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 |
|---|---|---|---|---|---|
Given the complex equation (1+2i)z = 4+3i, find the value of z. Express your answer as a complex number in the form a+bi. | 2-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Three of the vertices of a parallelogram are \((0,1)\), \((1,2)\), and \((2,1)\). Calculate the area of the parallelogram. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
The probability of a shooter hitting the 10, 9, and 8 rings in a single shot are 0.24, 0.28, and 0.19, respectively. Find the probability that the shooter scores less than 9 in a single shot. Express your answer as a decimal value between 0 and 1. | 0.48 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system, if $P(3,-2,1)$, find the coordinates of the point symmetric to $P$ with respect to the coordinate plane $xOz$. Express your answer as a set of coordinates (x, y, z). | (3,2,1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\frac{a+2i}{i}=b+i$ $(a,b \in \mathbb{R})$, where $i$ is the imaginary unit, calculate the value of $ab$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first 10 terms and the sum of the first 100 terms of an arithmetic sequence are 100 and 10 respectively, calculate the sum of the first 110 terms. Express your answer as a single integer. | -110 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The sum of three consecutive integers is 153. What is the largest of these three integers? Express your answer as a whole number. | 52 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The numbers 41, 35, 19, 9, 26, 45, 13, 28 are arranged in pairs so that the sum of the numbers in each pair is the same. Find the number paired with 13. Express your answer as a single integer. | 41 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an odd function with a period of 5 on $\mathbb{R}$, and it satisfies $f(1)=1$, $f(2)=2$, find the value of $f(3) - f(4)$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, the magnitudes of the sides are a, b, and c, as shown, with c ≤ b ≤ a. Through interior point P and the vertices A, B, C, lines are drawn meeting the opposite sides in A', B', C', respectively. Let s = AA' + BB' + CC'. Find the upper bound for s for all positions of point P. Express your answer as a si... | a+b+c | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, if A = 2π/3 and AB = 3, calculate the projection of AB onto AC. Express your answer as a single number. | -\frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{AB}=(2,3)$ and $\overrightarrow{AC}=(-1,-1)$, calculate the result of the operation $2\overrightarrow{AB}-\overrightarrow{AC}$. Express your answer as a vector in the form (x, y). | (5,7) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the product of the solutions to the equation $x^6 = -64$ that have a real part greater than 0. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A line forms an angle of 30° with plane a. Find the smallest angle formed between the line and all the lines within plane a. Express your answer in degrees as a whole number. | 30 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\{a_n\}$ is an arithmetic sequence with a common difference of -2, and the sum of the first 7 terms $S_7 = 14$, calculate the value of $a_2 + a_4 + a_6$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
If $\log_6 x = 2.5$, what is the value of $x$? Express your answer as a single value, possibly involving radicals. | 36\sqrt{6} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if S_4 = 25 and S_8 = 100, calculate the value of S_12. | 225 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In a mathematical test for grade 12 students in a certain city, it was found that the scores (X) approximately followed a normal distribution N(84, σ²) after a sample analysis. Also, P(78 < X ≤ 84) = 0.3. In this test, there were 400 participants from a particular school. Estimate the number of students from this schoo... | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 63,000 in scientific notation. Use the format $a \times 10^{b}$, where $a$ is the coefficient and $b$ is the exponent. | $6.3 \times 10^{4}$ | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Five people participate in a relay game, each randomly choosing between two options: left path and right path. If more than 2 people choose the same path, each of them gets 1 point; if fewer than 3 people choose the same path, each of them gets 0 points. Let $\xi$ denote the score of a player named Xiaoqiang. Calculate... | \frac{11}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The equation of circle C is $x^2+y^2+2x+4y-3=0$. Find the coordinates of the center of circle C. Express your answer as a coordinate pair (x, y). | (-1, -2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
If $n$ is an odd number, calculate the remainder of $7^n + \binom{n}{1}7^{n-1} + \binom{n}{2}7^{n-2} + \ldots + \binom{n}{n-1}7 + 7$ when divided by 9. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function \( f(x) = \frac{x^{4} + kx^{2} + 1}{x^{4} + x^{2} + 1} \) where \( k, x \in \mathbb{R} \), find the product of the maximum value \( f(x)_{\text{max}} \) and the minimum value \( f(x)_{\text{min}} \) of \( f(x) \). Express your answer as a mathematical expression involving \( k \). | \frac{k+2}{3} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that P is a point on side BC of an equilateral triangle ABC with side length 4, calculate the value of $\overrightarrow {AP}\cdot ( \overrightarrow {AB}+ \overrightarrow {AC})$. Express your answer as a single numerical value. | 24 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If the function $f(x) = a - \frac{1}{2^x + 1}$ is an odd function, find the value of the real number $a$. Express your answer as a single real number. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point A (1, 2) and point A' (a, b) are symmetric about the origin of the coordinates, find the values of the real numbers a and b. Express your answer as an ordered pair (a, b). | (-1, -2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first n terms of a positive arithmetic sequence {a_n} is S_n, if S_2012 = 2012, find the minimum value of 1/a_1 + 1/a_2012. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = a sin(x) + b cbrt(x) + c ln(x + sqrt(x^2 + 1)) + 4, where a, b, and c are real numbers, and f(lg log_3 10) = 5, find the value of f(lg lg 3). Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation in terms of $x$: $4x^{2}-px+q=0$, which can be transformed by completing the square to $(x-\frac{1}{4})^2=\frac{33}{16}$, find the value of $\frac{q}{p}$. Express your answer as a single number. | -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Suppose a random variable X follows the normal distribution N(4, σ^2). If P(X > m) = 0.3, then find the value of P(X > 8 - m). Express your answer as a decimal value between 0 and 1. | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Mathilde and Salah each had 100 coins at the start of this month. For Mathilde, this was 25% more coins than she had at the start of last month. For Salah, this was 20% fewer coins than he had at the start of last month. Find the total number of coins they had at the start of last month. Express your answer as a numeri... | 205 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\{-1, 0, \frac{1}{3}, \frac{1}{2}, 1, 2, 3, 4\}$, calculate the number of non-empty subsets that are companion relationship sets, where a companion relationship set is defined as a set A such that for any x in A, $\frac{1}{x}$ is also in A. | 15 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The animated film "Thirty Thousand Miles in Chang'an" is a high-grossing animated film based on the prosperous Tang Dynasty history of China. As of October 23, 2023, its box office revenue is 1,824,000,000 yuan. Express this number in scientific notation in the format $a \times 10^{b}$. | 1.824 \times 10^{9} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given an odd function f(x) defined over R such that f(x+2) = -f(x), calculate the value of f(6). | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A distance of 30 kilometers is represented by a line segment of 20 centimeters on a map. Calculate the scale of this map as a ratio in the form 1:x, where x is a whole number. | 1:150000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ be a real root of the equation $x^2+x-2009=0$. Find the value of $a^2+a-1$. Express your answer as a single integer. | 2008 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence $\{a_n\}$ is a geometric sequence with the first term being $1$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, and it is known that $9S_3=S_6$, calculate the sum of the first $5$ terms of the sequence $\left\{ \frac{1}{a_n} \right\}$. Express your answer as a numerical v... | \frac{31}{16} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression \((x \sqrt{x} - \frac{1}{x})^6\), the value of the 5th term in the expansion is 5. Calculate the limit \(\lim_{n \to \infty}\left(x^{-1} + x^{-3} + \cdots + x^{1-2n}\right)\). Express your answer as a simplified fraction. | \frac{3}{8} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Applied Mathematics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan 2\alpha = \frac{3}{4}$ and $\alpha \in (0, \frac{\pi}{4})$, calculate the value of $\frac{\sin \alpha + \cos \alpha}{\sin \alpha - \cos \alpha}$. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^2$ in the expansion of the binomial expression $(- \sqrt {x}+ \frac {1}{x})^{10}$. Express your answer as a single integer. | 45 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that 3 is a root of the function $$f(x)= \begin{cases} \log_{3}(x+t), & x\geq 3 \\ 3^{x}, & x<3 \end{cases},$$ calculate the value of f[f(6)]. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_2 + a_10 = 16. Find the value of a_4 + a_6 + a_8. Express your answer as a single numerical value. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given a right square pyramid with the side lengths of the top and bottom bases being 3 and 6, respectively, and the lateral area equal to the sum of the areas of the two bases, find the height of the pyramid. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence of equations:
${1}^{3}+{2}^{3}={\left( \frac{2\times 3}{2}\right)}^{2}$,
${1}^{3}+{2}^{3}+{3}^{3}={\left( \frac{3\times 4}{2}\right)}^{2}$,
${1}^{3}+{2}^{3}+{3}^{3}+{4}^{3}={\left( \frac{4\times 5}{2}\right)}^{2}$,
$⋮$
Find the value of n such that ${1}^{3}+{2}^{3}+{3}^{3}+...+{n}^{3}=3025$. Exp... | 10 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) and g(x) be functions defined on (0,+\infty), and the derivatives of f(x) and g(x) exist. If f'(x)g(x) + f(x)g'(x) < 1, and f(1) = 2, g(1) = 1, find the solution set of f(x)g(x) < x + 1. Express your answer as an interval. | (1,+\infty) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A school sends 5 students to Peking University, Tsinghua University, and Zhejiang University for study. Calculate the total number of different methods to send at least one student to each university. | 150 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3(4x + 5\pi) = P$, express $6(8x + 10\pi)$ in terms of $P$. Provide your answer in the form $\boxed{\text{your expression here}}$. | 4P | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} satisfies 3a_{n+1} + a_n = 4 (n ≥ 1), with a_1 = 9, and the sum of its first n terms is S_n, find the smallest integer n that satisfies the inequality |S_n - n - 6| < 1/125. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x \in \mathbb{R}$, vector $\vec{a} = (3, 2)$, vector $\vec{b} = (x, 4)$, and $\vec{a} \parallel \vec{b}$, find the value of $x$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the slope of the tangent line to the function f(x) = x^3 + x^2 + 1 at x = 1. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $ M = \{x | \ln(x^2) = \ln(x), x \in \mathbb{R}\} $ and the set $ N = \{n \in \mathbb{Z} | -1 \leq n \leq 3 \} $, find the intersection $ M \cap N $. Express your answer as a set of integers using standard set notation. | \{1\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = cx / (2x + 3), where x cannot equal -3/2, and the condition f[f(x)] = x, calculate the constant c. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|y=\sqrt{2-{2}^{x}}\}$ and $B=\{x|x^{2}-3x \leqslant 0\}$, find the intersection $A \cap B$. Express your answer as an interval. | [0, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the polar equation of a circle $\rho = \sqrt{2} (\cos \theta + \sin \theta)$, find the coordinates of the center of this circle in polar coordinates. Express your answer as (ρ, θ). | \left(1, \frac{\pi}{4}\right) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A woman is skilled at weaving and increases her weaving by an equal amount every day. In seven days, she weaves a total of 28 feet. The sum of the lengths she weaves on the second, fifth, and eighth days is 15 feet. What is the length she weaves on the 30th day? Express your answer as a whole number in feet. | 30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5 | 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, using curly braces and separating elements with commas. | {1,4} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {m}$ and $\overrightarrow {n}$ satisfy $\overrightarrow {m} = (2,0)$ and $\overrightarrow {n} = \left( \frac {3}{2}, \frac { \sqrt {3}}{2}
ight)$. For triangle $ABC$, $\overrightarrow {AB} = 2\overrightarrow {m} + 2\overrightarrow {n}$ and $\overrightarrow {AC} = 2\overrightarrow {m} - 6... | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and the area is denoted as $S$. Given $S+a^{2}=(b+c)^{2}$, find the value of $\cos A$. Express your answer as a fraction or decimal. | -\dfrac{15}{17} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\frac {a+2i}{i}=b+i$$ (where $a$, $b$ are real numbers, and $i$ is the imaginary unit), calculate the value of $a+b$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Integers x and y with x>y>0 satisfy the equation x+y+xy=80. Find the value of x. Express your answer as a single integer. | 26 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The gravitational potential energy of two stars separated by a distance d is U. Calculate the gravitational potential energy of the system when the stars are separated by a distance 2d. Express your answer in terms of U. | \frac{U}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
What is the remainder when $29 \cdot 169^{1990}$ is divided by 11? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Points A and B are located on a grid paper. If point B is used as the origin to establish a Cartesian coordinate system, then the coordinates of point A are (2, 5). Find the coordinates of point B if point A is used as the origin to establish a Cartesian coordinate system. Express your answer as an ordered pair (x, y),... | (-2, -5) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(0, σ^2), and P(ξ < -1) = 0.2, calculate the probability P(-1 < ξ < 1). Express your answer as a decimal value between 0 and 1. | 0.6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The surface area of a rectangular solid is 11, and the sum of the lengths of all its edges is 24. Find the length of one of its diagonals. Express your answer as a numerical value. | 5 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=1+i$ (where $i$ is the imaginary unit) and $\overline{z}$ is the conjugate of $z$, calculate the value of $\frac{\overline{z}}{z}$. Express your answer as a complex number in the form $a + bi$. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given a non-zero complex number $z$ satisfies $z\\cdot (2+2i)=|z|^{2}$, find the conjugate complex number of $z$. Express your answer as a complex number in the form $a+bi$. | 2+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given x > 0 and y > 0, and that x, a, b, y form an arithmetic sequence, while x, c, d, y form a geometric sequence, find the minimum value of ((a + b)^2) / (cd). | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The area of a circle increases at a rate of $10\pi \text{ cm}^2/\text{s}$. What is the rate of increase of the circle's radius $r$ when $r=20\text{ cm}$? Express your answer in cm/s. | \frac{1}{4} \text{ cm/s} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of lines that pass through the point (1, 2) and have equal intercepts on the coordinate axes. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the propositions p: "For any x in R, it always holds that 3^x > 0"; q: "x > 0 is a sufficient but not necessary condition for x > 1", determine the true proposition. Express your answer as a logical proposition (e.g. p ∧ ¬q). | p ∧ ¬q | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 213,000,000 in scientific notation. Your answer should be in the form a × 10^b, where a is a number between 1 and 10, and b is an integer. | 2.13 × 10^8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex numbers \( z_{1} \) and \( z_{2} \) correspond to points \( A \) and \( B \) respectively on the complex plane, and let \( \left|z_{1}\right| = 4 \). Given that \( 4z_{1}^{2} - 2z_{1}z_{2} + z_{2}^{2} = 0 \). Let \( O \) be the origin. Find the area of \( \triangle OAB \). Express your answer as a numer... | 8\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Aaron says his age is 50 years, 50 months, 50 weeks, and 50 days old. Calculate Aaron's age on his next birthday. Express your answer as a whole number. | 56 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Let $x$ be a real number, and consider the inequality $|x-5|+|x-3|<m$. What is the range of $m$ for which this inequality has solutions? Express your answer as an inequality involving $m$. | m>2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. Find the imaginary part of the complex number $3-4i$. Express your answer as a single number. | -4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The surface area of a cone is $3\pi$, and its lateral surface unfolds into a semicircle. Find the diameter of the base of the cone. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that O is the origin of the coordinate system, $\overrightarrow{OA} = (1, 3)$, $\overrightarrow{OB} = (3, -1)$, and $\overrightarrow{AP} = 2\overrightarrow{PB}$, find the coordinates of point P, denoted as (x, y). Express your answer as an ordered pair. | (\frac{7}{3}, \frac{1}{3}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \( 75^{x}=0.75^{y}=10^{-2} \), calculate the value of \( \frac{1}{x}-\frac{1}{y} \). Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the following system of inequalities and describe the solution set:
\[
\begin{cases}
2x-1 \geqslant x+1 \\
x+8 \leqslant 4x-1
\end{cases}
\]
Express your answer in terms of intervals or inequalities that describe the solution set. | x \geqslant 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
A chess tournament is planned for teams, each of which has exactly three players. Each player in a team will play exactly once against each player from all the other teams. Given that no more than 250 games can be played in total, what is the maximum number of teams that can enter the tournament? Express your answer as... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Two tourists, A and B, plan to visit Xiamen and choose to visit 3 out of 4 tourist attractions: Gulangyu, Zengcuo'an, Botanical Garden, and South Putuo. Calculate the probability that they choose exactly two identical attractions. Express your answer as a simplified fraction. | \dfrac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the left and right foci of the ellipse $\frac{x^{2}}{16}+ \frac{y^{2}}{12}=1$ are $F_{1}$ and $F_{2}$, and point $P$ is on the ellipse such that $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}=9$, find the value of $|\overrightarrow{PF_{1}}| \cdot |\overrightarrow{PF_{2}}|$. Express your answer as a s... | 15 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $$\sqrt[3]{(2-\pi )^{3}} + \sqrt {(3-\pi)^{2}}$$ and provide your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|x<1\}$ and $N=\{x|3^x < 1\}$, calculate the intersection $M \cap N$. Express your answer as a set definition. | \{x|x<0\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given **_a_** = (0,1,－1), **_b_** = (1,1,0), and (**_a_** + _λ_**b**) is perpendicular to **_a_**, find the value of the real number _λ_. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let $a\in R$, and consider the complex number $\frac{1-{i}^{2023}}{ai}$. If the imaginary part of this complex number is $3$, calculate the value of $a$. Express your answer as a single real number. | -\frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
A positive integer N divided by 60 leaves a remainder of 49. What is the remainder when N is divided by 15? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let $\overrightarrow{{e}_{1}}, \overrightarrow{{e}_{2}}$ be two non-collinear vectors in a plane, $\overrightarrow{AB}=(a-1) \overrightarrow{{e}_{1}}+ \overrightarrow{{e}_{2}}, \overrightarrow{AC}=b \overrightarrow{{e}_{1}}-2 \overrightarrow{{e}_{2}}$ where $a>0,b>0$. If points A, B, and C are collinear, find the minim... | 4 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Calculus -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
6 books of different types are to be divided into 3 groups, with one group containing 4 books and the other two groups containing 1 book each. How many different ways can this be done? Express your answer as a single numerical value. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse equation $x^{2}+4y^{2}=16$ and $|PF_{1}|=7$, find the exact value of $|PF_{2}|$. Use the property that the sum of the distances from any point on an ellipse to its two foci is constant and equal to $2a$. | 1 | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In a granary, there are 3318 stones of grain. A sample of grain is taken, which contains 168 grains, and 22 of them are imperfect. Using this information, calculate the approximate amount of imperfect grains in the granary. Express your answer in stones, rounded to the nearest whole number. | 434 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Five different awards are to be given to three students. Each student will receive at least one award. Calculate the total number of different ways the awards can be distributed. Express your answer as a single integer. | 150 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = 2sin(x) at point P(π, 0). Express your answer in slope-intercept form (y = mx + b). | y = -2x + 2π | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) is a monotonically increasing function on (0, +∞). For n ∈ N*, f(n) ∈ N*, and f(f(n)) = 3n. Find the value of f(1). Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Assign 7 senior high school students to two dormitories, A and B, with each dormitory accommodating at least 2 students. Given that students A and B cannot stay in the same dormitory, calculate the total number of different arrangements possible. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let $n$ be a natural number between 100 and 200. Count the number of $n$ such that $7n + 2$ is a multiple of 5. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x|0<x\leq3, x\in \mathbb{N}\}$ and $B=\{x|y= \sqrt{x^{2}-9}\}$, calculate the set $A\cap(\complement_{\mathbb{R}}B)$. Express your answer as a set of integers, including all elements of the resulting set. | \{1,2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{3\pi}{4}$, and $|\overrightarrow{a}|=\sqrt{2}$, $|\overrightarrow{b}|=3$, calculate the magnitude of the vector $\overrightarrow{a}+2\overrightarrow{b}$. Express your answer as a single value. | \sqrt{26} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence of 10 terms where each term is either 1 or -1, how many different possible values can the sum of the sequence take? Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.53125 | 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.