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 |
|---|---|---|---|---|---|
Let $i$ be the imaginary unit. Find the conjugate of the complex number $\frac{2}{1+i}$. Express your answer as a complex number in the form $a+bi$, and include the boxed notation around your final answer. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation ${C}_{5}^{x}={C}_{5}^{x-1}$, solve for the value of $x$. Provide your answer as a single integer. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} x^a & (x \geqslant 0) \\ |x - 2| & (x < 0) \end{cases}$, and $f(-2) = f(2)$, calculate the value of $f(4)$. Provide your answer as a single numerical value. | 16 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence where the first term is 2, the last term is 29, and the sum of all the terms is 155, find the common difference of the sequence. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {0,1,2,3,4}, set M = {0,2,3}, and the complement of N in U, ∁_U N = {1,2,4}, find the intersection of M and N, M ∩ N. Express your answer as a set of elements enclosed in curly brackets, with elements separated by commas. | {0,3} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of figures is formed by adding 4 unshaded squares to the previous figure. The first figure has 8 unshaded squares. Find the number of unshaded squares in the 10th figure. Express your answer as a whole number. | 44 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_2 + a_4 = 20 and a_3 + a_5 = 40. Find the value of a_6. Express your answer as a numerical value. | 64 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
The number \( \frac{16!}{2^k} \) is an odd integer. Note that \( n! = 1 \times 2 \times 3 \times \cdots \times (n-1) \times n \). Find the value of \( k \) that makes the expression an odd integer. Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
A box contains 15 red, 20 blue, and 16 green jelly beans. Jack first chooses a green jelly bean and eats it. Then he chooses a blue jelly bean and eats it. If each of the remaining jelly beans is equally likely to be chosen, what is the probability that Jack chooses a red jelly bean next? Express your answer as a fract... | \frac{15}{49} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let $x > 0$, $y > 0$, and $x + y = 18$. Find the maximum value of $xy$. Express your answer as a single integer. | 81 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, it is known that $a_1 = 2$ and $a_2 + a_3 = 13$. Find the common difference $d$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex number expression $i - \frac{1}{i}$. Express your answer as a simplified complex number. | 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
When rolling three fair dice (each face is numbered from 1 to 6), calculate the probability that exactly one die shows a number divisible by 3. Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$. | \frac{4}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given |a-c| < |b| (where a, b, and c are non-zero real numbers), what inequality must be true based on the properties of absolute values? Express your answer as the correct inequality. | |a| < |b| + |c| | big_math | [
"Mathematics -> Algebra -> Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( a \) and \( b \) are non-zero real numbers, there are three distinct points \( A \), \( B \), and \( C \) on the parabola \( y=ax^2 + b \) such that \( BC \) is parallel to the x-axis and \(\angle BAC = 90^\circ \). Find the distance from point \( A \) to the line \( BC \). Express your answer as a fracti... | \frac{1}{|a|} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the fractional equation $\frac{2x-m}{x+1}=3$, find the range of the variable $m$ such that the solution for $x$ is negative. Express your answer as an inequality, including any specific conditions or exclusions for $m$. | m > -3 \text{ and } m \neq -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A scout troop buys 1000 candy bars at a price of five for $2 dollars. They sell all the candy bars at the price of two for $1 dollar. Calculate the troop's profit from selling the candy bars. Express your answer in dollars. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} satisfies a_1 = 1, a_n - a_{n-1} = 2 for n >= 2, find the general term a_n. Express your answer as a formula or equation using 'n' as the variable for the term number. | a_n = 2n - 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\frac{\sin\alpha + \cos\alpha}{\sin\alpha - \cos\alpha} = 2$, find the value of $\tan2\alpha$. Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A geometric arrangement contains octagons and squares only. There are 4 rows of octagons with 5 octagons per row, and 3 rows of squares with 4 squares per row. Find the ratio of the number of octagons to the number of squares. Express your answer as a simplified ratio in the form a:b. | 5:3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} e^{x}+1, & x < 1 \\ f(\ln x), & x \geqslant 1 \end{cases}$, find the value of $f(e)$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation for heat generated by a current flowing through a conductor is Q=I^2Rt, where Q is the heat in joules (J), I is the current in amperes (A), R is the resistance in ohms (Ω), and t is the time in seconds (s). If the resistance of the conductor is 5Ω, and the conductor generates 30J of heat in 1s, calcu... | \sqrt{6} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(n)$ ($n\in\mathbb{N}_{+}$) that satisfies $f(n)=$ $$\begin{cases} n-3, &n\geq 100\\ f[f(n+5)], &n<100 \end{cases}$$, calculate the value of $f(1)$. Express your answer as a single integer. | 98 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
From a cabinet containing 3 pairs of different shoes, 2 shoes are randomly selected. Calculate the probability that the 2 selected shoes do not form a pair. Express your answer as a simplified fraction within the boxed format. | \dfrac{4}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given two sets $S=\{a_1, a_2\}$ and $T=\{b_1, b_2\}$, calculate the total number of mappings from set $S$ to set $T$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of lines that can be drawn parallel to a plane through a point outside the plane. Express your answer as a word describing quantity (e.g., 'one', 'two', 'infinite'). | infinite | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Let m and n be any two odd numbers, with n less than m. Find the largest integer that divides all possible numbers of the form m^2 - n^2. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x^2 + 3x + 8 = 7$, calculate the value of the algebraic expression $3x^2 + 9x - 2$. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations $2x-y=1$ and $xy=2$, calculate the value of the expression $4x^{3}y-4x^{2}y^{2}+xy^{3}$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\triangle PQR$ is isosceles. Find the value of $x$. Express your answer in degrees as a whole number. | 70 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In a right triangle \( \triangle ABC \), the lengths of the two legs are 5 and 12. Two arcs are drawn with centers at points \( A \) and \( B \) and radii 12 and 5, respectively. These arcs intersect the hypotenuse at points \( M \) and \( N \). Calculate the length of the segment \( MN \). Express your answer as a sin... | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence $\{a_{n}\}$, if $a_{3}=1$ and $a_{11}=25$, calculate the value of $a_{7}$. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 - 12x, find the value of x that maximizes f(x). Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = (m-1)x^{|m|-2}, find the value of m that makes this function an inverse proportion function. Ensure that your solution satisfies the conditions |m| - 2 = -1 and m - 1 ≠ 0. Provide your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $y=f(x)+\sin \frac {π}{6}x$ is an even function, and $f(\log _{ \sqrt {2}}2)= \sqrt {3}$, determine the value of $f(\log _{2} \frac {1}{4})$. Express your answer as a single value, using square roots if necessary. | 2 \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{1,2,3,4\}$ and the set $B$ defined by $y=3x-2$, where $x\in A$, calculate the intersection of sets $A$ and $B$, denoted as $A \cap B$. Express your answer as a set of numbers. | \{1, 4\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A frog starts jumping from the point (1,2) and makes a series of jumps, each of which is parallel to one of the coordinate axes and is 1 unit in length. The direction of each jump (up, down, right, or left) is chosen at random. The frog stops jumping when it reaches the boundary of the square with vertices (0,0), (0,4)... | \frac{5}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
7 people stand in a row. If person A and person B do not stand next to person C, find the total number of different arrangements. Express your answer as a single integer. | 2400 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the plane rectangular coordinate system, find the area of the region described by the system of inequalities:
\[
\left\{
\begin{array}{l}
\sqrt{3} x - y \leq 0, \\
x - \sqrt{3} y + 2 \geq 0, \\
y \geq 0
\end{array}
\right.
\]
Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\lg x - \lg y = a$, simplify the expression $\lg \left( \frac{x}{2} \right)^3 - \lg \left( \frac{y}{2} \right)^3$. Express your answer in terms of $a$. | 3a | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overset{→}{a}=(1,-2)$ and $\overset{→}{b}=(3,0)$, if $(2\overset{→}{a}+ \overset{→}{b}) \parallel (m\overset{→}{a}- \overset{→}{b})$, find the value of $m$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\\vec{a}$ and $\\vec{b}$ satisfying $|\\vec{a}|=1$, $|\\vec{b}|=2$, and $\\vec{a}-\\vec{b}=( \\sqrt{3}, \\sqrt{2} )$, calculate the magnitude of $\\vec{a}+2\\vec{b}$. Express your answer as a numerical value. | \\sqrt{17} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfying $$\frac {z}{2+ai} = \frac {2}{1+i}$$ ($a\in\mathbb{R}$), if the imaginary part of $z$ is -3, calculate the real part of $z$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $3^{x}=15$ and $3^{y}=5$, calculate the value of $3^{x-y}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an even function defined on \\(\\mathbb{R}\\), and for any x \\(\\in \\mathbb{R}\\), f(x+6) = f(x), and f(4) = 5, calculate the value of f(2018). | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{2+4i}{(1+i)^{2}}$. Express your answer as a simplified complex number. | 2-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 people, including A and B, lining up to take a photo. The requirement is that A and B must stand next to each other but not at the ends of the line. How many different arrangements are there? Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $$\frac {a+i}{1+2i}=ti$$, where $i$ is the imaginary unit, and $a$, $t$ are real numbers, find the value of $t+a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, PQR is a straight line segment and QS = QT. Also, ∠PQS = x° and ∠TQR = 3x°. If ∠QTS = 76°, find the value of x. Express your answer in degrees as a whole number. | 38 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\left\{ a_n \right\}$, $a_1+a_5=10$, $a_2+a_6=14$. Find the sum of the first 9 terms of the sequence. Express your answer as a single integer. | 81 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the result of the division $(-4a^{2}+12a^{3}b)\div (-4a^{2})$. Express your answer as a simplified algebraic expression. | 1 - 3ab | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\overrightarrow {a}=(1,2)$$ and $$\overrightarrow {b}=(m,1)$$, and $$\overrightarrow {a} \perp \overrightarrow {b}$$, find the value of $m$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the objective function z = 3x + 2y, and assuming the constraints x >= 0, y >= 0, and x + y <= 4, calculate the maximum value of z. Express your answer as a single number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A wire 52 cm long is exactly enough to solder into a rectangular solid teaching aid with a length of 6 cm and a width of 4 cm. Calculate the height of the rectangular solid. Express your answer in centimeters (cm) as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation: x + 2x + 3x + 4x = 5. Express your answer as a fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $e_1$ and $e_2$ are orthogonal unit vectors, and $a = 3e_1 + 2e_2$, $b = -3e_1 + 4e_2$, calculate the dot product $a \cdot b$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of zeros of the function f(x) = 2^x + (x-1)^3 - 2014 in the interval (10, 11). Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{0,1,2\}$ and $N=\{x|x=a^{2}, a \in M\}$, find the intersection of sets $M$ and $N$. Express your answer as a set using standard set notation. | \{0,1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that x and y are positive numbers satisfying the equation x + 3y = xy, find the minimum value of 3x + 4y. Express your answer as a single number. | 28 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality 2x - m > 4, where x = 3 is an integer solution and x = 2 is not an integer solution, find the range of values for m. Express your answer in interval notation, including any necessary mathematical notation (e.g., ≤, ≥, <, >). | [0, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Grace writes a sequence of 20 numbers. The first number is 43, and each number after the first is 4 less than the number before it. Find the number of positive terms in the sequence. Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, and the complex number $z_1=3-4i$, if in the complex plane, the points corresponding to the complex numbers $z_1$ and $z_2$ are symmetric about the imaginary axis, calculate the product $z_1 \cdot z_2$. Express your answer as a single integer. | -25 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the circumference of a circle with a diameter of 1 cm. Express your answer in centimeters, approximated to two decimal places. | 3.14 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The number $2.5252525\ldots$ can be written as a fraction in lowest terms. Find the sum of the numerator and denominator of this fraction. Express your answer as a whole number. | 349 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a \in \{1,2,3,5\}$ and $b \in \{1,2,3,5\}$, determine the number of distinct lines represented by the equation $y = \frac{b}{a}x$. Provide your answer as a whole number. | 13 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
If the arc length corresponding to a central angle of 2 radians is 4 cm, then calculate the area of the sector enclosed by this central angle. Express your answer in square centimeters (cm²). | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\sin(\pi-\alpha) = 2\cos\alpha$, calculate the constant term in the expansion of $\left(x+ \frac {\tan\alpha}{x}\right)^{6}$. Express your answer as a single number. | 160 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Trigonometry -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( AE \) and \( BD \) are straight lines that intersect at \( C \). Given that \( BD = 16 \), \( AB = 9 \), \( CE = 5 \), and \( DE = 3 \), calculate the length of \( AC \). Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given a function y = f(x) with an inverse function y = f^{-1}(x), determine the function obtained after rotating the graph of y = f(x) by 90 degrees counterclockwise about the origin. Express your answer as a function of x, using f^{-1}(x) notation if necessary. | y = f^{-1}(-x) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
How many integers are greater than \( \frac{5}{7} \) and less than \( \frac{28}{3} \)? Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan \alpha = 2$, calculate the value of $\sin \alpha \cos \alpha$. Express your answer as a fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $a_1=1$, and $a_{n+1}-3a_n+4=0$, where $n\in\mathbb{N}^+$, find the number of terms that are negative in the first 10 terms of the sequence $\{a_n\}$. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the inequality $(1-a)x^2 - 4x + 6 > 0$ is $\{x|-3 < x < 1\}$, and the solution set of $ax^2 + bx + 3 \geq 0$ is $\mathbb{R}$, find the range of values for $b$. Express your answer using interval notation. | [-6, 6] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 0$ and $b > 0$, if $2$ is the geometric mean of $4^{a}$ and $2^{b}$, find the minimum value of $\frac {2}{a}+ \frac {1}{b}$. Express your answer as a single numerical value in simplest form. | \frac{9}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|m| = 2$, $|n| = 4$, and $m > 0$, $n < 0$, calculate the value of $m - n$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the reciprocal of -3. Express your answer as a fraction. | -\frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets A = {1, 2, 3, 4} and B = {2, 4, 5, 6}, find A ∪ B. Express your answer as a set of elements, separated by commas and enclosed in curly brackets. | {1, 2, 3, 4, 5, 6} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the exponential function f(x) = (a+1)^x, which is a decreasing function on R, find the exact range of values for a. Express your answer as an inequality, including all necessary mathematical symbols and notations. | -1 < a < 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(x-2)^{2}+y^{2}=1$, find the maximum value of $\frac{y}{x}$. Express your answer as a single numerical value in the boxed format. | \frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Define the function f(x) on R such that it satisfies: (1) f(0) = 0, (2) f(x) + f(1-x) = 1, and (3) f(x/3) = (1/2)f(x). Additionally, when 0 <= x_1 < x_2 <= 1, f(x_1) <= f(x_2). Calculate the value of f(1/3) + f(1/8). Express your answer as a single fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In a convex polygon, exactly five of the interior angles are obtuse. Find the largest possible number of sides for this polygon. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let set A = {0,1,2,3}. An equation of the form x^2 - mx - n = 0, where m, n in A, is considered qualified if it has at least one root x in A. Find the total number of qualified equations. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A product is classified into three grades: Grade A, Grade B, and Grade C, where both Grade B and Grade C are considered defective. If the probability of producing a Grade B product is 0.03 and the probability of producing a Grade C product is 0.01, then what is the probability of obtaining a non-defective product in a ... | 0.96 | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(\frac{1}{x}) = \frac{1+x}{x}$, find the value of $f(2)$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Let $A_1 = \{0, 1\}$, $A_2 = \{1, 2\}$, $S = A_1 \cup A_2$, $T = A_1 \cap A_2$. Calculate the set $C_ST$. Express your answer as a set, including all elements of the set. | \{0, 2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
What is the simplified fraction when you multiply both x and y by 3 in the fraction x / (x - y)? Express your answer in terms of the original fraction. | \frac{x}{x-y} | big_math | [
"Mathematics -> Algebra -> Rational Expressions -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 + 2x^2 - ax + 1 is increasing on the interval [1, 2], find the range of the real number a. Express your answer in interval notation. | (-\infty, 7] | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
There are 9 table tennis players: 5 men and 4 women. The task is to select 2 male and 2 female players to pair up for a mixed doubles match. Each male player can pair with either of the two selected female players. Calculate the total number of different pairing methods. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
On October 4, 1957, the world’s first artificial Earth satellite was launched using a multistage rocket, marking the beginning of the space age for humanity. What country launched this satellite? Provide your answer as the name of the country. | Soviet Union | big_math | [
"Mathematics -> Other -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given three data points (x1, y1), (x2, y2), and (x3, y3) such that x1 < x2 < x3 and x3 - x2 = x2 - x1, find the slope of the line that best fits the data. Express your answer as a fraction in terms of the given variables. | \frac{y3 - y1}{x3 - x1} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of sets of positive integers (x, y, z) that satisfy the system of equations:
\[
\begin{cases}
xy + xz = 255 \\
xy + yz = 31
\end{cases}
\]
Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the necessary and sufficient condition for the circle $(x-a)^2+(y-b)^2=r^2$ to pass through the origin. Express your answer as an equation or a condition. | a^2+b^2=r^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A parabolic arch has a height of 16 inches and a span of 40 inches. Calculate the height of the arch at a point 5 inches from the center. Express your answer in inches. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Let set $A = \{x \,|\, |x-1| < 1\}$, and $B = \{x \,|\, x < 1\}$. Calculate the set $(C_{R}B) \cap A$ and express your answer in set notation, including any relevant intervals or bounds. | \{x \,|\, 1 \leq x < 2\} | big_math | [
"Mathematics -> Set Theory -> Set Operations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Assume the random variable $X\sim N(1,\delta^2)$. Given that $P(X > 2) = 0.3$, calculate the probability $P(X > 0)$. Express your answer as a decimal value between 0 and 1. | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function f(x) = (x+1)(x^2-x+1). Express your answer as a function of x. | 3x^2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression ${A}_{7}^{2}+{C}_{7}^{3}+{C}_{7}^{4}$, where ${A}_{n}^{r}$ represents the number of arrangements of $n$ items taken $r$ at a time, and ${C}_{n}^{r}$ represents the number of combinations of $n$ items taken $r$ at a time. Express your answer as a single integer. | 112 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Permutations",
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
From the 6 finalists, 1 first prize, 2 second prizes, and 3 third prizes are to be awarded. Calculate the total number of possible outcomes. Express your answer as a single numerical value. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
The circumference of a circle is $24\pi$. Express the area of the circle in the form $k\pi$ and find the value of $k$. Provide your answer as a single integer. | 144 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of the function $y={2^{\frac{1}{{{x^2}+1}}}}$. Express your answer in interval notation. | (1, 2] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
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