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 |
|---|---|---|---|---|---|
Find the sum of all zeros of the function $f(x)=\begin{cases} & {(\frac{1}{2})}^{x}-2, & x < 0, \\ & x-1, & x\geqslant 0 \\ \end{cases}$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given $z=\frac{1}{1+i}$ (where $i$ is the imaginary unit), find the conjugate of $z$, denoted as $\overline{z}$. Express your answer as a complex number in the form $a + bi$, where $a$ and $b$ are real numbers. | \frac{1}{2} + \frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^2$ in the expansion of $(x+1)^5(x-2)$. Express your answer as a single integer. | -15 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function $y=-(x+h)^2$, when $x<-3$, $y$ increases as $x$ increases, and when $x>-3$, $y$ decreases as $x$ increases. Find the value of $y$ when $x=0$. Express your answer as a single integer. | -9 | big_math | [
"Mathematics -> Algebra -> Quadratic Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $M$ is a point on the circle defined by the equation $(x-5)^2+(y-3)^2=4$, determine the minimum distance from $M$ to the line $4x+3y-4=0$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, PR and QS meet at X. Also, triangle PQX is right-angled at Q with angle QPX = 62 degrees, and triangle RXS is isosceles with RX = SX and angle XSR = y degrees. Find the value of y. Express your answer in degrees as a whole number. | 76 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
From a class of 4 boys and 2 girls, 4 people are to be selected to participate in a psychological survey. If at least one girl must be selected, calculate the number of different ways to choose the 4 people. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the formula $u=\frac{{S}_{1}-{S}_{2}}{t-1}$, solve for $t$ in terms of $u$, $S_1$, and $S_2$. Express your answer as a single algebraic expression. | \frac{{S}_{1}-{S}_{2}+u}{u} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Mike has two containers. One container is a rectangular prism with width \( 2 \text{ cm} \), length \( 4 \text{ cm} \), and height \( 10 \text{ cm} \). The other is a right cylinder with radius \( 1 \text{ cm} \) and height \( 10 \text{ cm} \). Both containers sit on a flat surface. Water has been poured into the two c... | 7.2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $ax + y - 1 - a = 0$ is parallel to the line $x - \frac{1}{2}y = 0$, find the value of $a$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Consider the function $f(x)=\begin{cases}
2^{1-x}, & \text{if } x \leq 1,\\
1-\log_{2}x, & \text{if } x > 1,
\end{cases}$
Find the range of x for which f(x) ≤ 2. Express your answer as an interval, e.g., [a, b] or [a, +∞). | [0,+\infty) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Li Ming drew a 20° angle on paper. When viewed through a 10x magnifying glass, what is the degree of the angle? Express your answer in degrees. | 20 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
The famous mathematician of the Ming Dynasty in China, Cheng Dawei, recorded the following mathematical problem in his classic book "Suanfa Tongzong": "From afar, the magnificent pagoda has seven layers, with red lights doubling on each ascending floor, totaling 381 lights. How many lights are there at the very top of ... | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (2, 1, 3)$ and $\overrightarrow{b} = (-1, 2, 1)$, find the value of λ such that $\overrightarrow{a}$ is perpendicular to $(\overrightarrow{a} - \lambda \overrightarrow{b})$. Express your answer as a single numerical value. | \frac{14}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given P(B) = 0.3, P(B|A) = 0.9, P(B|\\overline{A}) = 0.2, calculate P(\\overline{A}). Express your answer as a fraction. | \\frac{6}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In an equilateral triangle $ABC$ with side length $1$, $\\overrightarrow{BD} = \\frac{1}{3} \\overrightarrow{BA}$, and $E$ is the midpoint of $CA$. Calculate the value of $\\overrightarrow{CD} \\cdot \\overrightarrow{BE}$. Express your answer as a single numerical value. | -\\frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \(\left(a+\frac{1}{3a}\right)^{2}=3\), find the value of \(27a^{3}+\frac{1}{a^{3}}\). Express your answer as an algebraic expression with radicals if necessary. | \pm 54\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1=-2$, $a_{n+1}=a_n-2^{n}$. Calculate the value of $a_{2017}$. Express your answer as a single integer, including the negative sign if applicable. | -2^{2017} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
There are three volunteers, A, B, and C, who are scheduled to participate in a volunteer activity from Monday to Friday. Each person participates only once, and no more than one person is scheduled per day. It is required that A is scheduled before both B and C. Calculate the total number of different scheduling method... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with the common ratio q, if a_1 * a_3 = 16 and a_2 + a_4 = 12, find the value of q. Express your answer as a single value, including the ± symbol if applicable. | ±√2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let the sets A = {1, 2, ⋯, 10} and B = {1, 2, 3, 4}. Calculate the number of subsets C of A such that C ∩ B ≠ ∅. Express your answer as a single integer. | 960 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Estimate the value of $\sqrt{2}(\sqrt{8}+\sqrt{10})$ to the nearest integer range. Express your answer in the format "Between $a$ and $b$", where $a$ and $b$ are integers. | Between 8 and 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequalities |x - 1| ≤ 1 and |y - 2| ≤ 1, find the maximum value of |x - 2y + 1|. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Find the x-intercept of the line $2x + 3y + 6 = 0$. Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 shmacks in 2 shicks, 3 shicks in 5 shures, and 2 shures in 9 shneids. Calculate the number of shmacks in 6 shneids. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the average of the 10 data points ${a_1, a_2,..., a_{10}}$ is $\bar{a}$ and the variance is $1.1$, calculate the variance of the 11 data points ${a_1, a_2,..., a_{10}, \bar{a}}$. Express your answer as a decimal value. | 1.0 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A parabola is defined by the equation y = 2x^2. Points A(x1, y1) and B(x2, y2) on the parabola are symmetric about the line y = x + m, and x1 * x2 = -1/2. Determine the value of m. Express your answer as a single value. | m = 3/2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the exponential function f(x) = (3m-1)^x is a decreasing function on R, find the range of values for the real number m. Express your answer as an inequality. | \frac{1}{3} < m < \frac{2}{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Lili wants to know the height of the school flagpole. She found that the rope hanging from the top of the flagpole is 2 meters longer than the height of the flagpole. When she pulled the lower end of the rope 6 meters away from the flagpole, she found that the lower end just touched the ground. What is the height of th... | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The relationship between the current $I(A)$ and time $t(s)$ is given by $I=5\sin\left( 100\pi t+\frac{\pi}{3} \right)$. Calculate the current $I$ when $t=\frac{1}{200}$. Provide your answer as a single number. | \frac{5}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a real number $a$, the set $M$ is defined as $M=\{x| x^2-3x-a^2+2=0, x\in \mathbb{R}\}$. Calculate the number of subsets of the set $M$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A ball and a bat cost £90 in total. Three balls and two bats cost £210 in total. Calculate the cost of a bat. Express your answer in pounds (£). | 60 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_{n}\}$, where $a_{4}+a_{8}=8$, calculate the sum of the first $11$ terms $S_{11}$. Express your answer as a single integer. | 44 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a > 0$ and $b > 0$, if $\sqrt{2}$ is the geometric mean of $2^{a}$ and $2^{b}$, find the minimum value of $\frac{1}{a}+\frac{1}{b}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given a function f(x) defined on R, satisfying f(x) = -f(x + 3/2), and f(1) = 1, find the value of f(2017). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(4, 6^2), and P(ξ ≤ 5) = 0.89, find the probability P(ξ ≤ 3). Express your answer as a decimal value between 0 and 1. | 0.11 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The first row in a certain sports palace has 5 seats, the second row has 7 seats, the third row has 9 seats, and so on. The number of seats in each row forms an arithmetic sequence where each term increases by 2. Calculate the number of seats in the 15th row. | 33 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of the squares of the numbers 17, 19, 23, and 29. Express your answer as a single integer. | 2020 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
In the number "20220420", calculate the frequency of the digit "2". Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {0,1,2,3,4} and sets A = {1,2,3}, B = {2,4}, determine the set (C_U A) ∪ B. Express your answer as a set of elements using standard set notation. | {0,2,4} | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the binomial coefficient of the 7th term in the expansion of (a+b)^n is the largest, find the value of n. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The distance from the Earth to the surface of the moon is about 384,000 kilometers. Express this distance in scientific notation, keeping three significant figures. Provide your answer in the form $a \times 10^{n}$, where $1 \leq |a| < 10$ and $n$ is an integer. | 3.84 \times 10^{5} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
As shown in the figure, $A B \parallel E F \parallel C D$. Given that $A B = 20$, $C D = 80$, and $B C = 100$, find the value of $E F$. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with a common ratio q = 2, calculate the value of the expression (2a_1 + a_2) / (2a_3 + a_4). Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system $Oxyz$, if the distance from a point $M$ on the $y$-axis to two points $P(1, 0, 2)$ and $Q(1, -3, 1)$ is equal, then what are the coordinates of point $M$? Express your answer as coordinates (x, y, z) using whole numbers. | (0, -1, 0) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (x^2 - 4x + 9)/(x - 4) and x >= 5, find the minimum value of f(x). Express your answer as a single number. | 10 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$, $\overrightarrow{b}$ satisfy $|\overrightarrow{a}+ \overrightarrow{b}|=2 \sqrt {3}$, $\overrightarrow{a}\cdot \overrightarrow{b}=2$, calculate the magnitude of the vector difference $|\overrightarrow{a}- \overrightarrow{b}|$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition $p$: $\forall x_1,x_2\in \mathbb{R},(f(x_2)-f(x_1))(x_2-x_1)\geqslant 0$, what is the negation $\neg p$? Express your answer as a logical statement with the correct quantifier and inequality. | \exists x_1, x_2 \in \mathbb{R}, (f(x_2)-f(x_1))(x_2-x_1) < 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the definitions of binomial coefficients and the condition that $\binom{n}{1}, \binom{n}{2}, \binom{n}{3}$ form an arithmetic progression with $n > 3$, calculate the exact value of $n$. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the dimensions of the rectangular prism (2 x 2 x 1) and the cylinder (base radius: 1, height: 3), and the overlapping volume (pi/2), calculate the volume of the composite solid. Express your answer as a mathematical expression including pi. | 4 + \frac{5\pi}{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Let $a \in \mathbb{R}$, and $(a+i)^2i$ is a positive real number. Find the value of $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the binomial coefficients in the expansion of $(1+2x)^5$. Express your answer as a single integer. | 32 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that -4 ≤ x ≤ -2 and 2 ≤ y ≤ 4, find the largest possible value of the expression (x + y)/x. Express your answer as a single number. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_4 + a_8 = 16. Find the value of a_2 + a_6 + a_10. Express your answer as a single numerical value. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear relationship between the number of hot drinks sold in a day \((y)\) and the temperature of that day \((x℃)\), the regression equation is \( \hat y=-2.35x+147.77 \). If the temperature one day is \(2℃\), calculate the approximate number of hot drinks that the shop can sell using the regression equation.... | 143 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} e^{x},x < 0 \\ \ln x,x > 0 \end{cases}$, calculate $f[f( \frac {1}{e})]$. Express your answer as a single value. | \frac {1}{e} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for \( n \) in the equation \( 3 \times n = 6 \times 2 \). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(1,5,-1), B(2,4,1), and C(a,3,b+2) are collinear, use vector mathematics to determine the values of a and b, and then calculate a + b. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ where $a_1=1$, $a_2= \frac{1}{1+2}$, $a_3= \frac{1}{1+2+3}$, $a_4= \frac{1}{1+2+3+4}$,..., $a_n= \frac{1}{1+2+3+\ldots+n}$, find the sum of the first $n$ terms of the sequence $\{a_n\}$, denoted as $s_n$. Express your answer as a formula in terms of $n$. | \frac{2n}{n+1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
If moving east for 3 meters is denoted as $+3m$, then how would moving west for 4 meters be denoted? Express your answer in the format of a numerical value with a sign, including the unit'm'. | -4m | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|-1\leqslant x\leqslant 1\}$ and $N=\{y|y=x^{2},x\in M\}$, find the intersection $M\cap N$. Express your answer as a closed interval, including the endpoints. | [0,1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
If line $l_1: 2x-y-1=0$ is parallel to line $l_2: 2x+(a+1)y+2=0$, find the value of $a$ that makes the slopes of these two lines equal. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, where a_1 + a_3 + a_9 = 20, find the value of 4a_5 - a_7. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable ξ follows a normal distribution N(0,σ^2), if P(ξ > 3) = 0.023, calculate the probability P(-3 ≤ ξ ≤ 3). Express your answer as a decimal value between 0 and 1. | 0.954 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the perimeter of square PQRS is 120 and the perimeter of triangle PZS is 2x. Find the perimeter of pentagon PQRSZ in terms of x. | 60 + 2x | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(\{a_{n}\}\) is a geometric sequence, \(S_{n}\) is the sum of the first \(n\) terms of the sequence \(\{a_{n}\}\), and \(a_{n+1}=2S_{n}+2\), calculate the value of \(a_{4}\). Provide your answer as a single number. | 54 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number (1+ai)/(2-i), where a is a real number, find the value of 'a' such that the real part and the imaginary part of the complex number are equal. Express your answer as a single real number. | 1/3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of the real number $k$ for which the equation $2x^2 + ky^2 = 1$ represents an ellipse with foci on the $y$-axis. Express your answer as an interval using standard interval notation. | (0, 2) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 students standing in a row to take a photo, and students A and B must stand next to each other. Calculate the total number of different possible arrangements. Express your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given five numbers -9, $b_1$, $b_2$, $b_3$, -1 form a geometric sequence, and four numbers -9, $a_1$, $a_2$, -3 form an arithmetic sequence, find the value of $b_2(a_2-a_1)$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line mx+4y-2=0 is perpendicular to the line 2x-5y+n=0, and the foot of the perpendicular is (1,p), calculate the value of the real number n. | -12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function $f(x)=\cos ^{3}x+\sin ^{2}x-\cos x$. Express your answer as a single number. | \frac{32}{27} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overset{→}{a}=(1,2)$ and $\overset{→}{b}=(1,1)$, let $\overset{→}{c}=\overset{→}{a}+k\overset{→}{b}$. If $\overset{→}{b}⊥\overset{→}{c}$, then find the value of the real number $k$. Express your answer as a fraction or decimal. | -\dfrac{3}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A new drug's blood concentration peaks after 3 hours of administration. Subsequently, the concentration is tested every 2 hours, and each time it decreases to 40% of the previous test's value. Calculate the administration time when the blood drug concentration reaches 1.024% of its peak value. Provide your answer as a ... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the parabola y^2 = 16x, find the equation of its directrix in the form x = a. | x=-4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given sin(α) - 3cos(α) = 0, find the value of sin²(α) + sin(α)cos(α). Express your answer as a simplified fraction. | \frac{6}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A={x∈N||x|<3} and B={x|-2<x≤1}, calculate the intersection of sets A and B, denoted as A∩B. Express your answer as a set of elements that are common to both sets A and B. | \{0, 1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
An ellipse with its center at the origin and foci at (±√2, 0) is intersected by the line y = x + 1. The x-coordinate of the midpoint of the resulting chord is -2/3. Determine the equation of the ellipse in standard form. | \frac{x^2}{4} + \frac{y^2}{2} = 1 | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Conic Sections"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solutions (P, Q) of the equation P/Q - Q/P = (P+Q)/PQ, where P and Q are integers from 1 to 9 inclusive. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $m, n \in \mathbb{R}^+$, and $m^2 - amn + 2n^2 \geq 0$, find the maximum value of the real number $a$. Express your answer as a single value. | 2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given three points $A$, $B$, and $C$ on a plane that satisfy $| \overrightarrow{AB}|=3$, $| \overrightarrow{BC}|=4$, and $| \overrightarrow{CA}|=5$, calculate the value of $\overrightarrow{AB}\cdot \overrightarrow{BC}+ \overrightarrow{BC}\cdot \overrightarrow{CA}+ \overrightarrow{CA}\cdot \overrightarrow{AB}$. Express ... | -25 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The function \( f \) is given by the table
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| :---: | :--- | :--- | :--- | :--- | :--- |
| \( f(x) \) | 4 | 1 | 3 | 5 | 2 |
Calculate the value of \( \underbrace{f(f(f(f(\ldots f}_{2004 \text{ times }}(4) \ldots))) \). Provide your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
A class has four periods in one morning: Chinese, Mathematics, English, and Physical Education. The Physical Education class cannot be scheduled as the first or fourth period. Calculate the total number of different possible schedules for the four periods. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the angle that has the same terminal side as -30° by adding an appropriate multiple of 360° to -30°. Express your answer in degrees. | 330° | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Angle Measures"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $P$ is a point on the ellipse $\frac{x^2}{25} + \frac{y^2}{b^2} = 1$ ($0 < b < 5$) other than the vertices, and $F_1$ is the left focus of the ellipse. If $|\overrightarrow{OP} + \overrightarrow{OF_1}| = 8$, then find the distance from point $P$ to the left focus of the ellipse. Express your answer as a sing... | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given two lines $l_1: (3+m)x-4y=5-3m$ and $l_2: 2x-y=8$ are parallel, find the value of the real number $m$. Express your answer as a single real number. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The domain of the function f(x) is \u211d+. If f(x+y) = f(x) + f(y) and f(8) = 3, find the value of f(2). Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of linear equations $\left\{\begin{array}{l}{x+2y=m}\\{2x-3y=4}\end{array}\right.$, find the value of $m$ that satisfies the condition $x+y=7$. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A fair coin is tossed, with the probability of landing on heads or tails both being 1/2. The sequence {a_n} is defined as: a_n = 1 if the nth toss is heads, and a_n = -1 if the nth toss is tails. If S_n = a_1 + a_2 +... + a_n, then find the probability of the event S_4 > 0. | \frac{5}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given $i$ is the imaginary unit, calculate the product of $(1+i)(1-i)$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = √3cos(x) + sin(x), translate it to the left by m units (m > 0). Find the minimum value of m such that the resulting graph is symmetric with respect to the y-axis. Express your answer in radians. | \frac{\pi}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The ratio of the exterior angles of a triangle is 3:4:5. Find the ratio of the interior angles of this triangle. Express your answer as a simplified ratio. | 3:2:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given two complex numbers $z_1$ and $z_2$ in the complex plane, their corresponding points are symmetric about the imaginary axis, and $z_1=2-i$. Calculate the value of $({z}_{1} \cdot \overline{z_2})$. Express your answer as a complex number in the form a+bi. | -3+4i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given $m=-4$, $n=-5$, $c=-7$, calculate the value of $m-n-c$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer values of $a$ that satisfy the equation $|a+5|+|a-3|=8$. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $a$, $b$, $c$ that satisfy $3x+4y-5\leqslant ax+by+c\leqslant 3x+4y+5$ for any real numbers $x$, $y$, find the minimum value of $a+b-c$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Five students, A, B, C, D, and E, had an average score of 80 points in a math test. The average score of students A, B, and C was 78 points. Calculate the average score of students D and E. | 83 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the radius of sector $AOB$ is $3cm$, and the central angle measures $120^{\circ}$, calculate the radius of the base of the cone formed from this sector. Express your answer in centimeters (cm) as a whole number. | 1 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$. Express your answer as a single integer. | 64 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
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