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 |
|---|---|---|---|---|---|
Evaluate the integral $\int_{2}^{3}{\frac{x-2}{\left( x-1 \right)\left( x-4 \right)}dx}$. Express your answer as a single expression, including any necessary logarithmic or algebraic terms. | -\frac{1}{3}\ln 2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x)=\begin{cases}
\log_{2}(x+1), & x > 3 \\
2^{x-3}+1, & x\leqslant 3
\end{cases}$, and the condition $f(a)=3$, calculate the value of $f(a-5)$. Express your answer as a numerical value. | \dfrac{3}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, determine the equation of the line containing the chord with midpoint $M(-1, 1)$. Express your answer in the standard form of a line equation ($Ax + By + C = 0$). | 3x - 4y + 7 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the remainder when the polynomial x^{51} + 51 is divided by x + 1. Express your answer as a numerical value. | 50 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that points P and Q are on a unit circle with its center at the origin O of the coordinate system, located in the first and fourth quadrant respectively. The vertical coordinate of point P is 4/5, and the horizontal coordinate of point Q is 5/13. Find the value of cos(∠POQ). Express your answer as a fraction. | -\frac{33}{65} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi}{3}$, and $\overrightarrow{a}=(2, 0)$, $\|\overrightarrow{b}\|=1$, calculate the magnitude of $\|\overrightarrow{a}+2\overrightarrow{b}\|$. Express your answer as a simplified radical. | 2\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
A large classroom has 100 light bulbs and 100 switches, each marked with numbers from 1 to 100. Xiao Ming pulls the switches with even numbers, and Xiao Cong pulls the switches with numbers divisible by 3. Originally, all the lights in the classroom were off. Calculate the number of light bulbs that are on now. | 51 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let $n$ be a two-digit integer represented as $n = 10a + b$, where $a$ and $b$ are the tens and units digits respectively. If $n$ is $k$ times the sum of its digits, find the constant $x$ such that the number formed by interchanging the digits of $n$ is $x$ times the sum of its digits. Express your answer as an algebra... | 11-k | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the degree of each exterior angle of a regular octagon. Express your answer in degrees as a whole number. | 45 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Count the whole numbers from 1 through 46 that are divisible by either 3 or 5 or both. Express your answer as a whole number. | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{3}$, $\overrightarrow{a}•\overrightarrow{b}=1$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ in radians. Express your answer as a fraction of pi (e... | \frac{\pi}{3} | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $2a^{2x}b^{3y}$ and $-3a^2b^{2-x}$ are like terms, find the values of x and y. Express your answer in the format (x, y). | (1, \frac{1}{3}) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of the binomial \\((x- \frac {1}{x})^{6}\\). Express your answer as a single integer. | -20 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Expansion"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x\in Z|x^{2}-2x-3 \lt 0\}$ and $B=\{-2,-1,0,1\}$, find the union $A\cup B$. Express your answer as a set of integers. | \{-2, -1, 0, 1, 2\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $l: y = kx - \sqrt{3}$ intersects with the line $x + y - 3 = 0$ in the first quadrant, determine the range of the slope angle of line $l$. Express your answer as an interval of angles in degrees. | (30^\circ, 90^\circ) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_8=30$ and $S_4=7$, then find the value of $a_4$. Express your answer as a fraction in simplest form. | \dfrac{13}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The probability of selecting a first-grade product is 0.65, and the probability of selecting a second-grade product is 0.3. Calculate the probability of selecting a defective product. Express your answer as a decimal value between 0 and 1. | 0.05 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. The total amount paid to the guest speaker is $1A2. What is the missing digit A of this 3-digit number? Express your answer as a single digit. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function y = 4^x + 2^(x+1) + 5, where x ∈ [1,2]. Express your answer as a single numerical value. | 29 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the combinatorial expression $$C_n^0 - 2C_n^1 + 4C_n^2 - 8C_n^3 + \ldots + (-2)^n C_n^n.$$ Express your answer as a mathematical expression. | (-1)^n | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2x f'(1) + x^2, find the value of f'(0). Express your answer as a single number. | -4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Twelve girls met in a cafe. On average, they ate 1.5 cupcakes each, although no cupcakes were actually divided. None of them ate more than two cupcakes and two of them ate no cupcakes at all. Calculate the number of girls who ate two cupcakes. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(α) - cos(α) = 1/5 and 0 < α < 180°, find the value of tan(α). Express your answer as a fraction. | \frac{4}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Among 8 consecutive natural numbers, there are k numbers such that any two of them are relatively prime. Find the maximum value of k. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vertical coordinate of a moving point P is always 2, describe the type of geometric shape formed by the trajectory of the moving point P in a three-dimensional Cartesian coordinate system. | Plane | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $(1+3x)^n$ where $n \in \mathbb{N}$ and $n \geq 6$, the coefficients of $x^5$ and $x^6$ are equal. Find the value of $n$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given a function f(x) (x ∈ ℝ) that satisfies f(-x) = 8 - f(4 + x), and a function g(x) = (4x + 3)/(x - 2), if the graphs of functions f(x) and g(x) have 168 common points, denoted as P_i(x_i, y_i) (i = 1, 2, …, 168), then calculate the sum of (x + y) for all common points. | 1008 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Functions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the digit A such that the numeral 1AA is a perfect square in base-5 and a perfect cube in base-6. Express your answer as a single digit (0-4). | 4 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the context of statistical analysis, what term best describes the entire group of scores of the 5,000 students who participated in a computer proficiency test? Provide your answer as a specific statistical term. | Population | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In a certain community with 1000 households, the monthly electricity consumption approximately follows a normal distribution $N(300, 10^2)$. Using the reference data provided, calculate the estimated number of households with electricity consumption above 320 units. Round your answer to the nearest whole number. | 23 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, satisfying $a^2 + b^2 - 3c^2 = 0$ where $c$ is the semi-focal length, calculate the value of $\frac{a+c}{a-c}$. Express your answer as a simplified mathematical expression. | 3 + 2\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
There are 8,100 people in the northern village, 7,488 people in the western village, and 6,912 people in the southern village. A total of 300 people are to be conscripted from these three villages. Calculate the number of people the northern village should provide based on proportional distribution. | 108 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the largest prime number less than 30 that can be expressed as the sum of two prime numbers. Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression $i + i^2 + i^3 + \ldots + i^{2015}$. Express your answer as a single value. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The school plans to select 3 students from 5 applicants to serve as volunteers for the high jump, long jump, and shot put at the sports meeting. It is known that student A cannot serve as a volunteer for the high jump event. Find the total number of different arrangements. Express your answer as a numerical value. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Any permutation of the digits in the number 12345 results in a five-digit number. How many of these permutations are prime numbers? Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = i(-2 + i), where i is the imaginary unit, determine the quadrant in which the point representing z is located. Express your answer as the name of the quadrant (e.g., first, second, third, fourth). | third | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the value of $C^{2}_{100}+C^{97}_{100}$. Express your answer as a combination in the form $C^{r}_{n}$. | C^{3}_{101} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A large square is divided into smaller squares. In one of the smaller squares, a diagonal is also drawn. What fraction of the large square is shaded? Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = |x+1| - 2|x-1|, find the solution set of the inequality f(x) > 1. Express your answer in interval notation. | \left( \frac{2}{3}, 2 \right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
What is the least positive integer m such that, given m integers between 1 and 2023 inclusive, there must exist two of them a, b such that 1 < a/b <= 2? Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle OAB$, let $\overrightarrow{OA}=(2\cos \alpha, 2\sin \alpha)$ and $\overrightarrow{OB}=(5\cos \beta, 5\sin \beta)$. If $\overrightarrow{OA} \cdot \overrightarrow{OB} = -5$, then calculate the area of $\triangle OAB$, $S_{\triangle OAB}$. Express your answer as a numerical value, including the squa... | \dfrac{5\sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequalities $1 \leq a-b \leq 2$ and $2 \leq a+b \leq 4$, find the range of values for the expression $4a-2b$. Express your answer as an interval, using square brackets for inclusive endpoints. | [5,10] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $A(2,4)$ is symmetric to point $B$ with respect to the origin, find the coordinates of point $B$. Express your answer as a coordinate pair in the format $\boxed{(x, y)}$. | (-2,-4) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic-geometric sequence with first term $a_1=2$ and common ratio $q=3$, calculate the third term $a_3$ using the formula $a_n = a_1 \cdot q^{n-1}$. Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Arrange the 5 natural numbers from 1 to 5 in a row from left to right, in such a way that starting from the third number, each number is either the sum or difference of the previous two numbers. How many valid arrangements are there? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of real numbers m for which the function f(x) = 2x^3 - 6x + m has three zeros. Express your answer as an interval of real numbers. | (-4, 4) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A certain type of bacteria divides once every 10 minutes (from one bacterium to two). After 2 hours, how many bacteria will there be? Express your answer in exponential form (e.g., 2^n). | 2^{12} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Explore the basic pattern of the following sequence: 0, 2, 6, 14, 30,... Write down the sixth number that follows the pattern. Express your answer as a single integer. | 62 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, if S_6 > S_7 > S_5, find the smallest positive integer value of n such that S_n < 0. Express your answer as a positive integer. | 13 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The participation in the local soccer league this year is 10% higher than last year. The number of males increased by 5% and the number of females increased by 20%. What is the fraction of the soccer league that is female this year? Express your answer as a simplified fraction. | \frac{4}{11} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation: x(x(x+1)+2)+3 = x^3 + x^2 + x - 6. Express your answer as a single integer. | -9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies the equation $z(1+i)=3+2i$, find the imaginary part of $z$. Express your answer as a single fraction or decimal. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the closed figure formed by the curve y = sqrt(x), the lines y = x - 2, and the y-axis. Express your answer as a single number. | 16/3 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that positive numbers a and b satisfy a = b^2 and log_b(a) = a/b, find the domain of the function f(x) = sqrt(1/b - log_a(x-1)). Express your answer as an interval. | (1,3] | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Every time these two wheels are spun, two numbers are selected by the pointers. Calculate the probability that the sum of the two selected numbers is even. Express your answer as a fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
For how many integers a with 1 ≤ a ≤ 10 is the expression a^{2014} + a^{2015} divisible by 5? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z = 2 + i$, where $i$ is the imaginary unit, which quadrant does the corresponding point of complex number $z$ lie in on the complex plane? Express your answer as the name of the quadrant (e.g., 'first quadrant'). | first quadrant | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} where a_5 = 11 and the sum of the first 12 terms S_{12} = 186, find the value of a_8. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex fraction $$\frac{3+i}{1+i}$$ and express your answer in the form a+bi, where a and b are real numbers. | 2-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
How many non-negative integers can be expressed as a sum of the form
$$
\sum_{i=0}^{7} 3^{i} a_{i}
$$
where each $a_i$ belongs to $\{-1, 0, 1\}$, and $0 \leqslant i \leqslant 7$? Express your answer as a single integer. | 3281 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_3 + a_5 = 10. Using the properties of the arithmetic sequence, find the value of a_4. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$, $b$, and $c$ form a geometric sequence, $m$ is the arithmetic mean of $a$ and $b$, and $n$ is the arithmetic mean of $b$ and $c$, calculate the value of $\frac{a}{m} + \frac{c}{n}$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the total number of subsets of the set $A = \{-1, 0, 1\}$. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given point A(0, –1), and point B is on the line $x-y+1=0$. If line AB is perpendicular to the line $x+2y-3=0$, then calculate the coordinates of point B. Express your answer as an ordered pair (x, y). | (2, 3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
A group of boys and girls undergoes two changes. First, 15 girls leave, resulting in two boys for each remaining girl. Then, 45 boys leave, resulting in five girls for each remaining boy. Calculate the initial number of girls in the group. Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (x, 1)$ and $\overrightarrow{b} = (3, 6)$, and $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of the real number $x$. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{x^2}{3-k} - \frac{y^2}{k-1} = 1$, find the range of the real number k such that the equation represents a hyperbola. Express your answer as an interval or inequality. | (1, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $sinα=2sin(\\frac{3π}{2}-α)$, calculate the value of $tan(\\frac{π}{4}-α)$. Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $S=\mathbb{R}$ and $A=\{x|x^2-2x-3\leq0\}$, find the complement of set $A$ in $S$, denoted as $C_{S}A$. Express your answer in set builder notation. | \{x|x<-1 \text{ or } x>3\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
There are 7 points on a plane. Except for 3 points that are collinear, no other three points are on the same line. How many different lines can be formed by selecting 2 points among them? Express your answer as a whole number. | 19 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
If $\sin 2t = -\int_{0}^{t}{\cos xdx}$, where $t \in (0, \pi)$, calculate the exact value of $t$ in radians as a fraction of $\pi$. | \frac{2\pi}{3} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = 1 - 3i, and its conjugate complex number \\overline{z} = 1 + 3i, calculate the modulus of the ratio \\left| \\frac{z}{\\overline{z}} \\right|. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In a certain middle school in Jianping County, the ninth-grade classes held a basketball competition. Each pair of classes had to play a match, totaling 10 matches. Let x be the total number of classes participating. Write an equation that represents the relationship between the total number of classes x and the total ... | \frac{x(x-1)}{2} = 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{15^{30}}{45^{15}}$. Express your answer in a simplified form, including any necessary exponents or bases. | 5^{15} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex numbers z1 and z2 corresponding to the points (1,-1) and (-2,1) in the complex plane, calculate the value of z2/z1. Express your answer in the form a + bi. | -\frac{3}{2}-\frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the variance of the sample $a_1, a_2, a_3$ is $a$, find the variance of the sample $3a_1+1, 3a_2+1, 3a_3+1$. Express your answer as a function of $a$. | 9a | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In the subtraction shown, \( K, L, M \), and \( N \) are digits. Calculate the sum of \( K, L, M, \) and \( N \).
\[
\begin{array}{llll}
5 & K & 3 & L \\
\end{array}
\]
\[
\begin{array}{r}
M & 4 & N & 1 \\
\hline
4 & 4 & 5 & 1 \\
\end{array}
\]
Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A certain beverage is packed in boxes, with each box containing 6 cans. Among them, there are 2 cans that are not up to standard. A quality inspector randomly selects 2 cans from the box and finds that both are acceptable products. Calculate the probability of this occurrence. Express your answer as a simplified fracti... | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfies $\left(1+i\right)z=3+2i$, calculate the imaginary part of $\overline{z}$. Express your answer as a single number. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $x^{2}+2x=3$ for its roots. Express your answer in the format $x_{1}=$ and $x_{2}=$. | x_{1}=1, x_{2}=-3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of ordered triplets (x, y, z) that satisfy the system of equations:
\[
\begin{cases}
x + 2y + 4z = 12, \\
xy + 4yz + 2xz = 22, \\
xyz = 6.
\end{cases}
\]
Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{1, 3, \sqrt{m}\}$ and $B=\{1, m\}$, if $A\cup B=A$, then find the value(s) of $m$ that satisfy this condition. Express your answer as a numerical value or a list of values. | 0 \text{ or } 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{\cos^{2} 5^{\circ} - \sin^{2} 5^{\circ}}{\sin 40^{\circ} \cos 40^{\circ}}$. Express your answer as a simplified numerical value. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the following expression:
$$
\sqrt{2-\sqrt{3}} \cdot \sqrt{2-\sqrt{2-\sqrt{3}}} \cdot \sqrt{2-\sqrt{2-\sqrt{2-\sqrt{3}}}} \cdot \sqrt{2+\sqrt{2-\sqrt{2-\sqrt{3}}}}
$$
Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given two circles with radii of 2 and 3, respectively, find the range of the distance d between their centers such that the two circles have common points. Express your answer as an inequality involving d. | 1 \leq d \leq 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Convert the quinary (base-5) number $324_{(5)}$ into a binary number. Express your answer in binary (base-2), including the base subscript (e.g., _{2}). | 1011001_{(2)} | big_math | [
"Mathematics -> Other -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
6 different books are to be divided into 3 groups, one group containing 4 books and the other two groups containing 1 book each. Calculate the total number of ways to do this. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that P is a point moving on the circle $x^2+y^2=1$, calculate the minimum distance from point P to the line $l: x+y-2\sqrt{2}=0$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
There are 160 students in grade 8 at Murray Public School. If exactly 10% of these students are on the school's chess team, how many grade 8 students are on the team? Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the equation \(\sqrt{2} + \sqrt{8} + \sqrt{18} = \sqrt{k}\) and find the value of \(k\). Express your answer as a single integer. | 72 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the expression \( x^2 - 6x + 13 \). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A store sells two different types of goods at the same price of 135 yuan each. One type of goods makes a profit of 25%, and the other type incurs a loss of 25%. Calculate the overall profit or loss from this transaction. Express your answer in yuan as a whole number, including a negative sign if the result is a loss. | -18 | big_math | [
"Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A bank employee is filling an empty cash machine with bundles of $5.00, $10.00, and $20.00 bills. Each bundle has 100 bills in it, and the machine holds 10 bundles of each type. Calculate the total amount of money required to fill the machine. Express your answer in dollars as a whole number. | 35000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Let vector $\overrightarrow{a} =(x,x+1)$ and $\overrightarrow{b} =(1,2)$, where $\overrightarrow{a} \perp \overrightarrow{b}$. Find the value of $x$. | -\dfrac{2}{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \\begin{cases} 2^x, & x \\leqslant 0 \\\\ f(x-3), & x > 0 \\end{cases}$, find the value of $f(5)$. Express your answer as a single value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Using the formula of perfect square trinomial (a+b)^2 = a^2 + 2ab + b^2, calculate the value of 2ab for the given expression (x + 1/2)^2. Express your answer as a single algebraic expression. | x | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation cos 60° = cos 45° cos θ, where 0° ≤ θ ≤ 90°, calculate the value of θ in degrees. | 45° | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a=(1, 2)$ and $b=(x, -6)$, if $a \parallel b$, find the value of $x$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex fraction: $\dfrac{1+3i}{1-i}$. Express your answer in the form a+bi, where a and b are real numbers. | -1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
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