instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain sequencelengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
In a sequence, each term after the first two terms is the mean of all the terms which come before that term. Given that the first term is 8 and the tenth term is 26, calculate the value of the second term. | 44 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation Ax^2 + Bx + C = 0 with roots r and s, find the value of p in the new quadratic equation x^2 + px + q = 0 such that its roots are r^2 and s^2. Express your answer in terms of A, B, and C. | \frac{2AC - B^2}{A^2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the definition: $\left|\overrightarrow{a}\times\overrightarrow{b}\right|=\left|\overrightarrow{a}\right|\cdot\left|\overrightarrow{b}\right|\cdot\sin\theta$, where $\theta$ is the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. If $\left|\overrightarrow{a}\right|=2$, $\left|\overrightarrow{b}... | 8 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2(2-x), determine the interval where f(x) is increasing. Express your answer as an interval in the form (a, b). | (0, 4/3) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Four points in space \( A, B, C, D \) satisfy \( |AB| = 3, |BC| = 7, |CD| = 11, |DA| = 9 \). Calculate the dot product \( \overrightarrow{AC} \cdot \overrightarrow{BD} \). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Line $l_1$: $y=k(x-4)$ is symmetric to line $l_2$ about the point $(2,1)$. Given that line $l_1$ passes through the fixed point $(4,0)$, find the fixed point that line $l_2$ always passes through. Express your answer as an ordered pair $(x,y)$. | (0,2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let $ABC$ be a triangle such that $|AB|=8$ and $|AC|=2|BC|$. Calculate the largest value of altitude from side $[AB]$. Express your answer as a single numerical value. | \frac{16}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the monotonically decreasing interval of the function y = 2^(x^2 - 5x - 6). Express your answer as an interval using standard interval notation. | (-\infty, \frac {5}{2}) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Express "98.83 million" in scientific notation. Provide your answer in the form of a coefficient multiplied by a base with an exponent, like $\boxed{a \times 10^b}$. | 9.883 \times 10^7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations:
\[
\left\{
\begin{array}{l}
x + 2y = 1 + m \\
2x + y = -3
\end{array}
\right.
\]
and the condition x + y > 0, find the range of values for m that satisfies this condition. Express your answer using inequality notation (e.g., m > value). | m > 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given two non-zero real numbers $a$ and $b$ with $x^{2a}= \frac{1}{x^b}$ where $x > 0$, find the constant term in the expansion of $(x^a-2x^b)^9$. Express your answer as a single numerical value, including the negative sign if applicable. | -672 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A person has a probability of 0.6 to hit the target with one shot. After 3 shots, calculate the probability that this person hits the target at least twice. Express your answer as a simplified fraction. | \frac {81}{125} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = a*sin(ฯx + ฮฑ) + b*cos(ฯx + ฮฒ), where a, b, ฮฑ, ฮฒ are all non-zero real numbers, and f(2015) = -1, calculate the value of f(2016). | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of regular hexagons is formed by adding an additional layer to the existing point lattice of the original regular hexagon. The number of points in each layer follows a recursive sequence. Let \( h_n \) be the number of points inside the \( n \)-th hexagon layer, where \( h_1 = 1 \) and \( h_n = h_{n-1} + 6(n... | 37 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The school cafeteria offers lunch boxes at three different prices: 10 yuan, 12 yuan, and 15 yuan. After statistics, it was found that the sales of the 10 yuan, 12 yuan, and 15 yuan lunch boxes accounted for 50%, 30%, and 20% respectively. Calculate the median price of the sold lunch boxes. Express your answer in yuan. | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let $M=\{x|\lg x > 0\}$, $N=\{x|-3 \leq x-1 \leq 1\}$. Calculate the intersection of sets M and N, expressing your answer in standard interval notation. | (1,2] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the definite integral $\int_{0}^{\sqrt{2}} \sqrt{2-x^{2}}dx$. Express your answer as a fraction or decimal multiple of $\pi$. | \frac{\pi}{2} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
From 1000 to 9999, a four-digit number is randomly chosen. Calculate the probability that all the digits in this number are different and the units digit is an odd number. Express your answer as a simplified fraction. | \frac{56}{225} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = x^3 at the origin (0, 0). Express your answer in the form y = mx + b or x = a. | y=0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
On the planet Magne, people use migs, mags, and mogs as units of currency. Given that 1 mags = 8 migs and 1 mogs = 6 mags, what is the total value of 10 mogs + 6 mags in migs? Express your answer as a whole number. | 528 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the ellipse \\( \\frac{x^{2}}{16} + \\frac{y^{2}}{25} = 1 \\), calculate the eccentricity of the ellipse using the formula \\( e = \\frac{c}{a} \\). Express your answer as a fraction in the form \\( \\frac{a}{b} \\), where a and b are integers. Provide your answer in the boxed notation. | \\frac{3}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
The Zhuzhou City Committee for the Care of the Next Generation conducted a vision test on 500 randomly selected ninth-grade students out of a total of 30,000 in the city to understand their vision health. It was found that 100 of these students had poor vision. Estimate the total number of ninth-grade students with poo... | 6000 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
An object moves in the left and right directions. It is defined that moving $3m$ to the right is denoted as $+3m$. What is the notation for moving $3m$ to the left? Express your answer as a notation (e.g. $+3m$ or $-3m$). | -3m | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \( (x-4 \cos \theta)^{2}+(y-4 \sin \theta)^{2}=4(\theta \in \mathbf{R}) \), find the area of the region where the point \( P(x, y) \) can be located. Express your answer in terms of ฯ. | 32\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $2x-1=3x+2$ for x. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $2-3$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the domain of the function $f\left( x \right)=\frac{3{{x}^{2}}}{\sqrt{1-x}}+ \log \left( 3x+1 \right)$. Express your answer in interval notation. | \left(- \frac{1}{3}, 1\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence ${a_{n}}$ with sum $S_{n}$ of the first $n$ terms, we have the following equations:
$a_{2}+S_{3}=4$
$a_{3}+S_{5}=12$
Using the properties of arithmetic sequences, find the value of $a_{4}+S_{7}$. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the numbers 10000, 1, 10, 100, and 1000 from largest to smallest. What is the middle number in this ordered sequence? | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if a_3 + a_4 + a_5 = 12 and a_6 = 2, then what is the value of a_2 + a_8? Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $$\frac{i}{1+ai}$$ is a pure imaginary number, find the value of the real number $a$. Express your answer as a single real number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{1+i}{1-i}$. Express your answer as a simplified complex number. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The state income tax where Kristin lives is levied at the rate of $p\%$ of the first $\textdollar 28000$ of annual income plus $(p + 2)\%$ of any amount above $\textdollar 28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. Find Kristin's annual income. Express yo... | 32000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \(5, 6x, 7x^2, 8x^3, 9x^4, \ldots\), find the 50th term. Express your answer in the form of a mathematical expression with the coefficient and exponent of \(x\), i.e., \(\boxed{Ax^B}\), where \(A\) is the coefficient and \(B\) is the exponent. | 54x^{49} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Exactly 120 tickets were sold for a concert. The tickets cost $12 each for adults, $10 each for seniors, and $6 each for children. The number of adult tickets sold was equal to the number of child tickets sold. Given that the total revenue from the ticket sales was $1100, calculate the number of senior tickets sold. Ex... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = a^(x-2) + log_a(x-1) + 1 (a > 0, a!= 1), find the point that the graph must pass through. Express your answer as an ordered pair (x, y). | (2, 2) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z = (a-3) + (a^2 - 2a - 3)i$, where $i$ is the unit imaginary number, find the real values of $a$ such that $z$ is a real number. Express your answer as real numbers. | a = 3 \text{ or } a = -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function $f(x)=\sin \left( 2x+\frac{\pi}{3} \right)$. Express your answer in terms of ฯ. | \pi | big_math | [
"Mathematics -> Precalculus -> Functions -> Trigonometric Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = sqrt(x) at the point (4, 2). Express your answer in the standard form of a line equation (Ax + By + C = 0). | x - 4y + 4 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of non-negative integer solutions $(x, y)$ to the equation $3x^2 + y^2 = 3x - 2y$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Five numbers in increasing order are 2, 5, x, 10, and y. Given that the median of the numbers is 7 and the mean (average) is 8, calculate the value of y. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The sequence {a_n} satisfies a_1 = 1, a_2 = 1/2, and a_n(a_{n-1} + a_{n+1}) = 2 a_{n+1} a_{n-1} for n >= 2. Calculate the 2012th term of the sequence {a_n}. Express your answer as a fraction. | \frac{1}{2012} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a function f(x) = ax^3 - 3x^2 + 1, if f(-a), f(a), and f(3a) form an arithmetic sequence with a non-zero common difference, determine the number of tangent lines to the curve y = f(x) that can be drawn from the origin. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
If $f(\lg x) = x$, then calculate the value of $f(3)$. Express your answer as a single numerical value. | 1000 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\dfrac{\sin ({{180}^{\circ }}-\alpha )}{\cos ({{180}^{\circ }}+\alpha )}\cdot \dfrac{\cos (-\alpha )\cdot \cos ({{360}^{\circ }}-\alpha )}{\sin ({{90}^{\circ }}+\alpha )}$. Express your answer as a simplified trigonometric expression. | -\sin \alpha | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the interior angles of an n-sided polygon is three times the sum of its exterior angles, find the value of n. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In the sum shown, \( P, Q \) and \( R \) represent three different single digits. Find the value of \( P+Q+R \).
\[
\begin{array}{r}
P7R \\
+ 39R \\
\hline
RQ0
\end{array}
\] | 13 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\theta \in \left(\frac{\pi}{2}, \pi\right)$ and $\sin\theta = \frac{3}{5}$, calculate $\sin\left(\theta + \frac{5\pi}{2}\right)$. Express your answer as a fraction, including a negative sign if applicable. | -\frac{4}{5} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In an equilateral triangle $ABC$ with side length 1, let $\overrightarrow{BC} = \vec{a}$, $\overrightarrow{CA} = \vec{b}$, and $\overrightarrow{AB} = \vec{c}$. Calculate the value of $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$. Express your answer as a fraction or decimal. | -\frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $R$, the set $M=\{x|x>1\}$, and $N=\{x||x|\leq2\}$, calculate the intersection $(\complement_R M) \cap N$. Express your answer as an interval. | [-2,1] | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Let the even function $f(x)$ satisfy $f(x) = 2^x - 4$ for $x \geqslant 0$. Find the solution set of the inequality $f(x-2) > 0$. Express your answer as an interval or set of intervals, using any necessary mathematical notation (e.g. union, intersection). | (-\infty, 0) \cup (4, \infty) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{e_1}$ and $\overrightarrow{e_2}$ are unit vectors with an angle of $60^{\circ}$ between them, $\overrightarrow{a} = 2\overrightarrow{e_1} + \overrightarrow{e_2}$, and $\overrightarrow{b} = -3\overrightarrow{e_1} + 2\overrightarrow{e_2}$, find the angle between $\overrightarrow{a}$ and $\over... | 120 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_{n}}$, where $a_{1}=a_{2}=1$, and $a_{n+2}-a_{n}=1$, find the sum of the first $100$ terms of the sequence. Express your answer as a numerical value. | 2550 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function f(x) = x^2 + 2x - 3 on the interval [-2, 2]. Express your answer as an interval, using square brackets for closed intervals and parentheses for open intervals. | [-4, 5] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {2+ai}{3-i}$ is a pure imaginary number (where $i$ is the imaginary unit, $a\in\mathbb{R}$), compute the imaginary part of $z$. Express your answer as a real number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Convex quadrilateral ABCD is given, and lines are extended from points A, B, C, and D to points B', C', D', and A' respectively, such that AB = BB' = 6, BC = CC' = 7, CD = DD' = 8, and DA = AA' = 9. If the area of quadrilateral ABCD is 10, calculate the area of quadrilateral A'B'C'D'. Express your answer as a single in... | 50 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence $\{a_n\}$ with the general term formula $a_n = 3 - 2n$, calculate the common difference $d$ using the definition $d = a_n - a_{n-1}$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$, where $a_1 = 2$, $a_2 = 3$, and $a_n + a_{n+2} = 2a_{n+1}$ for each $n \geq 1$, find the sum $S_{18} = \sum_{k=1}^{18} a_k$. Express your answer as a single integer. | 189 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ satisfies $a_1=0$, $a_{n+1}=a_n+2\sqrt{a_n}+1$, calculate the value of $a_{13}$. Express your answer as a single integer. | 144 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
The parabola y = x^2 - 1 is shifted 3 units upwards. What is the equation of the resulting parabola? Express your answer in the form y = ax^2 + bx + c. | y = x^2 + 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Three circles \( P, Q \) and \( R \) are all tangent to each other. Their centers all lie on a diameter of \( P \), as shown in the figure. Find the value of \(\frac{\text{circumference of } Q + \text{circumference of } R}{\text{circumference of } P}\). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the parabola with its vertex at (4, -3). Express your answer in vertex form. | y = (x - 4)^2 - 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {2+i}{1+i}$, where $i$ is the imaginary unit, calculate the imaginary part of $z$. Express your answer as a single number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ฮพ follows a normal distribution N(1,1) and P(ฮพ < 3) = 0.977, calculate the probability P(-1 < ฮพ < 3). Express your answer as a decimal value rounded to three decimal places. | 0.954 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Among the four-digit numbers formed by the digits 2, 4, 5, 7, with each digit appearing only once, these numbers are arranged in ascending order. Identify the four-digit number that is in the 13th position. Express your answer as a four-digit number using the digits 2, 4, 5, 7 without repetition. | 5247 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x(x-1)/2, find the expression for f(x+2) in terms of x. | \frac{(x+2)(x+1)}{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $f(x)$ is differentiable at $x=x_0$, and $\lim\limits_{\Delta x\to0} \frac {f(x_0+3\Delta x)-f(x_0)}{\Delta x}=1$. Calculate $f'(x_0)$. Express your answer as a single value in the form $\boxed{[answer]}$. | \frac{1}{3} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
The polynomial $x^{2}+mx+25$ is a perfect square trinomial. Find the value of $m$. Express your answer as a single numerical value or a range of values, including the ยฑ symbol if applicable. | \pm 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x-2y+2=0$, find the value of the algebraic expression $\left(2y-x\right)^{2}-2x+4y-1$. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3x + (2k-1) = x - 6(3k+2)$ and the solution $x = 1$, calculate the value of $k$. Express your answer as a fraction. | -\dfrac{13}{20} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y^2 = 2x, the focus F is at (1/2, 0). A line l passes through the focus F and intersects the parabola at two different points A(x1, y1) and B(x2, y2). If x1 + x2 = 3, calculate the length of chord AB. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, $\angle A=55^\circ$, $\angle C=75^\circ, D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$. If $DB=BE$, then calculate the measure of $\angle{BED}$ in degrees. | 65^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the points (1,3) and (-4,-2) are on opposite sides of the line 2x + y + m = 0, find the range of values for m. Express your answer as an interval, including the endpoints if they are included. | (-5, 10) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Convert -630ยฐ to radians. Express your answer as a fraction of ฯ, if possible. | -\frac {7\pi}{2} | big_math | [
"Mathematics -> Precalculus -> Functions -> Angle Conversion"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=(x^{2}-6x+5)+(x-2)i$, determine the range of values for the real number $x$ such that the corresponding point in the complex plane is located in the third quadrant. Express your answer as an interval. | (1,2) | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given a line (l) with a slope of 1 and a hyperbola (C): x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0), they intersect at points B and D. The midpoint of BD is M(1, 3). Calculate the eccentricity of the hyperbola (C). Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
How many solutions are there to the equation 1 + 2 sin X - 4 sin^2 X - 8 sin^3 X = 0 within the interval 0ยฐ < X < 360ยฐ? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Three identical circles touch each other, and the circumference of each circle is 36 units. Calculate the perimeter of the shaded region formed by these circles. Express your answer in units. | 18 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sin 71^{\circ}\cos 26^{\circ}-\sin 19^{\circ}\sin 26^{\circ}$. Express your answer as a simplified fraction or radical. | \dfrac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all numbers of the form 2k + 1, where k takes on integral values from 1 to n. Express your answer as a formula or expression in terms of n. | n(n+2) | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
If $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=6$ has area 6, calculate the product $ab$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Solve for y in the equation: 40% of 50% of x is equal to 20% of 30% of y, where x is not equal to 0. Express your answer as an equation in terms of x, with y on the left-hand side. | y = \frac{10x}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the zeros of the function $f(x) = \begin{cases} -x-4, & (x < 0) \\ x^{2}-4, & (x > 0) \end{cases}$. Express your answer as a list of zeros, separated by commas. | -4, 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola x^2 - y^2 = 1 with foci F1 and F2, and point P on the hyperbola with angle F1PF2 = 60 degrees, calculate the distance from P to the x-axis. | \dfrac{\sqrt{6}}{2} | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The equation \( y = ax^2 + bx + c \) was used to create the table of values below:
\[
\begin{array}{c|c|c}
x & 0 & 1 \\
\hline
y & 8 & 9
\end{array}
\]
Find the value of \( a + b \). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A seven-layered tower is adorned with lights, with the number of lights doubling on each successive layer, totaling 381 lights. Calculate the number of lights on the top layer of the tower. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the sum of the coefficients of all terms in the expansion of $(1-2x)^{10}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse with its foci on the $x$-axis, described by the equation $\frac{x^2}{2} + \frac{y^2}{m} = 1$, and its eccentricity is $\frac{1}{2}$, calculate the length of the minor axis of this ellipse. Express your answer as a numerical value. | \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given two lines: $l_{1}$: $y=ax+3$ and $l_{2}$ symmetric about the line $y=x$, $l_{2}$ is parallel to $l_{3}$: $x+2y-1=0$. Find the value of $a$ that satisfies these conditions. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that points \(A, B, C, D\) are on the surface of a sphere centered at point \(O\), the line segments \(AB\), \(AC\), and \(AD\) are mutually perpendicular to each other, and \(AB = 3 \mathrm{~cm}\), \(AC = 4 \mathrm{~cm}\), and \(AD = \sqrt{11} \mathrm{~cm}\), calculate the radius of the sphere. Express your answ... | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The mean of five numbers is 25. Abbie adds 5 to the first number, 10 to the second number, 15 to the third number, 20 to the fourth number, and 25 to the fifth number to obtain a new set of five numbers. Calculate the mean of the numbers in the new set. Express your answer as a single numerical value. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = e^x / x, find the range of real number m such that f(x) has a local minimum on the interval (m, 6-m^2). Express your answer in interval notation. | (-\sqrt{5}, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function f(x) = (โ3 sin x + cos x)(โ3 cos x - sin x). Express your answer in radians. | ฯ | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A school library bought 140 new books, ending up with $\frac{27}{25}$ of the number of books it had before the purchase. Find the original number of books in the library. Express your answer as a whole number. | 1750 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set U = {x | x < 3} and A = {x | x < 1}, describe the complement of A within U, denoted as โ_U A, in the format {x | condition}. | โ_U A = {x | 1 โค x < 3} | big_math | [
"Mathematics -> Other -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
If $\{a_n\}$ is an arithmetic sequence, and $a_2 + a_5 + a_8 = 39$, then find the value of $a_1 + a_2 + \ldots + a_9$. Express your answer as a single numerical value. | 117 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the derivative of the function f(x) as f'(x)=x^{2}-4x+3, find the monotonically decreasing interval of f(x+1). Express your answer as an interval (a, b). | (0, 2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that an exterior angle of a regular n-sided polygon is 40 degrees, find the number of sides n. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vertex of angle ฮธ is at the origin, its initial side coincides with the positive x-axis, and the terminal side lies on the line 2x - y = 0, calculate the value of the expression:
$$\frac{\sin\left(\frac{3\pi}{2} + \theta\right) + \cos\left(\pi - \theta\right)}{\sin\left(\frac{\pi}{2} - \theta\right) - \... | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Using the method of simple random sampling to draw a sample of size 5 from a population containing 100 individuals, what is the probability of a specified individual being drawn? Express your answer as a simplified fraction. | \frac{1}{20} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
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