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 coterminal angle of 1560° in radians, within the standard range of 0 to 2π. Express your answer as a fraction of π (e.g., π/6, 2π/3). | \frac{2\pi}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n} with a_n > 0, given that a_3 * a_7 = 64, find the value of a_5. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular block, with dimensions 4 cm, 5 cm, and 6 cm, is made up of cubes each with side length 1 cm. If 1 cm³ cubes are removed from this larger rectangular block, what is the minimum number of these cubes that must be removed so that the resulting solid is itself a cube? Express your answer as a whole number. | 56 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
For 30 consecutive days, the daily high temperature was recorded. On each of the first 25 days, the temperature recorded was 21°C. On each of the remaining 5 days, the temperature recorded was 15°C. Calculate the mean (average) temperature for the 30 days. Express your answer in degrees Celsius (°C) as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In an earthquake relief operation in Sichuan, a rescue site needs to be blasted. After igniting the fuse, the operator must run to a safe area 400 meters away before the explosive detonates. Given that the burning speed of the fuse is 1.2 cm/s and the running speed of the operator is 5 m/s, calculate the minimum length... | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given $3a=2$, express $\log_3 8 - 2\log_3 6$ as an algebraic expression in terms of $a$. Provide your answer in the form $oxed{\text{your expression here}}$. | a-2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A class has 60 students. After an exam, the math scores $\xi$ are normally distributed with mean 110 and variance 102. If $P(100 \leq \xi \leq 110) = 0.35$, then estimate the number of students scoring above 120 in math. Provide your answer as a whole number, noting that an approximate answer is acceptable. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the minimum positive period of the function \( y = \sin x \left(1 + \tan x \tan \frac{x}{2} \right) \). Express your answer in terms of π. | 2\pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is differentiable at $x$, evaluate the limit $$\lim_{h \to 0} \frac{f(x+h) - f(x-h)}{2h}.$$ Express your answer as a single mathematical expression. | f'(x) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A circle is inscribed in a square of side m, then a square is inscribed in that circle, then a circle is inscribed in the latter square, and so on. Calculate the limit of the sum of the areas of the circles as the number of iterations approaches infinity. Express your answer as a mathematical expression involving pi an... | \frac{\pi m^2}{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 3m + 3/(n + 1/p) = 17, where m, n, and p are positive integers, find the value of p. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = cos(4x - π/3), the horizontal coordinates of each point on its graph are stretched to twice their original length, while the vertical coordinates remain unchanged, resulting in the graph of the function y = g(x). Calculate the smallest positive period of g(x) in radians. | π | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation |x-8y| + (4y-1)^2 = 0, solve for x and y, and then calculate the value of log_2 y^x. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
A moving point P lies on the circle $C: (x - 3)^2 + (y + 4)^2 = 4$. If point O is the origin of the coordinate system, find the maximum value of $|OP|$. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Point P is a moving point on the parabola $y^2=4x$. The distance from point P to the line $x=-1$ is 4. Calculate the distance from P to the focus of the parabola $y^2=4x$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
The crafty rabbit and the foolish fox made an agreement: every time the fox crosses the bridge in front of the rabbit's house, the rabbit would double the fox's money. However, each time the fox crosses the bridge, he has to pay the rabbit a toll of 40 cents. After crossing the bridge three times, the fox discovered th... | 35 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Find the opposite number of the polynomial $-a+b-c$. Express your answer as a polynomial. | a-b+c | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\left\{{x|{x=\frac{k}{6}+1,k∈Z}}\right\}$, $B=\left\{{x|{x=\frac{k}{3}+\frac{1}{2},k∈Z}}\right\}$, and $C=\left\{{x|{x=\frac{{2k}}{3}+\frac{1}{2},k∈Z}}\right\}$, determine the subset relationships among A, B, and C. Express your answer using ⊆ to denote subset relationships, and include all sets in yo... | C \subseteq B \subseteq A | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the monomial $2a^{5}b^{2m+4}$ and $a^{2n-3}b^{8}$ is still a monomial, find the values of $m$ and $n$. Express your answer as a pair of whole numbers (m, n). | (2, 4) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations of lines $l_1: 2x-5y+20=0$ and $l_2: mx+2y-10=0$, find the value of the real number $m$ such that the lines are perpendicular and form a quadrilateral with the coordinate axes that has a circumscribed circle. Express your answer as a single real number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} \ln x, & x > 0 \\ -x, & x < 0 \end{cases}$, solve for the real number $a$ in the equation $f(\frac{1}{3}) = \frac{1}{3}f(a)$. | \frac{1}{27} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
When \( x = -5 \), calculate the value of \( -2 x^{2} + \frac{5}{x} \). Express your answer as a single integer. | -51 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^2 + (a + 2)y^2 + 2ax + a = 0$, find the value of $a$ that makes this equation represent a circle. Express your answer as a single value. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex numbers $z_1$ and $z_2$ correspond to points in the complex plane that are symmetric with respect to the imaginary axis. Given $z_1 = 2 + i$, calculate the value of $z_2$. Express your answer as a complex number in the form $a + bi$. | -2 + i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the interval where the function f(x) = 2ln(x) - x is monotonically increasing. Express your answer as an interval in standard notation. | (0, 2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A type of bulrush grows 3 feet on the first day and subsequently decreases its growth by half each day; while a type of reed grows 1 foot on the first day and subsequently doubles its height each day. Using the reference data: lg 2 = 0.3010, lg 3 = 0.4771, calculate the number of days until the height of the bulrush an... | 2.6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems",
"Mathematics -> Calculus -> Other",
"Mathematics -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The lengths of the three sides of a triangle are \(7\), \(x+4\), and \(2x+1\). The perimeter of the triangle is 36. Find the length of the longest side of the triangle. Express your answer as a whole number. | 17 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the equation a_3 + 3a_8 + a_13 = 120 holds. Determine the value of a_8. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|\overrightarrow{a}|=|\overrightarrow{b}|=2$ and $|\overrightarrow{a}+\overrightarrow{b}|=2\sqrt{3}$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer in radians. | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition ∀x > 1, x² + x + 1 > 0, what is the negation of this proposition? Express your answer as a logical statement. | ∃x > 1, x² + x + 1 ≤ 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A cultural evening is composed of 6 acts, with the following requirements for the order of performance: Act A must be arranged in the first two positions, Act B cannot be arranged in the first position, and Act C must be arranged in the last position. Calculate the total number of different arrangements of the performa... | 42 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 3 different movie tickets among 3 out of 10 people, with each person receiving one ticket. Calculate the total number of different ways to distribute the tickets. Express your answer as a single integer. | 720 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{-1,0,1\}$ and $N=\{-2,-1,0,2\}$, determine the intersection $M\cap N$. Express your answer as a set. | \{-1,0\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve $f(x)=\frac{1}{3}x^3-x^2-\frac{16}{x-1}\ (x>1)$, find the minimum value of the slope of the tangent line at the point $(x_0, f(x_0))$. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for the complex number z in the equation (2+i)z = 5i. Express your answer in the form a + bi. | 1 + 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
To find the greatest common divisor (GCD) of $16$ and $12$, the entire operation is: $(16,12)\rightarrow(12,4)\rightarrow(8,4)\rightarrow(4,4)$. From this operation, determine the greatest common divisor of $16$ and $12$. Provide your answer as a single whole number. | 4 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given $(a+i)^{2}=bi$, where $a$ and $b$ are real numbers, find the value of $|z|$ where $z=a+bi$. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( AB = 25 \text{ cm} \), \( AC = 20 \text{ cm} \) and \(\angle A = 90^\circ\). Calculate the area of triangle \(ABC\). Express your answer in square centimeters (cm^2) as a whole number. | 250 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {1,2,3,4}, set M = {1,2,3}, and set N = {2,3}, find the complement of M ∩ N in U. Express your answer as a set, with elements separated by commas and enclosed in curly brackets. | {1,4} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
When A and B play chess, the probability of A winning is 30%, and the probability of A not losing is 80%. Find the probability of A and B drawing. Express your answer as a percentage. | 50% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular garden has dimensions 15 m by 2 m. Each patio stone measures 0.5 m by 0.5 m. How many stones are needed to cover the entire garden? Express your answer as a whole number. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function $f(x) = (\sin x + \cos x)^2 + 1$. Express your answer in terms of π. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given $x^{a}=3$, $x^{b}=5$, find the value of $x^{a-b}$. Express your answer as a fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Algebra -> Exponents and Indices -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $y = a^{x-3} + 1$ where $a > 0$ and $a \neq 1$, find a point $(x, y)$ that the graph of the function certainly passes through. Express your answer as an ordered pair with exact numerical values. | (3, 2) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the formula $1+a+a^{2}+\ldots+a^{n}= \dfrac {1-a^{n+1}}{1-a}$ ($a\neq 1,n\in\mathbb{N}^{*}$), calculate the left side of the equation when $n=1$. Express your answer as an algebraic expression. | 1+a | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the reciprocal of -6. Express your answer as a fraction. | -\frac{1}{6} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(5π/2 + α) = 1/5, find the value of cos(α). Express your answer as a single fraction or decimal value. | \frac{1}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
What is the correct relationship between the number 2 and the set {0, 1, 2}? Express your answer using the appropriate symbol (e.g., ∈, ⊆, =). | 2 ∈ {0, 1, 2} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A right triangular pyramid has three mutually perpendicular side edges of equal length. The surface area of its circumscribed sphere is $4π$. Find the length of the side edge. Express your answer as a simplified fraction or exact decimal if necessary, including any necessary mathematical notation (e.g., square roots). | \frac{2\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin\left(\frac{\pi}{6} - \alpha\right) = \frac{\sqrt{2}}{3}$, calculate the value of $\sin\left(2\alpha + \frac{\pi}{6}\right)$. Express your answer as a single fraction. | \frac{5}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let the function y = f(x) satisfy f(3 + x) = f(3 - x) for all real numbers x. Given that the equation f(x) = 0 has exactly 6 distinct real roots, find the sum of these 6 roots. Express your answer as a single numerical value. | 18 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
How many centimeters are in 3.5 meters? Express your answer as a whole number. | 350 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A solid cube of side length 4 cm is cut into two pieces by a plane that passes through the midpoints of six edges. Calculate the surface area of each half-cube to the nearest square centimeter. Express your answer in square centimeters. | 69 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\dot{z}$ is the conjugate of $z$, and it satisfies $\dot{z}(1-i)=3+i$, find the value of $z$. Express your answer in the form $a+bi$, where $a$ and $b$ are real numbers. | 1-2i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the focus of the parabola $x^{2}=20y$. Express your answer as an ordered pair (x, y). | (0,5) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given two arithmetic sequences {a_n} and {b_n}, the sums of the first n terms are S_n and T_n, respectively. It is known that for all positive integers n, S_n / T_n = (5n + 3) / (2n + 7). Calculate a_9 / b_9 using this ratio. Express your answer as a simplified fraction. | \frac{88}{41} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a bag, there are 2 red, 2 yellow, and 2 blue balls. If 3 balls are drawn from the bag without replacement, calculate the probability that exactly two of the balls are of the same color. Express your answer as a simplified fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
There are 3 students and 2 teachers standing in a row for a group photo. Find the number of ways they can stand such that the 2 teachers are not adjacent to each other. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The letters of the word 'GAUSS' and the digits in the number '1998' are each cycled separately. If the pattern continues in this way, how many cycles will it take for both sequences to return to their original configurations? Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Define the operation $\oplus$ on $\mathbb{R}$: $a \oplus b = ab + 2a + b$. Solve the inequality $x \oplus (x - 2) < 0$ and express the solution set as an interval or a combination of intervals. | (-2,1) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets:
\[ A = \{-1, 0, 2\} \]
\[ B = \{-x \mid x \in A \text{ and } 2-x \notin A\} \]
Determine the elements of set \( B \). Express your answer as a set, including all elements of the set. | \{1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A product is originally priced at $a$ dollars. Its price is reduced twice, each time by 10%, and then increased by 20%. Calculate the final price of the product after these adjustments, expressing your answer in terms of 'a'. | 0.972a | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of $a$ such that the result of $(x^{2}+ax+1)(x-2)$ does not contain the term $x^{2}$? Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $P=\{y|y=x^2+1\}$, $E=\{x|y=x^2+1\}$, $F=\{x|x\geq1\}$, and $G=\{(x,y)|y=x^2+1\}$, identify the equivalent sets among these. Express your answer as a set equation (e.g., P=F) using standard set notation. | P=F | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
What is the time 45 minutes after 10:20? Express your answer in 12-hour clock format (HH:MM). | 11:05 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If $(2a+i)(1-2i)$ is a pure imaginary number, find the real number $a$ that satisfies this condition. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. Calculate the average value of $S$ if all possible orders of these 20 people are considered. Express your answer as a single number, rounded to the nearest whole number i... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ whose one asymptote has the equation $x-2y=0$, calculate the eccentricity of the hyperbola. Express your answer as a single value, including the square root symbol if necessary. | \frac{\sqrt{5}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y=2(x-3)^2-5, determine its axis of symmetry. Express your answer in the form x = h. | x = 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, find the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\). Express your answer as a single integer. | 42 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In 2005, the per capita net income of rural residents in our city increased by 14.2% compared to the previous year. If the per capita net income of rural residents in 2004 was $a$ yuan, then express the per capita net income of rural residents in 2005 in terms of $a$. Include the unit 'yuan' in your answer. | 1.142a\text{ yuan} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given an increasing sequence of positive integers $\left\{a_{n}\right\}$ with the property $a_{n+2}=a_{n+1}+a_{n}$, and $a_{7}=120$, calculate the value of $a_{8}$. Express your answer as a single integer. | 194 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial equation x^2+x^{10}=a_0+a_1(x+1)+a_2(x+1)^2+...+a_9(x+1)^9+a_{10}(x+1)^{10}, calculate the value of the coefficient a_9. | -10 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A collector offers to buy state quarters for 2000% of their face value. If Bryden has four state quarters, how much will he get for them in total? Express your answer in dollars. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations:
1. x + y + z = 25
2. x + y = 19
3. y + z = 18
Find the value of y. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Systems of Equations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. Find the smallest possible difference between the two 2-digit numbers. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
When A and B play chess, the probability of A winning is 40%, and the probability of A not losing is 90%. Calculate the probability of A and B drawing. Express your answer as a percentage. | 50\% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given $i$ is the imaginary unit, $\overline{z}$ is the conjugate of $z$, and $(2-i) \overline{z}=3-4i$, find the imaginary part of $z$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the interior angles of a polygon is three times the sum of its exterior angles, find the number of sides of the polygon. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $991+993+995+997+999=5000-N$, calculate the value of $N$. Express your answer as a single integer. | 25 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, calculate the coordinates corresponding to the complex number $i(1-i)$. Express your answer as coordinates $(x, y)$. | (1,1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos \left( \frac{\pi }{2}-\alpha \right)=\frac{\sqrt{2}}{3}$, find the value of $\cos \left( \pi -2\alpha \right)$. Express your answer as a fraction in simplest form. | -\frac{5}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that a line $l$ with a slope of 2 passes through the focus $F$ of the parabola $y^2 = ax$ and intersects the $y$-axis at point $A$, if the area of triangle $\triangle OAF$ (where $O$ is the origin) is 4, then find the equation of the parabola in the form $y^2 = ax$. Include all possible solutions. | y^2 = 8x \text{ or } y^2 = -8x | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set I = {0,1,2,3,4}, set M = {1,2,3}, and set N = {0,3,4}, find the result of the operation (∁_I M) ∩ N. Express your answer as a set using standard set notation. | {0,4} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function $y=ax+b$ passes through the first, second, and third quadrants, and intersects the x-axis at the point $(-2, 0)$. Find the solution set for $ax > b$, expressing your answer as an inequality in terms of $x$. | x > 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then find the number of articles produced by $y$ men working $y$ hours a day for $y$ days. Express your answer as an algebraic expression in terms of $x$ and $y$. | \frac{y^3}{x^2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given positive numbers a and b such that a^b = b^a and b = 9a, find the exact value of a. Express your answer as a mathematical expression, including any necessary radicals or exponents. | \sqrt[4]{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\vec{AB} = (3,1)$, and the vector $\vec{a} = (2,\lambda)$, if $\vec{a}$ is parallel to $\vec{AB}$, find the value of the real number $\lambda$. Express your answer as a fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A semicircle with radius 1 is centered at 0 and has AB as its diameter. A point C moves along the entire circumference of the circle. Extend AC to point D so that CD/CB = k (where k > 0 and k is a constant). Calculate the length of the curve traced by point D. Express your answer in terms of pi and the square root of 1... | \pi \sqrt{1 + k^2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( \angle A > \angle B > \angle C \) and \( \angle A \neq 90^{\circ} \). Determine the number of non-parallel lines that can divide \( \triangle ABC \) into two parts such that one of the parts is similar to \( \triangle ABC \). Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence \\(\\{a_n\\}\\), if \\(a_2 + a_7 + a_{15} = 12\\), then find the value of \\(a_8\\). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, find the value of the complex number $\frac{1+3i}{1-i}$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. | -1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations: $$\begin{cases} x+y=5 \\ 2x-y=1 \end{cases}$$ and represent the solution set in the form of enumeration, using curly brackets to denote the set and expressing the answer as an ordered pair. | \{(2, 3)\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-2,1)$ and $\overrightarrow{b}=(m,3)$, and the condition that $\overrightarrow{a}∥\overrightarrow{b}$, find the vector $\overrightarrow{a}-\overrightarrow{b}$. Express your answer as a vector in the form (x, y). | (4, -2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that n is a positive integer, find the minimum value of the sum |n-1| + |n-2| +... + |n-100|. | 2500 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of sets P that satisfy the condition {a} ⊂ P ⊆ {a, b, c}. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Five volunteers are randomly assigned to conduct promotional activities in three communities, $A$, $B$, and $C$. At least 2 volunteers must be assigned to community $A$, and at least 1 volunteer must be assigned to each of communities $B$ and $C$. Calculate the total number of different arrangements. | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions $a=\frac{\ln 2}{2}$, $b=\frac{\ln 3}{3}$, and $c=\frac{\ln 5}{5}$, determine the correct ordering of the variables a, b, and c. Express your answer as a sequence of variables using the less than (<) symbol (e.g., a < b < c). | c < a < b | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the angle of inclination of the tangent line to the curve y = (1/3)x^3 - 2 at the point (-1, -7/3). Express your answer in degrees. | 45° | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 0$ and $a \neq 1$, the graph of the function $y=a^{x}$ and $y=\log _{a}x$ is symmetric about a specific line. What is the equation of this line? | y=x | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.