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 space vectors $\overrightarrow{a}=(0,2,0)$ and $\overrightarrow{b}=(1,0,-1)$, calculate the dot product $({\overrightarrow{a}+\overrightarrow{b}})•\overrightarrow{b}$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $2x-3=-1$ for $x$. Provide your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac {x^{2}}{25}+ \frac {y^{2}}{9}=1$, a line passing through one of its foci intersects the ellipse at points $A$ and $B$. If the sum of the distances from the other focus to points $A$ and $B$ is 12, calculate the length of segment $|AB|$. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with a first term of 12 and a common difference of -2, find the probability that a randomly chosen number from this sequence is not greater than 4. Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the constant term in the expansion of ${\left( {{x}^{3}}+{\frac{1}{{{x}^{2}}}} \right)}^{5}$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Among all factors of \(201^9\), how many of them are either perfect squares or perfect cubes? Express your answer as a single integer. | 37 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\{a_n\}$ is an arithmetic sequence with the first term $2$ and a non-zero common difference, if $a_1$, $a_3$, $a_6$ form a geometric sequence, calculate the sum of the first $9$ terms of $\{a_n\}$. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x$ and $y$ are positive real numbers, find the minimum value of $\frac{4x}{x + 3y} + \frac{3y}{x}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If $\alpha$ is in the third quadrant, simplify the expression $$\frac {\cos\alpha}{ \sqrt {1-\sin^{2}\alpha}}+ \frac {2\sin\alpha}{ \sqrt {1-\cos^{2}\alpha}}.$$ Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = cos(x), if we shrink the x-coordinates of all points on its graph to half of their original value (while keeping the y-coordinates unchanged), and then translate the graph to the left by π/4 units, what is the equation of the resulting function? Express your answer as a function equation in the f... | y = cos(2x + π/2) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The distance from the center of the circle (x+1)^2 + y^2 = 1 to the line y = sqrt(3)x - sqrt(3) is to be calculated. Find this distance. Express your answer as a single exact value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given $(1+x)^{10} = a + a_1(1-x) + a_2(1-x)^2 + \ldots + a_{10}(1-x)^{10}$, calculate the value of the coefficient $a_8$. Express your answer as an integer. | 180 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a binomial distribution B(n, p), with E(ξ) = 1.6 and D(ξ) = 1.28, determine the explicit values of n and p. Use the formulas for the expectation and variance of a binomial distribution to establish a system of equations and solve for n and p. Express your answer as explicit valu... | n = 8, p = 0.2 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,1)$, $\overrightarrow{b}=(0,1)$, $\overrightarrow{c}=(3,6)$, and a real number $\lambda$, if $(\overrightarrow{a}+ \lambda \overrightarrow{b}) \parallel \overrightarrow{c}$, find the value of $\lambda$. Express your answer as a single real number. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the exact value of $\tan {75}^{{o}}$ using the formula for the tangent of the sum of two angles. Express your answer as a simple algebraic expression. | 2+\sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(ωx - π/3), points A(m, n) and B(m+π, n) (|n| ≠ 1) both lie on the curve y = f(x), and the line segment AB has five common points with the curve y = f(x). Find the value of ω. | ω = 4 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Consider the ellipse given by the equation x^2/12 + y^2/3 = 1. Let F1 denote the left focus of the ellipse, and let P be a point on the ellipse. If the midpoint M of the line segment PF1 lies on the y-axis, then find the y-coordinate of point P. Express your answer as a numerical value, including the ± symbol to indica... | ±√3/2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that the three points $A(-1,-2)$, $B(4,8)$, and $C(5,x)$ are on the same line, find the value of the real number $x$. Express your answer as a single number. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the real numbers x_i (i = 1, 2,..., n) satisfy the equation sum(x_i) = n, find the minimum value of the sum of 3^x_i. Express your answer as a single mathematical expression in terms of n. | 3n | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = [0,1] and the set A = {1}, find the complement of A in U, denoted as ∁_U A. Express your answer in interval notation. | ∁_U A = [0,1) | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the equation 2(a_1+a_4+a_7) + 3(a_9+a_{11}) = 24 holds. Find the value of S_{13} + 2a_7. Express your answer as a single numerical value. | 30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $z = (1 - i)(-2 + ai)$ is purely imaginary, then find the value of the real number $a$. Express your answer as a single real number. | a = 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=2-i$, calculate the magnitude of its conjugate, denoted as $|\overline{z}|$. Express your answer as a single numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{x|-3 < x < 3\}$ and set $A=\{x|x^{2}+x-2 < 0\}$, describe the complement of $A$ in $U$, denoted as $\complement _{U}A$, in interval notation. | (-3,-2]\cup [1,3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)$ defined as follows:
$$
f(x) =
\begin{cases}
f(x+2), & \text{if } x < 3 \\
(\frac{1}{2})^{x}, & \text{if } x \geq 3
\end{cases}
$$
Calculate the value of $f(-4)$. Express your answer as a fraction. | \frac{1}{16} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x)= \begin{cases} x-3 \ (x\geqslant 9) \\ f(x+6) \ (x < 9) \end{cases}$, calculate the value of $f(5)$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
The equation of a circle is given as $x^2-6x+y^2+8=0$, and a line is defined as $y=x+1$. Find the minimum length of the tangent drawn from a point on the line to the circle. Express your answer as a numerical value. | \sqrt{7} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the binomial expansion of $(x+ \\frac {1}{x})^{2n}$, the coefficients of the fourth and sixth terms are equal. Calculate the constant term in the expansion. Express your answer as a single integer. | 70 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $f(x) = x^{2} - 4x + 4$. Provide your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, find the value of $a$ such that the real part and the imaginary part of the complex number $-3i(a+i)$ (where $a \in \mathbb{R}$) are equal. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, if $|\overrightarrow{AB}+\overrightarrow{AC}|=2|\overrightarrow{AB}-\overrightarrow{AC}|$, calculate the maximum value of $\sin A$. Express your answer as a numerical value. | \frac{4}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1=1$, and $a_{n+1}=2a_n+2$. Calculate the value of $a_7$. Express your answer as a single integer. | 190 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the planar vectors $\overrightarrow{PA}, \overrightarrow{PB}$ that satisfy $|\overrightarrow{PA}| = |\overrightarrow{PB}| = 1, \overrightarrow{PA} \cdot \overrightarrow{PB} = -\frac{1}{2}$, if $|\overrightarrow{BC}| = 1$, find the maximum value of $|\overrightarrow{AC}|$. Express your answer as a mathematical exp... | \sqrt{3}+1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}$ is a non-zero vector, simplify the expression $(-4) \times 3\overrightarrow{a}$. Express your answer as a vector in the form $\boxed{[answer]}$. | -12\overrightarrow{a} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
A clock strikes the number of the hour at every whole hour and strikes once at every half hour. Calculate the total number of times the clock strikes in a full day and night. Express your answer as a whole number. | 180 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If \( u = -6 \) and \( x = \frac{1}{3}(3 - 4u) \), calculate the value of \( x \). Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
On the number line, point P starts at -2, moves 5 units to the right, then moves 4 units to the left. Calculate the final position of point P. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, given that a_1 + a_4 = 10 and a_2 - a_3 = 2, find the sum of the first n terms, S_n. Express your answer as an algebraic expression in terms of n. | -n^2 + 9n | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $$\frac {1}{i}+ \frac {1}{i^{3}}+ \frac {1}{i^{5}}+ \frac {1}{i^{7}}.$$ Express your answer as a simplified complex number, if necessary, in the form a + bi. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Point M is a moving point on the number line. If it first moves 7 units to the left and then moves 4 units to the right, it will be exactly at the origin. What is the initial position of point M? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Consider the system of equations:
\[
\begin{cases}
x^2 - y z = 1 \\
y^2 - x z = 1 \\
z^2 - x y = 1
\end{cases}
\]
Find the number of possible sets of rational solutions $(x, y, z)$ that satisfy the system. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The line $x+ay-7=0$ is parallel to the line $(a+1)x+2y-14=0$. Find the value of $a$ that makes the lines parallel. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 23,766,400 in scientific notation. Provide your answer in the form of a coefficient multiplied by 10 raised to an exponent, like this: $\boxed{a \times 10^{b}}$. | 2.37664 \times 10^{7} | big_math | [
"Mathematics -> Other -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Let L(m) be the x-coordinate of the left endpoint of the intersection of the graphs of y = x^2 - 6 and y = m, where -6 < m < 6. Let r = [L(-m) - L(m)]/m. Calculate the limit of r as m approaches zero. Express your answer as a fraction with a square root in the denominator. | \frac{1}{\sqrt{6}} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(2,3)$ and $\overrightarrow{b}=(-1,2)$, calculate the value of $(\overrightarrow{a}+2\overrightarrow{b})\cdot \overrightarrow{b}$. Express your answer as a single number. | 14 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Let the set M = {m ∈ Z | -3 < m < 2}, and the set N = {n ∈ Z | -1 ≤ n ≤ 3}. Find the intersection set M ∩ N. Express your answer as a set of integers using standard set notation. | {-1, 0, 1} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given x = 15°, calculate the value of sin^4 x - cos^4 x. Express your answer as a fraction or decimal, including any necessary negative sign or radical. | -\dfrac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets M = {0,1,2,3,4} and N = {1,3,5}, find the number of subsets of the set P, where P is the intersection of M and N. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In the rectangular coordinate system, the vertex of angle α coincides with the origin, and its initial side coincides with the non-negative semi-axis of the x-axis. The terminal side passes through point P(√3, -1). Calculate sin(2α - π/2). Express your answer as a single value, including a negative sign if applicable. | -\frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation x^2 + 1994x + 7 = 0 with roots m and n, calculate the value of (m^2 + 1993m + 6)(n^2 + 1995n + 8). Express your answer as a single integer. | 1986 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations 2x - 1 = 3x - 2 and 4 - (kx + 2)/3 = 3k - (2 - 2x)/4, and that the solutions of these equations are the same, find the value of k. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 19672 in scientific notation. Use the format $a \times 10^{b}$, where $a$ is a number between 1 and 10, and $b$ is an integer. | 1.9672 \times 10^{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $f(x)$ satisfies $f(ab) = f(a) + f(b)$, and $f(2) = m$, $f(3) = n$, find the value of $f(72)$. Express your answer as an algebraic expression in terms of $m$ and $n$. | 3m+2n | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Nasrin paddled to her camp, which was 4.5 km away, and then returned. The trip to the camp took 2 hours and 30 minutes, and the return trip took 1/3 of the time. Calculate Nasrin's mean speed for the round trip. Express your answer in km/h as a decimal number. | 2.7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}+mx-2=0$ with one solution $x=1$, find the other solution of this equation. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A merchant has 15 silver coins, among which one is counterfeit and lighter. Using a balance scale (without a measuring code), what is the minimum number of weighings required to find the counterfeit coin? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Two wheels rotate, and each rotation results in two pointers pointing to two different numbers. Wheel A has the numbers 3, 4, 5, and 8, while Wheel B has the numbers 6, 7, and 9. What is the probability that the sum of the two numbers indicated by the arrows is even? Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the two circles $x^2+y^2=m$ and $x^2+y^2+6x-8y-11=0$, find the range of values for the real number $m$ such that the two circles have a common point. Express your answer as an interval, e.g., [a, b]. | [1, 121] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
How many types of positional relationships are there between a line and a plane? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=({\sin\alpha,1})$ and $\overrightarrow{b}=({1,2\cos\alpha})$, and that $\overrightarrow{a} \perp \overrightarrow{b}$, calculate $\tan (\alpha -\frac{\pi}{4})$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Six athletes are preparing to compete on 6 tracks, where athlete A cannot stand on the first or second track, and athlete B must stand on either the fifth or sixth track. Find the total number of different arrangements. Express your answer as a whole number. | 144 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the conjugate of the complex number \(z=\dfrac{-3+i}{1-i}\). 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"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( AE \) and \( BD \) are straight lines that intersect at \( C \). If \( BD = 16 \), \( AB = 9 \), \( CE = 5 \), and \( DE = 3 \), calculate the length of \( AC \). Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let the universal set $U=\{-1,-2,-3,-4,0\}$, and the sets $A=\{-1,-2,0\}$, $B=\{-3,-4,0\}$. Find the intersection of the complement of $A$ in $U$ with $B$. Express your answer as a set using proper set notation. | \{-3,-4\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve \((x + 2y + a)(x^2 - y^2) = 0\), find the condition on \(a\) such that the curve represents three lines intersecting at a single point. Express your answer as a specific value or condition on \(a\), including any necessary mathematical notation or explanation. | a = 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial $3xy^{2}-2y-1$, find the coefficients of the linear term and the constant term. Express your answer as a pair of coefficients, with the linear term coefficient first, followed by the constant term coefficient. | (-2, -1) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A stone is dropped into a well, and the sound of the stone hitting the bottom is heard 7.7 seconds later. If the distance fallen by the stone in \( t \) seconds is \( 16t^2 \) feet, and the speed of sound is 1120 feet per second, calculate the depth of the well. Express your answer in feet as a whole number. | 784 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
When three uniformly dense coins are tossed, what is the probability of observing both heads and tails facing up? Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the integral $\int_{0}^{\frac{\pi}{4}} \frac{\cos 2x}{\cos x + \sin x} dx$. Express your answer as a single mathematical expression, using the boxed notation around your final answer. | \sqrt{2} - 1 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A person has a probability of 0.6 to hit the target with one shot. Calculate the probability that this person hits the target exactly twice in three shots. Express your answer as a simplified fraction. | \frac{54}{125} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation mx + 2y = 6 and the solution x = 1, y = 2, calculate the exact value of m. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(1,0)$, $\overrightarrow{b}=(1,1)$, and $( \overrightarrow{a}+\lambda \overrightarrow{b})\perp \overrightarrow{a}$, find the value of $\lambda$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The solution set of the inequality $x^2+ax+b\leq0$ is $\{x|-1\leq x\leq 3\}$. Find the product of $a$ and $b$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Four different products are arranged in a row for an exhibition, with the requirement that products A and B must have at least one other product between them. Calculate the number of different arrangement methods that satisfy this condition. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(\\frac{1}{x}) = \\frac{x}{1-x}$, find the expression for $f(x)$ when $x \\neq 0$ and $x \\neq 1$. Express your answer as a function of x. | \\frac{1}{x-1} | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, and $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, calculate the magnitude of the vector $|2\overrightarrow{a} - \overrightarrow{b}|$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} with the sum of its first n terms being S_n, where a_1 is the maximum value of the function f(x) = sqrt(3)sin(x) + cos(x) and a_n - a_nS_{n+1} = a_1/2 - a_nS_n, calculate the product of the first 2018 terms of the sequence {a_n}, denoted as A_{2018}. Express your answer as a single integer or fra... | 1 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of $a$ such that the inequality $x^{2}+ax+1\geqslant 0$ holds for all $x\in(0, \frac{1}{2}]$. Express your answer as a single number. | -\frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A shape is given where adjacent edges are at right angles, the four longer edges are equal in length, and all shorter edges are also equal in length. The area of the shape is 528. The shape can be subdivided into 33 smaller squares. Find the perimeter of the shape. Express your answer as a whole number. | 144 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\{(x, y) \mid 3x+4y-12<0, x, y \in \mathbb{N}^*\}$, calculate the number of proper subsets of $M$. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Abdul is 9 years older than Susie, and Binh is 2 years older than Susie. How many years older is Abdul than Binh? Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In a bag, there are 6 red balls and 4 white balls. If 4 balls are drawn from the bag, find the probability of drawing at least 2 white balls. Express your answer as a simplified fraction. | \frac {23}{42} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x + 2y = 30, find the value of the expression (x/5) + (2y/3) + (2y/5) + (x/3). Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Express the decimal number 0.000000907 in scientific notation. Include the coefficient and the exponent in your answer. | 9.07 \times 10^{-7} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The curve is defined by the parametric equations $egin{cases} x=t-1 \ y=t+2 \end{cases}$ (where $t$ is the parameter). Find the coordinates of the intersection points of the curve with the coordinate axes. Express your answer as two coordinates (x, y). | (0,3), (-3,0) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that e1 and e2 are two unit vectors with an angle of 60 degrees between them, what is the angle between vectors a = 2e1 + e2 and b = -3e1 + 2e2? Express your answer in degrees as a whole number. | 120 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In the plane rectangular coordinate system $xOy$, find the point symmetric about the $x$-axis to the point $A(2,1)$. Express your answer as an ordered pair $(x, y)$. | (2,-1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
What is the smallest number of red edges required for a cube such that every face of the cube has at least one red edge? Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Students A, B, and C independently solve the same problem. The probabilities of each student solving the problem correctly are $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{4}$, respectively. Find the probability that the problem is solved by someone. Express your answer as a fraction or decimal between 0 and 1. | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given a finite sequence \(P = \left(p_1, p_2, \cdots, p_n\right)\), the Cesaro sum is defined as \(\frac{1}{n}(S_1 + S_2 + \cdots + S_n)\), where \(S_k = p_1 + p_2 + \cdots + p_k\) for \(1 \leq k \leq n\). If a sequence \(\left(p_1, p_2, \cdots, p_{99}\right)\) of 99 terms has a Cesaro sum of 1000, then find the Cesaro... | 991 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
If $9-mx+x^2$ is a perfect square, find the value of $m$ that satisfies this condition. Express your answer as a numerical value, including the $\pm$ symbol if applicable. | \pm6 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^{2}y^{2}$ in the expansion of $\left( \frac{x}{ \sqrt{y}}- \frac{y}{ \sqrt{x}}\right)^{8}$. Express your answer as a single integer. | 70 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
When $x \in (0, +\infty)$, the power function $y = (m^2 - m - 1)x^m$ is a decreasing function. Find the exact value of the real number $m$. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
If the terminal side of angle α passes through point P(-1,3), calculate the value of cos α. Express your answer as a simplified fraction or exact value, including any necessary mathematical notation. | -\frac{\sqrt{10}}{10} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given a square with side length $s$, a triangle with three unequal sides is constructed on the diagonal of the square so that its area equals that of the square. Calculate the length of the altitude drawn to the base of the triangle. Express your answer in terms of $s$ and include any necessary square roots. | s\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ is $120^{\circ}$, and the magnitudes of both vectors are $|\overrightarrow{a}|=|\overrightarrow{b}|=1$, calculate the magnitude of the vector $\overrightarrow{a}+2\overrightarrow{b}$. Express your answer as a numerical value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\left( \frac {\pi}{6}-\alpha \right)=\cos\left( \frac {\pi}{6}+\alpha \right)$, find the value of $\tan\alpha$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The coordinates of point P are (2, 5). Find the coordinates of the point symmetric to P with respect to the x-axis. Express your answer as an ordered pair (x, y). | (2, -5) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A company has 430 employees, consisting of young, middle-aged, and elderly workers. There are 160 young workers, and the number of middle-aged workers is twice that of elderly workers. To understand the health status of the workers, a stratified sampling method is used. If there are 32 young workers in the sample, calc... | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\left(-a\right)^{3}\cdot \left(-a\right)^{2}$ using the rules of exponents. Express your answer as a simplified algebraic expression. | -a^{5} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
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