetic sequence $13$, $16$, $19$, $\dotsc$, $70$, $73$. Express your answer as a whole number. | 21 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $i + i^2 + i^3$. Express your answer as a single value. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given two polynomials A and B, where B=3x-2y, a student mistakenly calculated A+B instead of A-B and found the result to be x-y. Using this information, calculate the correct value of A-B. Express your answer as an algebraic expression in terms of x and y. | -5x+3y | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $4a\cdot 3a^{2}b\div 2ab$. Express your answer as a simplified algebraic expression. | 6a^{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x^{2}+x+1=8$, calculate the value of $4x^{2}+4x+9$. Express your answer as a single integer. | 37 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the lengths of two sides of a triangle are 3 and 5, find the range of the third side length, a. Express your answer as an inequality in the format (lower bound) < a < (upper bound). | 2 < a < 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The set $M=\{1, 2, 3\}$. Find the number of subsets of the set M. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the polar equation
$$
\rho = \frac{1}{1 - C_{m}^{n} \cos \theta}
$$
where \( 1 \leq n \leq m \leq 5 \), how many different hyperbolas are represented by this equation when \( C_{m}^{n} > 1 \)? Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|x^{2}-x-2\leqslant 0\}$ and $B=\{x|\sqrt{x-1}<1\}$, find the union $A \cup B$. Express your answer as an interval, using square brackets or parentheses as necessary. | \left[-1,2\right] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Two places, A and B, are 1600 meters apart. On a map, the distance between these two places is represented by 8 centimeters. What is the scale of this map? Express your answer as a ratio in the form 1:x. | 1:20000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (2x+3)/(x-1) and the domain x ∈ [2, +∞), find the range of the function. Express your answer in interval notation, including any relevant parentheses or brackets. | (2, 7] | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the maximum binomial coefficient in the expansion of \\((x-2)^{5}\\). Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfying $z \cdot \overline{z} + z + \overline{z} = 17$, find the minimum value of $|z + 2 - i|$. Express your answer as a numerical value. | 2 \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the similarity ratio of two similar triangles is 1:4, find the ratio of their corresponding heights. Express your answer as a ratio. | 1:4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a,b > 0$ and $a+b=5$, find the maximum value of $\sqrt{a+1} + \sqrt{b+3}$. Express your answer as a numerical value. | 3\sqrt{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
William has four cards with different integers written on them. Three of these integers are 2, 3, and 4. He puts one card in each cell of the 2x2 grid shown. The sum of the two integers in the second row is 6. The sum of the two integers in the second column is 10. What is the number on the card he places in the top le... | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the "Treasure Box" interactive segment of the "Lucky 52" program on CCTV, there is a guessing game with 20 logos, where 5 logos have a certain prize amount indicated on their back, and the rest show a sad face, meaning no prize if turned over. Participants in this game have three chances to flip the logos. If a part... | \frac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In a round-robin tournament, each team plays every other team exactly once. If a total of 21 games are played, how many teams are participating in the tournament? Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $2^{0}-1$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of values for $x$ that makes the expression $\sqrt{x+2}$ meaningful. Express your answer as an inequality in the format $x \geqslant...$ or $x >...$. | x \geqslant -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $S_{n}$ is the sum of the first $n$ terms of the arithmetic sequence ${a_{n}}$, if $a_{2}+a_{5}+a_{8}=12$, calculate the value of $S_{9}$. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given a point P(2, -5π/3) in the polar coordinate plane, find the polar and Cartesian coordinates of the point symmetric to P with respect to the pole. Express your answer in the format (r, θ) for polar coordinates and (x, y) for Cartesian coordinates. | (2, -2π/3), (-1, -√3) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
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