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 |
|---|---|---|---|---|---|
A number x is randomly selected from the interval [-1,1]. Find the probability that the value of x^2 falls between 0 and 1/4. Express your answer as a fraction or decimal between 0 and 1. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $z=1+i$ (where $i$ is the imaginary unit), calculate the value of $\frac{2}{z}$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. | 1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Point A on the number line represents the number -4. If point A is moved 6 units to the right on the number line to point B, what is the number represented by point B? Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A charity sells 140 benefit tickets for a total of $2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. Calculate the revenue raised by the full-price tickets. Express your answer in dollars as a whole number. | 782 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $x \in \left(-\frac{\pi}{2}, 0\right)$ and $\cos x = \frac{4}{5}$, find the value of $\tan 2x$. Express your answer as a single fraction or decimal value. | -\frac{24}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Express the number $2023000$ in scientific notation. Provide your answer in the form $\boxed{a \times 10^{b}}$, where $a$ is the coefficient rounded to 4 decimal places if necessary. | 2.023 \times 10^{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, given that $AB = 3$, $BC = 6$, and $AC = 4$, let $D$ be a point on side $BC$ such that $AD = 3$. Find the value of $\frac{BD}{DC}$. Express your answer as a simplified fraction. | \frac{29}{7} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that x and y are real numbers with -4 ≤ x ≤ -2 and 2 ≤ y ≤ 4, find the greatest possible value of (x+y)/x. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In a plane rectangular coordinate system, two transformations are defined: f(m, n) = (m, -n) and g(m, n) = (-m, -n). Apply the transformation f to the point (-3, 2), and then apply the transformation g to the result. Express your answer as a coordinate pair (x, y). | (3, 2) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the maximum number of lines in space for which there exists a plane that forms equal angles with all the lines. Provide your answer as a single integer. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
The manager of a company planned to distribute a $50 bonus to each employee from the company fund, but the fund contained $5 less than what was needed. Instead, the manager gave each employee a $45 bonus and kept the remaining $95 in the company fund. Calculate the amount of money in the company fund before any bonuses... | 995 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {an}, where the sum of the first 5 terms (S5) is 15 and the sum of the first 9 terms (S9) is 63, find the value of the 4th term (a4). Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A power boat and a raft both left dock A on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock B downriver, then immediately turned and traveled back upriver. It eventually met the raft on t... | 4.5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In a circle with center O, the shaded sector represents 20% of the area of the circle. What is the size of angle AOB in degrees? | 72 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear function y = (k+1)x + k-2, find the range of values for k such that the function passes through the first, third, and fourth quadrants. Express your answer as an inequality or interval. | -1 < k < 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If for any real number $x$, we have $x^3 = a + a_1(x-2) + a_2(x-2)^2 + a_3(x-2)^3$, then calculate the value of $a_2$ using the binomial theorem. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions a = 2^0.1, b = ln(1/2), and c = (2/3)^π, order these expressions from largest to smallest. Use the format a > b > c or equivalent. | a > c > b | big_math | [
"Mathematics -> Algebra -> Exponential Functions",
"Mathematics -> Algebra -> Logarithmic Functions",
"Mathematics -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain sophomore class, there are 36 boys and 28 girls. If one student is randomly selected as the math representative, how many different selection methods are there in total? Express your answer as a whole number. | 64 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Sixty men working on a construction job have done 1/3 of the work in 18 days. The project is behind schedule and must be completed in the next twelve days. Calculate the number of additional workers that need to be hired to meet the new deadline. Express your answer as a whole number. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the $x^{3}$ term in the expansion of $(\sqrt{x}-2x)^{5}$. Express your answer as an integer. | -10 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
A number is tripled and then decreased by 5, resulting in 16. What is the original number? Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The quadratic inequality $2kx^2 + kx - \frac{3}{8} < 0$ holds for all real numbers $x$. Determine the range of $k$ for which this inequality is true. Express your answer as an interval or a range of values for $k$. | (-3, 0) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a circle $x^{2}+y^{2}-2x-1=0$, find its radius. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the lines $ax + 4y - 2 = 0$ and $2x - 5y + b = 0$, what is the condition on $a$ for these lines to be perpendicular to each other? Express your answer as a specific value or equation for $a$. | 10 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)$ has a domain of $[0,4]$, find the domain of the function $y=f(x+3)+f(x^2)$. Express your answer in interval notation. | [-2,1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
A solid sphere is cut in half, resulting in an increase in surface area of $4\pi cm^2$. What was the original surface area of the solid sphere? Express your answer in terms of $\pi$ and units of $cm^2$. | 8\pi cm^2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Express 2.6 million in scientific notation, using the format $a×10^n$, where $1≤|a|＜10$ and $n$ is an integer. | 2.6×10^6 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $z$ satisfies $(2-3i)z=5-i$, calculate the value of $z$. Express your answer as a complex number in the form $a+bi$. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The point $(3,-5)$ lies on the graph of the proportional function $y=kx$ $(k\neq 0)$. Find the value of $k$. Express your answer as a fraction. | -\frac{5}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 - 2x, find the sum of the first n terms of the sequence a_n = f'(n), denoted as S_n. Express your answer as a formula in terms of n. | n^2 - n | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Maria must work 8 hours each day, excluding her 45-minute lunch break. If she starts working at 7:25 A.M. and takes her lunch break at noon, what time will her workday end? Express your answer in the format of time (e.g., 4:10 P.M.). | 4:10 P.M. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a=e^{0.25}$, $b=1$, and $c=-4\ln 0.75$. Determine the correct ordering of the values a, b, and c. Express your answer as an ordering of the variables (e.g., a < b < c). | b < c < a | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A ball is floating on the surface of a lake, and when it is removed, a hole with a diameter of 24 cm and a depth of 8 cm is left on the ice. What is the radius of the ball? Express your answer in centimeters (cm) as a whole number. | 13 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Four students register for interest groups in Chinese, Mathematics, and English, with each person choosing one subject. Calculate the total number of different possible selections. Express your answer as a whole number. | 81 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Student A needs half an hour to complete a batch of paper cranes alone, and Student B needs 45 minutes to complete it alone. Calculate the time it takes for both students, A and B, to complete the batch together. Express your answer in minutes as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{|x|-2}{x+2} = 0$, find the value of $x$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\left|\overrightarrow{a}+\overrightarrow{b}\right|=\sqrt{10}$ and $\left|\overrightarrow{a}-\overrightarrow{b}\right|=\sqrt{6}$, calculate the value of $\overrightarrow{a} \cdot \overrightarrow{b}$. Express your answer as a single numerical value... | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Angelo wrote down the word TEAM. He then swapped two adjacent letters around and wrote down the new order of the letters. He proceeded in this way until he obtained the word MATE. What is the minimum number of swaps that Angelo could have used to transform the word "TEAM" into "MATE"? Express your answer as a single in... | 5 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If θ ∈ [π/4, π/2] and sin(2θ) = 3√7/8, find the value of sin(θ). Express your answer as a fraction. | 3/4 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In a certain city, there are three drill sites where fire drills are being conducted simultaneously. There are four fire brigades to be assigned to these three drill sites. If each drill site must be assigned at least one fire brigade, calculate the number of different distribution schemes. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let S_n and T_n be the sums of the first n terms of the arithmetic sequences {a_n} and {b_n}, respectively. If a_5 = 2b_5, calculate the value of S_9 / T_9. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the cross-section of a cylinder is a square and its area is $4S$, find the area of one of its bases. Express your answer in terms of $S$ and $\pi$. | \pi S | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number z corresponds to the point (-1,1) in the complex plane, calculate the modulus of the complex number (z+3)/(z+2). Express your answer as a single value, including the square root symbol if necessary. | \frac{\sqrt{10}}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x+1)=2x^{2}+5x+2$, determine the analytical expression of $f(x)$. Express your answer as a function of x using standard function notation. | f(x)=2x^{2}+x-1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In a box, there are 5 black balls and 4 white balls. Now, two balls are drawn successively without replacement. Let event A be "drawing a black ball on the first draw" and event B be "drawing a white ball on the second draw." Calculate P(AB) + P(B|A). Express your answer as a simplified fraction. | \frac{7}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given an isosceles triangle with two sides of lengths 2 and 5 units, determine its perimeter if it exists. Apply the triangle inequality theorem to validate the triangle's existence and calculate the perimeter by summing the lengths of all sides. Express your answer in units as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A={x|2≤x<4} and B={x|x≥3}, find the intersection A∩B. Express your answer as an interval in standard notation (e.g., [a,b)). | [3,4) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given a triangle ∆ABC with side a = √3, side b = 1, and angle A = 130°, determine the number of solutions for this triangle. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given a triangle with side lengths satisfying a = 3 and b + c = 5, find the maximum value of the area of this triangle using the Heron-Qin Jiushao formula. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are hinged together as shown, then folded to form a cube. What is the color of the face opposite the white face? Express your answer as a single letter representing the color. | B | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of zeros at the end of the product \(25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 25 \times 8 \times 8 \times 8\). Express your answer as a single integer representing the count of trailing zeros. | 9 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $a^{3}\\cdot (a^{3})^{2}$ using the rules of exponents. Express your answer as a simplified exponential expression. | $a^{9}$ | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $M$ is a point inside $\triangle ABC$ (excluding the boundary) with an area of 1, and the areas of $\triangle MBC$, $\triangle MCA$, and $\triangle MAB$ are $x$, $y$, and $z$ respectively, find the minimum value of $$\frac {1}{x+y} + \frac {x+y}{z}.$$ Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a hyperbola $C: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ where $a > 0, b > 0$, and it has an asymptote with the equation $x + y = 0$, calculate the eccentricity of the hyperbola $C$. Express your answer as a single numerical value. | \\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality system about x: x > a and x ≥ 3, with the solution set x ≥ 3, determine the relationship between a and 3 that must hold for the solution set to be valid. Express your answer as a relationship (e.g., <, >, ≤, ≥) between a and 3. | a < 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\max \left\{ 3,8-2x,2x-5 \right\}=3$, find the range of values for $x$ that satisfies this condition. Express your answer as an interval (e.g., $[a, b]$). | \left[\frac{5}{2}, 4\right] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Let $A,M$, and $C$ be digits with
\[(100A+10M+C)(A+M+C) = 2005\]
What is the value of $A$? Express your answer as a single digit. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of 11 consecutive odd numbers is 1991, arrange these numbers in order. Calculate the sixth number in the series using the arithmetic series formula. Express your answer as a single integer. | 181 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
From a group of 10 college graduates, select 3 individuals to serve as village assistants. The selection must meet the conditions that at least one of individuals A or B is selected, and individual C is not selected. Calculate the total number of different selection methods that meet these criteria. | 49 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the cube $ABCD-A_1B_1C_1D_1$, $M$ is the midpoint of $DD_1$, $O$ is the center of $ABCD$, and $P$ is any point on the edge $A_1B_1$. Calculate the angle formed by line $OP$ and $AM$. Express your answer in degrees, including the degree symbol. | 90^\circ | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the coordinate half-plane, there is a point P located in the second quadrant. The distance from point P to the x-axis is 4, and the distance to the y-axis is 5. Find the coordinates of point P. Express your answer as an ordered pair (x, y), including the negative sign for the x-coordinate if applicable. | (-5, 4) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the complex number expression $i^3 (1 - i)^2$. Express your answer as a single number, which may be real or complex. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x^2 + y^2 = 14x + 6y + 6, find the maximum value of 3x + 4y. Express your answer as a single integer. | 73 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z=(1+ai)(1-2i)$ ($a\in\mathbb{R}$) is a purely imaginary number, find the real number $a$. Express your answer as a single real number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\left(-a\right)^{3}\cdot a^{2}$ by applying the appropriate exponent rules. Provide your answer as a simplified algebraic expression in the format $\boxed{[answer]}$. | -a^{5} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of intersection points of the curves $x=y^2$ and $y=x^2$. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose x and y satisfy the linear constraints \begin{cases} 2x - y + 2 \geqslant 0 \\ x - 3y + 1 \leqslant 0 \\ x + y - 2 \leqslant 0 \end{cases}, and the maximum value of z = ax - y (a > 0) has multiple optimal solutions. Find the value of the real number a. | \dfrac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the figure enclosed by the curve $y=x^{2}$ and the line $y-x-2=0$. Express your answer as a fraction in the form $\boxed{\frac{a}{b}}$, where $a$ and $b$ are integers. | \frac{9}{2} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of coefficients of all rational terms in the expansion of ${(x-\frac{1}{{\sqrt[3]{x}}})}^8$. Express your answer as a single integer. | -27 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In the rectangle \(ABCD\), \(AB = 4 \text{ cm}\) and \(AD = 12 \text{ cm}\). Point \(P\) is on side \(AD\) and moves from \(A\) towards \(D\) at a speed of \(1 \text{ cm/sec}\). Point \(Q\) is on side \(BC\) and moves from point \(C\) at a speed of \(3 \text{ cm/sec}\) back and forth between \(C\) and \(B\). Both point... | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\left(-ab
ight)^{3}(-3b)^{2}$. Express your answer as a simplified algebraic expression. | -9a^{3}b^{5} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let $A=\{x|x^{2}-x-2 < 0\}$ and $B=\{x|y=\ln x\}$. Find the intersection $A\cap B$ and express your answer in interval notation. | (0,2) | big_math | [
"Mathematics -> Algebra -> Inequalities",
"Mathematics -> Precalculus -> Functions"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12 square units, what is the area of the equilateral triangle? Express your answer in square units. | 18 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If \triangle ABC \sim \triangle DEF, and the ratio of AB to DE is 2:1, and the perimeter of \triangle ABC is 16, calculate the perimeter of \triangle DEF. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A protractor with a radius of 5 cm is rolled along the edge of a ruler without slipping, starting with the 0 cm mark on the ruler aligned with the 0° mark on the protractor. Calculate the central angle in degrees corresponding to the 10 cm mark on the ruler. Round your answer to the nearest degree if necessary. | 115 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex number z satisfy iz = 1, where i is the imaginary unit. Find the value of z. Express your answer as a complex number. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, calculate $(-2i)^2$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
There are 4 students attending 3 simultaneous extracurricular lectures, and each student can freely choose one of the lectures. Calculate the number of different selection methods. | 81 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
The hour hand on a clock rotates a full circle of -360° in 12 hours. Calculate the rotation of the hour hand after 2 hours. Express your answer in degrees, including the direction of rotation (use a negative sign for clockwise rotation). | -60° | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Let $X$ be a set containing 10 elements, and $A$, $B$ be two disjoint subsets of $X$, containing 3 and 4 elements respectively. Find the number of subsets of $X$ that contain neither $A$ nor $B$. Express your answer as a numerical value. | 840 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In the binomial expansion of $(x^2 - \frac{1}{x})^5$, find the coefficient of the second term. Express your answer as an integer. | -5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(n) = \tan \frac{n\pi}{3}$, where $n \in \mathbb{N}^{*}$, calculate the value of $f(1) + f(2) + \ldots + f(100)$. Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let the even function $f(x)$ be such that $f'(x) < 0$ on $(0, +\infty)$, and $f(2) = 0$. Find the solution set of the inequality $$\frac {f(x) + f(-x)}{x} > 0.$$ Express your answer as an interval or a union of intervals. | (-\infty, -2) \cup (0, 2) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A class has a budget of 300 yuan to purchase thermometers for its 53 students, with each student needing one thermometer. Mercury thermometers cost 3 yuan each, and electronic thermometers cost 10 yuan each. The class wants to buy as many electronic thermometers as possible due to the fragility and toxicity of mercury ... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\{a_n\}$ is a geometric sequence with all positive terms, and if $a_4 \cdot a_8 = 4$, then find the product $a_5 \cdot a_6 \cdot a_7$. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A real number x is randomly selected from the interval [-2, 4]. If the probability that x satisfies x^2 <= m is 5/6, find the value of the real number m. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the monomials $a^{m-1}b^{2}$ and $\frac{1}{2}ab^{n+5}$ is still a monomial, calculate the value of $n^{m}$. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f\left(x\right)=\left\{\begin{array}{l}x+5,x≥4,\\ x-2,x＜4,\end{array}\right.$, calculate the value of $f\left(3\right)$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the length of \( DC \) is twice the length of \( BD \), where \( BD \) is 3 units. The height from point \( A \) to the base \( BC \) is 4 units. Calculate the area of the triangle \( ABC \). Express your answer in square units as a whole number. | 18 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the first $n$ terms of the geometric sequence $\{a_n\}$, denoted by $S_n$, is given by $S_n = \frac{1}{2}3^{n+1} - a$. Using the property $a_2^2 = a_1 a_3$, find the value of $a$. Express your answer as a single value. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In a mathematics competition, 1000 students are numbered as follows: 0001, 0002, 0003, …, 1000. A sample of size 50 is to be drawn using systematic sampling, divided into 50 parts. If a number 0015 is randomly selected from the first part, calculate the 20th sample number. Express your answer as a four-digit number, in... | 0395 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the parabola y = -x^2 + 2022, find the axis of symmetry. Express your answer as an equation of a line in the form x = value. | x = 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given \(a, b, c \in \mathbf{Z}_{+}\), \([a, b] = 12\) and \([b, c] = 15\). Find the minimum possible value of \([a, c]\). Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Number Theory -> Least Common Multiples (LCM)"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In order to understand the learning situation of 800 students, a systematic sampling method is used to select a sample of size 40. Calculate the interval between segments by dividing the population size by the sample size. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The distance from the vertex of the hyperbola $\frac{x^2}{4}-y^2=1$ to its asymptote can be calculated using the standard equation of the hyperbola and the distance from a point to a line formula. Find this distance. Express your answer as a fraction or decimal. | \frac{2\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence \(1, 3, 2, \cdots\), each term after the first two is equal to the difference of the two preceding terms (the previous term minus the term before it). Find the sum of the first 100 terms of this sequence. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies the equation $z(\sqrt{3}+i)=1-\sqrt{3}i$, determine the value of $|z|$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{x|\frac{{x+1}}{{x-3}}\leqslant 0,x\in Z\}$, calculate the number of proper subsets of set $A$. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
There are 25 people arranged in a 5x5 square matrix. Calculate the number of different ways to select three people such that any two of the three are neither from the same row nor from the same column. | 600 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $c\\cos B + b\\cos C = 2a\\cos A$, $a = 2$, and the area of $\\triangle ABC$ is $\\sqrt{3}$, calculate the perimeter of $\\triangle ABC$. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
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