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 that $-2 < a < -1$ and $-1 < b < 0$, find the range of $M = a + b$. Express your answer as an inequality, including the upper and lower bounds of the range. | -3 < M < -1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $(1+bi)(2+i)$ is purely imaginary, calculate the value of $|\frac{2b+3i}{1+bi}|$. Express your answer as a single value. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 distinct red balls and 4 distinct white balls. How many different ways are there to extract two balls of different colors? Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The foreign trade income of a certain place in 2001 was $m$ billion yuan. If the annual growth rate is $x$, then the foreign trade income in 2003 reached $n$ billion yuan. Establish an equation that relates these variables. Express your answer as an equation using $m$, $x$, and $n$. | m(1+x)^2=n | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let $M=\{x|\log _{2}x \gt 1\}$, $N=\{x|\frac{x+3}{x-3}＜0\}$. Calculate the intersection $M\cap N$ and express your answer as an interval in standard interval notation. | (2,3) | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first n terms of a geometric sequence is S_n, if S_2 = 4 and S_4 = 16, find the value of S_8. Express your answer as a single numerical value. | 160 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A systematic sampling method is used to draw a sample of size 40 from a population of 2,000 students. Calculate the interval of segmentation used in this method. Express your answer as a whole number. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The 26 letters of the English alphabet are listed in an infinite, repeating loop:
ABCDEFGHIJKLMNOPQRSTUVWXYZABC...
Find the letter at the 258th position in this sequence. Express your answer as a single letter. | X | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression: $(1+11+21+31+41)+(9+19+29+39+49)$. Express your answer as a single integer. | 250 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
The eight vertices of a cube are randomly labelled with the integers from 1 to 8 inclusive. Judith looks at the labels of the four vertices of one of the faces of the cube. She lists these four labels in increasing order. After doing this for all six faces, she gets the following six lists: $(1,2,5,8),(3,4,6,7),(2,4,5,... | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
If \(a, b,\) and \(c\) are positive integers such that \(3a = 4b = 7c\), what is the smallest possible value of \(a + b + c\)? Express your answer as a single integer. | 61 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A town's vehicle license plate numbers consist of four characters. The first character must be a vowel (A, E, I, O, U), the second and third characters must be different non-vowel letters, and the fourth character must be a digit from 0 to 9. Calculate the probability of randomly selecting a valid car plate number that... | \frac{1}{21000} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} where a_1 = 4/5, and a_{n+1} is defined by
$$
a_{n+1}=
\begin{cases}
2a_n, & \text{if } 0 \leq a_n \leq \frac{1}{2} \\
2a_n - 1, & \text{if } \frac{1}{2} < a_n \leq 1
\end{cases}
$$
find the value of a_{2015}. Express your answer as a fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $i \cdot z = 1 + i$, find the imaginary part of $z$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line passing through point P (-4, 3) with a slope angle of 45°. Express your answer in standard form (Ax + By + C = 0). | x-y+7=0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\overline{z}(1-i)=i$, calculate the imaginary part of $z$. Express your answer as a single number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Point P is located in the second quadrant, and its distance to the x-axis is 3 units, and its distance to the y-axis is 1 unit. Find the coordinates of point P. Express your answer as an ordered pair (x, y), including negative signs if applicable. | (-1, 3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
When washing clothes with pure water, if each washing removes 3/4 of the dirt, find the minimum number of washes required to ensure that the remaining dirt does not exceed 1%. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 2x - 3y = m and the solution set x = 2, y = -1, calculate the value of m by substituting the given values into the equation. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
What is the negation of the proposition "$\forall x\geqslant 1$, $x^{2}-1 \lt 0$"? Express your answer as a logical proposition. | $\exists x\geqslant 1$, $x^{2}-1\geqslant 0$ | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
What is the number whose absolute value is 2? Express your answer using the ± symbol if necessary. | \pm2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number Z = (m+2i)/(1+i) is an imaginary number, find the value of the real number m. Express your answer as a single real number. | m = -2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^{4}=a_{0}+a_{1}(x-2)+a_{2}(x-2)^{2}+a_{3}(x-2)^{3}+a_{4}(x-2)^{4}$, calculate the value of $a_{2}$ using the binomial theorem. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex numbers $z_1=1+i$ and $z_2=2i$, calculate the value of $\frac{z_2}{z_1}$. Express your answer in the form $a+bi$, where $a$ and $b$ are the real and imaginary parts of the result, respectively. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the algebraic expression px^3 + qx + 3, when x = 3, the value of the expression is 2005. What is the value of the expression when x = -3? Express your answer as a single integer. | -1999 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
In the independence test, suppose $H_0$: Variable $X$ and variable $Y$ are unrelated. Given that $P(K^2 \geq 6.635) \approx 0.010$ under the condition that $H_0$ is true, interpret this probability in terms of the relationship between variable $X$ and variable $Y$. Express your answer as the probability that $X$ and $Y... | 99\% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of \(\frac{1}{2} + \frac{1}{2} + \frac{1}{3} + \frac{1}{3} + \frac{1}{3}\). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, the point $Z_1$ corresponds to the complex number $4+i$. If the midpoint of the line segment $Z_1Z_2$ corresponds to the complex number $1+2i$, calculate the complex number corresponding to the point $Z_2$. Express your answer in the form $\boxed{a+bi}$. | -2+3i | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given an even function f(x) defined on R satisfying f(x) = -1/f(x+3) and f(4) = -2018, calculate the value of f(2018). Express your answer as an integer. | -2018 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Let $Z$ be the set of integers. The set $A$ is defined as $A = \{x \in Z | x^2 - 5x < 6\}$. Find the number of elements in set $A$. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Inequalities"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a line passing through point M(1, -2) that intersects the x-axis at point P and the y-axis at point Q, and M is exactly the midpoint of line segment PQ, calculate the equation of line PQ in the standard form Ax + By + C = 0. | 2x - y - 4 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, the sides opposite to angle A, angle B, and angle C are a, b, and c respectively. If angle A : angle B : angle C = 1 : 2 : 3, then find the ratio a : b : c. Express your answer in the format a : b : c. | 1 : √3 : 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given line $l_1: 2x-y+1=0$, and line $l_2$ passes through the point $(1,1)$ with its angle of inclination being twice that of line $l_1$'s angle of inclination, calculate the equation of line $l_2$ in standard form (Ax + By + C = 0). | 4x+3y-7=0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation y = x^2 + bx + c, which intersects the x-axis at only one point and passes through the points A(m, n) and B(m-8, n), find the exact value of n. | 16 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the vertex of the parabola given by the equation y = 2(x-1)^2 + 5. Express your answer as an ordered pair (x, y). | (1,5) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A map represents a distance of 30 kilometers with a line segment of 20 centimeters. Calculate the scale of this map as a ratio in the form 1:x, where x is a whole number representing the scale. Express your answer in the simplest form. | 1:150000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, if the sum of the first 5 terms is S_5 = 25, find the value of a_3. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, if 2a_7 = a_9 + 7, then find the sum of the first 9 terms of the sequence (S_9). Express your answer as a single integer. | 63 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{x\in\mathbb{N}^*|-1\leq x<6\}$, $A=\{1,3\}$, and $B=\{3,5\}$, find the complement of $A\cup B$ in $U$. Express your answer as a set of elements using standard set notation. | \{2,4\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the interval where the function $f(x) = x \ln x$ is monotonically decreasing. Express your answer in interval notation. | \left(0, \frac{1}{e}\right) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function $y = \tan(\frac{x}{3} + \frac{\pi}{4})$. Express your answer as a single value with pi (if necessary). | 3\pi | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
If \( x \) and \( y \) are positive numbers and the average of 4, 20, and \( x \) is equal to the average of \( y \) and 16, then what is the ratio \( x: y \)? Express your answer in simplest form. | 3:2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
What is the maximum number of lines in space for which there exists a plane that forms equal angles with all the lines? Provide your answer as a single integer. | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the equation $\sqrt{100-x}=9$. Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(\sqrt{2}-\sqrt{3})^{2020} \cdot (\sqrt{2}+\sqrt{3})^{2021}$. Express your answer as a simplified algebraic expression involving square roots. | \sqrt{2}+\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.328125 | 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"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Express the decimal number 0.0000084 in scientific notation, with the coefficient between 1 and 10, and the exponent as a negative integer. | 8.4 \times 10^{-6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex number $Z$ satisfy $(1+i)Z=i$. Find the modulus of $Z$, denoted $|Z|$. Express your answer as a simplified fraction or decimal, including any necessary square roots or other mathematical operations. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A store sells jellybeans at a fixed price per gram. The price for 250g of jellybeans is $7.50. What mass of jellybeans sells for $1.80? Express your answer in grams as a whole number. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Let $m$ be a positive integer, the maximum binomial coefficient in the expansion of $(x+y)^{2m}$ is $a$, and the maximum binomial coefficient in the expansion of $(x+y)^{2m+1}$ is $b$. If $13a=7b$, calculate the value of $m$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | 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. Calculate the total number of different arrangements. Express your answer as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the total number of ways to arrange 15 playing cards, consisting of 13 hearts and the inserted queen and king of clubs. Express your answer as a single integer. | 210 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Permutations"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number (2+i)i. Express your answer as a single integer, the coefficient of i. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Daniel's age is one-ninth of his father's age. One year from now, Daniel's father's age will be seven times Daniel's age. Calculate the difference between their current ages. Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\min \{\sqrt{x},x^{2},x\}=\frac{1}{16}$, find the exact value of x. Express your answer as a single fraction or decimal value. | \frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Express 1.1 billion in scientific notation. Use the format $a \\times 10^{b}$. | 1.1 \times 10^{9} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=(x^{2}-6x+5)+(x-2)i$, find the range of values for the real number $x$ such that the corresponding point in the complex plane is located in the third quadrant. Express your answer as an interval (e.g., (a, b)). | (1,2) | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given $$x \in \left(-\frac{\pi}{2}, 0\right)$$ and $\cos 2x = a$, calculate $\sin x$ using the double-angle formula for cosine. Express your answer as a mathematical expression involving $a$. | -\sqrt{\frac{1-a}{2}} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Li Yong purchased a total of 16 stamps, with denominations of 80 cents and 100 cents, spending 14 yuan and 60 cents. Calculate the numbers of 80-cent and 100-cent stamps purchased. Express your answer as two integers separated by a comma, where the first integer represents the number of 80-cent stamps and the second in... | 7, 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x)$ satisfies $2f(x) - f(-x) = 3x + 1$ for any real number $x$, then find $f(x)$. Express your answer as a function of $x$. | x + 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} where a_1 = 5 and S_{n-1} = a_n for n >= 2, derive the general formula for S_n. Express your answer as a formula in terms of n. | 5 \cdot 2^{n-1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation |x-1| = 1-x, find the range of x that satisfies the equation. Express your answer as an inequality in the format x [inequality symbol] [value]. | x ≤ 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{AB}=({4,2})$ and $\overrightarrow{AC}=({1,4})$, calculate the dot product $\overrightarrow{AB}⋅\overrightarrow{BC}$. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=x^{2}-2x+2$, whose domain and range are both $[1,b]$ $(b > 1)$, find the value of $b$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If for all \(x_1\), \(x_2 \in (-\infty, m)\), and \(x_1 < x_2\), it holds that \( \dfrac {x_2e^{x_1}-x_1e^{x_2}}{e^{x_2}-e^{x_1}} > 1\), then find the maximum value of \(m\). Express your answer as a single value. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, two circles with centres \( A \) and \( B \) intersect at points \( P \) and \( Q \) so that \( \angle PAQ = 60^\circ \) and \( \angle PBQ = 90^\circ \). Calculate the ratio of the area of the circle with centre \( A \) to the area of the circle with centre \( B \). Express your answer as a ratio (e.g.,... | 2:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $C: \frac{x^2}{9} + \frac{y^2}{4} = 1$, with foci $F_1$ and $F_2$, and a point $M$ on $C$, find the maximum value of $|MF_1| \cdot |MF_2|$. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
If the symmetry center of the function $y=\sin(x+\phi)$ is $(\frac{\pi}{6}, 0)$, find the equation of the symmetry axis of the function $y=\cos(x+\phi)$. Express your answer in the form $x =...$. | x=\frac{\pi}{6} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)= \frac{1}{\ln (x+1)}+ \sqrt{9-x^{2}}$. Express your answer as an interval or union of intervals using interval notation. | (-1,0)\cup(0,3] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In quadrilateral \(ABCD\), given that \(\overrightarrow{AB} = \vec{a}\), \(\overrightarrow{AD} = \vec{b}\), and \(\overrightarrow{BC} = \vec{c}\), express \(\overrightarrow{CD}\) in terms of \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) using vector operations. | \vec{b} - \vec{a} - \vec{c} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the exact number of real roots for the equation sin(x) = lg(x) within the domain 0 < x <= 10. | 3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_n} where each term is positive, a_1 = 2 and a_2, a_4+2, a_5 form an arithmetic sequence. Let S_n be the sum of the first n terms of the sequence {a_n}. Calculate the value of S_5. | 62 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Let the sets be defined as
$$
\begin{array}{l}
M=\left\{y \left\lvert\, y=x^{\frac{1}{2}}\right., x \in[1,4]\right\}, \\
N=\left\{x \mid y=\log _{2}(1-x)\right\}.
\end{array}
$$
Calculate the intersection of $M$ and the complement of $N$, denoted as $M \cap \complement_{\mathbf{R}} N$. Express your answer in interval o... | [1, 4] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If $i$ is the imaginary unit, calculate the imaginary part of the complex number $i(1+i)$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the maximum number of possible points of intersection between a circle and a triangle. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
If \( x \) is \( 20\% \) of \( y \) and \( x \) is \( 50\% \) of \( z \), then what percentage is \( z \) of \( y \)? Express your answer as a percentage, including the percentage sign (%). | 40% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(X \sim B(10,0.8)\), calculate \(D(2X+1)\), the variance of the linear transformation \(Y = 2X + 1\). Express your answer as a single numerical value. | 6.4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
The scale of a map is 1:10000. After measuring, the area of a certain place on the map is found to be 10cm^2. Calculate the actual area of this place in square meters (m^2). Express your answer as a whole number. | 100000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of real roots of the equation $x^3 - 6x^2 + 9x - 4 = 0$. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangle inscribed in a triangle has its base coinciding with the base b of the triangle. If the altitude of the triangle is h, and the altitude x of the rectangle is half the base of the rectangle, express x in terms of b and h. | x = \frac{bh}{2h + b} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a decreasing arithmetic sequence {a_n}, where a_3 = -1, and a_1, a_4, -a_6 form a geometric sequence. If S_n represents the sum of the first n terms of {a_n}, calculate the value of S_7. Express your answer as a single integer. | -14 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $l$: $ax - by + 2 = 0$ ($a > 0$, $b > 0$) passes through the point $(-1, 2)$, find the slope of line $l$ when $\frac{2}{a} + \frac{1}{b}$ reaches its minimum value. Express your answer as a single number, the slope of the line. | 2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Point B divides the line segment AC into two parts. If BC/AB = AB/AC = k, then find the value of k. Express your answer as a single value. | \frac{\sqrt{5} - 1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( \frac{x}{3} = \frac{y}{-4} = \frac{z}{7} \), calculate the value of \( \frac{3x + y + z}{y} \). Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f$ where $f(1) = 1$, $f(2) = 3$, $f(3) = 4$, $f(4) = 7$, $f(5) = 11$, and each subsequent term is the sum of the two preceding terms, calculate the value of $f(10)$. Express your answer as a single integer. | 123 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain college, there are a total of 1200 students in majors A, B, and C. To investigate the situation of students working part-time to support their studies, it is planned to use stratified sampling to draw a sample of 120 students. It is known that there are 380 students in major A and 420 students in major B. ... | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\sin 13^\circ \cos 17^\circ - \cos 13^\circ \sin 17^\circ$. Express your answer as a trigonometric function value. | -\sin 4^\circ | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
The absolute value of the negative number $a$ is $2$. What is the value of $a$? Express your answer as an integer. | -2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the plane region represented by the system of inequalities
\[
\begin{cases}
y \leqslant -x + 2, \\
y \leqslant x - 1, \\
y \geqslant 0
\end{cases}
\]
Express your answer as a single number. | \dfrac{1}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ with two asymptotes that are perpendicular to each other, and the distance from a vertex to one of the asymptotes is $1$, find the distance from one of the foci of the hyperbola to an asymptote. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x) = \lg(-x + 4)$. Express your answer in interval notation. | (-\infty, 4) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
If the central angle of a sector is 72° and its radius is 20cm, calculate the area of the sector. Express your answer in square centimeters (cm²) and include π if necessary. | 80\pi | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(n)= \begin{cases} n-3(n\geqslant 10) \\ f[f(n+5)](n < 10) \end{cases}$, where $n \in N$, find the value of $f(8)$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
An equilateral triangle has a side length of $s$. A circle is inscribed in the triangle, and a square is inscribed in the circle. Calculate the area of the square in terms of $s^2$. Express your answer as a fraction of $s^2$ and include the boxed notation around the final answer. | \frac{s^2}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases}x^{2} \\\\ x+1\\end{cases} \quad \begin{matrix}-2\leqslant x\leqslant 0 \\\\ 0 < x\leqslant 2\\end{matrix}$, calculate the value of $\int\_{-2}^{2}f\left(x\right)dx$. Express your answer as a single fraction or decimal value. | \dfrac{20}{3} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer that, when divided by 10, leaves a remainder of 9; when divided by 9, leaves a remainder of 8; when divided by 8, leaves a remainder of 7; and so on, until when divided by 2, leaves a remainder of 1. Express your answer as a single integer. | 2519 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the coefficients of the terms that do not contain x in the expansion of (1+ax+by)^n is 243, where a and b are positive integers, calculate the exact value of n. | 5 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If the complex number $\frac{2 - mi}{1 + i}$ is a pure imaginary number, then find the value of the real number $m$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the two linear equations in one variable $y$: $y+3m=32$ and $y-4=1$, and knowing that the solutions are the same, solve for the value of $m$. Provide your answer as an integer. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = xe^x + 2x + 1 at the point (0, 1). Express your answer in the slope-intercept form (y = mx + b). | y = 3x + 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
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