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 an arithmetic sequence {a_n} that satisfies a_1 = 1 and a_{n+2} - a_n = 6, calculate the value of a_{11}. | 31 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, $a_1 + a_4 + a_7 = 48$, $a_2 + a_5 + a_8 = 40$. Calculate the value of $a_3 + a_6 + a_9$. Express your answer as a single numerical value. | 32 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation \( x^2 + px - 444p = 0 \) with integer roots, and knowing that \( p \) is a prime number, find the specific value of \( p \) that satisfies these conditions. Express your answer as a single prime number. | 37 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the coefficients of the terms in the expansion of $( \sqrt {x}- \dfrac {3}{x})^{9}$ that do not contain $x^{3}$. Express your answer as an integer. | -485 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the series 2 + 5 + 8 + 11 + 14 +... + 50. Express your answer as a single numerical value. | 442 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y^2 = 4x and points A(1,0), B(-1,0), if point M is on the parabola, calculate the maximum value of angle MBA in radians, using pi notation for precision. | \dfrac {\pi}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $-21\div 3$. Express your answer as an integer. | -7 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
What is the largest number of T-shaped pieces that can be placed on a 4x5 grid without any overlap of the pieces? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $z \in \mathbb{C}$ and $|z + 2 - 2i| = 1$, find the range of values for $|z - 2 - 2i|$. Express your answer as an interval, including the minimum and maximum values. | [3, 5] | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. Calculate the number of calories in 200 grams of her lemonade. | 137 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, the complex number $z=(m-1)(m-2)+(m-2)i$, where $m\in\mathbb{R}$. If $z$ is a purely imaginary number, find the value of $m$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
A school has 600, 500, and 400 students in the first, second, and third grades, respectively. If 30 first-year students are sampled using stratified sampling, find the total number of students sampled, denoted as $n$. Express your answer as a whole number. | 75 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=m\left(m-2\right)+(m^{2}-4)i$, find the real number $m$ such that $z$ is purely imaginary. Express your answer as a real number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
On the number line, what is the integer point that is closest to the point representing $\sqrt {6}$? Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first, third, fifth, seventh, and ninth terms of an arithmetic sequence equals 55, find the sum of the first 9 terms (S9) of the sequence. | 99 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of the function $y= \sqrt {16-4^{x}}$. Express your answer as an interval, using square brackets to denote inclusion of an endpoint and parentheses for exclusion. | [0, 4) | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system, find the distance from point $P(-1,-2,-3)$ to the plane $xOz$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$, $a+1$, $a+2$ are the side lengths of an obtuse triangle, find the range of values for $a$ that satisfy the conditions for an obtuse triangle. Express your answer as an inequality in the format $\boxed{lower bound < a < upper bound}$. | 1 < a < 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
From a group of 4 boys and 3 girls, select 3 people to undertake three different tasks. If there is at least one girl among these 3 people, calculate the total number of possible arrangements. Provide your answer as a single number. | 186 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of the linear function y = x - 1 passes through the point (m, 2), find the value of m. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that one of the foci of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ is $F(2,0)$, and the hyperbola is tangent to the circle $(x-2)^2 + y^2 = 1$, calculate the eccentricity of the hyperbola. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(A C = C B = 10 \, \text{m}\), where \(A C\) and \(C B\) are each the diameter of the small equal semi-circles. The diameter of the larger semi-circle is \(A B\). Calculate the difference in lengths between the semi-circular arc from \(A\) to \(B\) and the combined semi-circular arcs from \(A\) to \(C\... | 0 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation x^2 + (1 - 2i)x + 3m - i = 0, where m is a real number, and the equation has one real root, find the other root. Express your answer as a complex number in the form a + bi. | -\frac{1}{2} + 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Arrange students A, B, C, D, and E from left to right. Calculate the total number of arrangements where A and B are adjacent and there is exactly one student between A and C. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^{2}-3y-5=0$, calculate the value of $2x^{2}-6y-6$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Mr. Wang and his friends plan to play badminton continuously for 6 hours at a venue that offers two billing schemes. The package billing scheme charges $90 per hour plus a $10 entrance fee per person. The per person billing scheme costs $54 per person for the first 3 hours, and $8 per person for each additional hour. F... | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(-1,3)$ and $\overrightarrow{b}=(m,m-2)$, if $\overrightarrow{a}$ is collinear with $\overrightarrow{b}$, find the value of $m$. Express your answer as a single numerical value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
There are 100 people in a room. Some are wise and some are optimists. A wise person can look at someone and know if they are wise or if they are an optimist. An optimist thinks everyone is wise (including themselves). Everyone in the room writes down what they think is the number of wise people in the room. Calculate t... | 75 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the slope angle of the line x = 2. Express your answer in radians. | \frac{\pi}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function y = f(x) is differentiable in the interval (a, b), and x_0 is in the interval (a, b), evaluate the limit:
$$\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0 - h)}{h}$$
Express your answer in terms of f'(x_0). | 2f'(x_0) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
A farmer bought 749 sheep. He sold 700 of them for the price paid for the 749 sheep. The remaining 49 sheep were sold at the same price per head as the other 700. Calculate the percent gain on the entire transaction. Express your answer as a percentage, rounded to two decimal places if necessary. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the largest possible sum of three 2-digit numbers, where all six digits are different. Express your answer as a single integer. | 255 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x) = \begin{cases} \ln x, & x > 1 \\ 2x + \int_{0}^{m} 3t^{2} \, dt, & x \leq 1 \end{cases}$, and the equation $f(f(e)) = 10$, calculate the value of $m$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x|y=\sqrt{\frac{3-x}{x+1}}\}$ and $B=\{x|\lg x < 1\}$, calculate the intersection of sets A and B, expressing your answer as an interval. Include the endpoints of the interval if they are included in the intersection. | (0,3] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The regular price for a T-shirt is $25 and the regular price for a pair of jeans is $75. If the T-shirt is sold at a 30% discount and the jeans are sold at a 10% discount, calculate the total discount. Express your answer in dollars, rounded to two decimal places. | 15.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the matrix operation definition:
$$\begin{bmatrix} a & b \\ c & d\end{bmatrix}$$• $$\begin{bmatrix} e \\ f\end{bmatrix}$$= $$\begin{bmatrix} ae+bf \\ ce+df\end{bmatrix}$$
and the values α+β=π and α-β=π/2, calculate the result of the matrix operation:
$$\begin{bmatrix} \sinα & \cosα \\ \cosα & \sinα\end{bmatrix... | \begin{bmatrix} 0 \\ 0\end{bmatrix} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Matrices"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
If \( x = -2 \), evaluate the expression \( (x) \left(x^2\right) \left(\frac{1}{x}\right) \) and provide your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Divide 3 teachers and 3 students, a total of 6 people, evenly into 3 groups, each assigned to three communities to participate in social practice activities. Calculate the probability that each group has exactly 1 teacher and 1 student. Express your answer as a simplified fraction. | \dfrac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sample divided into several groups, where a specific group's frequency is 30 and its relative frequency is 0.25, calculate the sample capacity n. Express your answer as a whole number. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A sample consisting of five observations has an arithmetic mean of 10 and a median of 12. Find the smallest possible range (largest observation minus smallest) for such a sample. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations $4^{a}=6^{b}=24$, find the value of $\frac{1}{a}+\frac{1}{b}$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $$\overrightarrow {a}$$ and $$\overrightarrow {b}$$ are both unit vectors, and their angle is 60°, calculate the magnitude of $$\overrightarrow {a}+3 \overrightarrow {b}$$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{13} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If $f(x)$ is an odd function on $\mathbb{R}$ with a period of 5, and it satisfies $f(1) = 1$ and $f(2) = 2$, then find the value of $f(3) - f(4)$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Consider the recursive function:
\[ f(n) = \log(n + f(n-1)) \]
with the base case:
\[ f(2) = \log 2 \]
Estimate the interval that contains the value of f(2013). Express your answer as an interval, e.g., (log 2016, log 2017). | (\log 2016, \log 2017) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
The radius of the circle obtained by the plane $\alpha$ intersecting the sphere of center O is $\sqrt{2}$, and the distance from the center O to the plane $\alpha$ is 1. Find the radius of this sphere. Express your answer as a numerical value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\frac{\sin 70^{\circ} \sin 20^{\circ}}{\cos^2 155^{\circ} - \sin^2 155^{\circ}}$. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} with each term positive, the sum of the first n terms is S_n = 2, and the sum of the first 3n terms is S_{3n} = 14. Calculate the sum of the first 4n terms, S_{4n}. Express your answer as a single numerical value. | 30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function y = x^2, find the expression of the new graph after shifting it 3 units to the right and 2 units down. Express your answer as a function of x. | y = (x - 3)^2 - 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of terms in the expansion of $(\sqrt{x} - \frac{1}{x})^{10}$ that contain positive integer powers of $x$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then what is the height of the block in centimeters? Express your answer in centimeters. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the complex conjugate $\overline{z}$ of the complex number $z = \frac{4i}{1-i}$, where $i$ is the imaginary unit. Express your answer in the standard form $a + bi$. | -2 - 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $R$, the set $M=\{x|x>1\}$, and $N=\{x||x|\leq2\}$, find the intersection $(\complement_R M) \cap N$. Express your answer as an interval, including the endpoints if they are part of the solution. | [-2,1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression \(2x^2 \cdot 3x^2\), apply the rules of exponents to simplify it. Express your answer as a simplified algebraic expression. | 6x^4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a circle $C: x^{2}+y^{2}-6x+8=0$, if one of the asymptotes of the hyperbola ${y^2}-\frac{{{x^2}}}{{{m^2}}}=1(m＞0)$ is tangent to the circle $C$, calculate the value of $m$. | 2\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given a scalene triangle $ABC$ with two altitudes of lengths 4 and 12, if the length of the third altitude is also an integer, what is the largest possible integer value for the length of the third altitude? | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations of curve C are $\begin{cases}x=4\cos \theta \\ y=2\sin \theta\end{cases}$, calculate the eccentricity of this curve. Express your answer as a simplified fraction. | \dfrac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the hyperbola passes through the point (2, 3) and its asymptotes are given by the equations y = ±√3x, derive the standard equation of the hyperbola. Express your answer in the standard form of a hyperbola equation, including all necessary variables and coefficients. | x^2 - \frac{y^2}{3} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be an odd function with a period of 2. When 0 <= x <= 1, f(x) = 2x(1-x). Calculate the value of f(-5/2). Express your answer as a fraction. | - \frac {1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the zeros of the function f(x) = x^3 - 2x^2 - x + 2. Express your answer as a list of exact values. | -1, 1, 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of distinct real roots of the equation $|x^2 - 6x| = 9$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
The common difference of the arithmetic sequence $\{a_n\}$ is not zero, the first term $a_1=1$, and $a_2$ is the geometric mean of $a_1$ and $a_5$. Calculate the sum of the first 10 terms of the sequence. Express your answer as a single integer. | 100 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
In parallelogram ABCD, diagonals AC and BD intersect at point O. Given $\overrightarrow {AB} + \overrightarrow {AD} = \lambda \overrightarrow {AO}$, find the real number $\lambda$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the shadow of a tree is 30m long, and at the same moment, a 1.5m tall pole has a shadow of 3m, calculate the height of the tree. Express your answer in meters. | 15 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Similarity"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
A two-digit number, when three times the sum of its units and tens digits is subtracted by -2, still results in the original number. Find this two-digit number. Express your answer as a two-digit number. | 28 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (-3, 2)$ and $\overrightarrow{b} = (-1, \lambda)$, if vector $\overrightarrow{a}$ is perpendicular to vector $\overrightarrow{b}$, find the value of the real number $\lambda$. Express your answer as a real number. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
There are five different books, of which two are Chinese language books, two are math books, and one is a physics book. If they are randomly placed side by side on the same shelf, calculate the probability that the books of the same subject are adjacent to each other. Express your answer as a simplified fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, if $\frac{1+i}{1-i}=a+bi$ $(a,b\in \mathbb{R})$, then find the value of $a+b$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression \(\left(\frac{\sqrt{3}}{2}+\frac{x}{2} \mathrm{i}\right)^{2008} = f(x) + \mathrm{i} g(x)\), where \(f(x)\) and \(g(x)\) are polynomials with real coefficients, find the sum of the coefficients of \(f(x)\). Express your answer as a real number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the definition of an "ideal complex number" as a complex number where the real part and the imaginary part are opposites of each other, and the complex number $z= \frac {a}{1-2i}+bi$ ($a, b \in \mathbb{R}$) is an "ideal complex number", find the relationship between $a$ and $b$. Express your answer as an equation... | 3a+5b=0 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the reciprocal of -4. Express your answer as a fraction. | -\dfrac{1}{4} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, if ∠A = 30°, AC = 2, and BC = √2, calculate the value of sinB. Express your answer as a fraction or decimal. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The lengths of the three edges of a rectangular solid at one vertex are 3, 4, and 5, respectively, and all 8 vertices of the rectangular solid lie on the same spherical surface. Find the surface area of this sphere. Express your answer in terms of π. | 50\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that an isosceles triangle with legs of 1 kilometer each and a base length of 4.848 millimeters has a vertex angle of 1″, calculate the base length of a similar isosceles triangle with legs equal to the average distance from the sun to the earth (1.5×10^8 kilometers) and the same vertex angle. Express your answer... | 727.2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Similar Triangles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition: "If \( n \) is a positive odd number, and the sum of its digits is 4, and none of the digits are zero, then \( n \) is a prime number." How many counterexamples exist for this proposition? Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the angle of inclination of the tangent to the curve y = (1/3)x^3 - 2 at the point (1, -5/3). Express your answer in degrees, including the degree symbol (°). | 45° | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{1+i}{3-4i}$, find the value of $|\bar{z}|$. Express your answer as a fraction with a square root. | \frac{\sqrt{2}}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Mr. Anderson has more than 25 students in his class. He has more than 2 but fewer than 10 boys and more than 14 but fewer than 23 girls in his class. Calculate the number of different class sizes that would satisfy these conditions. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of the function f(x) is tangent to the line 2x+y-1=0 at the point (1,f(1)), find the value of f(1) + f'(1). Express your answer as a single number. | -3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A haunted house has six windows. Calculate the total number of ways Georgie the Ghost can enter the house by one window and leave by a different window. Express your answer as a numerical value. | 30 | big_math | [
"Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $1010_2$ to a decimal number. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Other -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=(a-1)+i$, if $z$ is a purely imaginary number, find the value of the real number $a$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies the equation $\frac{\overline{z}}{1 - i} = i$, where $i$ is the imaginary unit, determine the value of $z$. Express your answer as a complex number in the form $a + bi$. | 1 - i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A rope that is initially 1 meter long is repeatedly cut in half. After the sixth cut, what is the length of the remaining rope? Express your answer as a power of 1/2. | (\frac{1}{2})^6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_{n}\}$ satisfies ${a}_{n+1} = \begin{cases} 2{a}_{n}, & 0 \leqslant {a}_{n} < \frac{1}{2} \\ 2{a}_{n}-1, & \frac{1}{2} \leqslant {a}_{n} < 1 \end{cases}$, and $a_{1} = \frac{3}{5}$. Find the value of ${a}_{2015}$. Express your answer as a fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(3x) = \log_{2} \sqrt{\frac{9x + 5}{2}}$, find the value of $f(1)$. Express your answer as a single value. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\ln ( \sqrt {1+9x^{2}}-3x)+1$, calculate the value of $f(\ln 2)+f(\ln \dfrac {1}{2})$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions to the equation $2^{2x} - 3^{2y} = 55$. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x) = \begin{cases} \log_2(1-x^2) & \text{if } -1 < x < 1 \\ \sin(\frac{\pi x}{3}) & \text{if } x \geq 1 \end{cases}$, find the value of $f(\frac{31}{2}) + f(\frac{\sqrt{3}}{2})$. Express your answer as a single number. | -\frac{5}{2} | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)$ defined as:
$f(x)=\left\{\begin{array}{ll}
\ln({x}^{2}+a), & \text{for } x < 0 \\
f(x-2), & \text{for } x \geq 0
\end{array}\right.$
with the condition that $f\left(2023\right)=1$, find the value of the positive number $a$. Express your answer as a single mathematical expression. | e-1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Rewrite the equation $x^2 - 12x + 33 = 0$ in the form $(x+m)^2 = n$. Calculate the values of $m$ and $n$ and express your answer in the format $(m, n)$. | (-6, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic inequality about $x$, $ax^{2}-ax-2a^{2} > 1 (a > 0, a
eq 1)$, the solution set is $\{x|-a < x < 2a\}$. The domain of the function $f(x)=\sqrt{\left(\frac{1}{a}
ight)^{x^{2}+2mx-m} -1}$ is $\mathbb{R}$. Determine the range of values for $m$. Express your answer as an interval or a specific numerical... | [-1,0] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities",
"Mathematics -> Precalculus -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The shortest path from Atown to Cetown runs through Betown. Two signposts on this path are shown, but one of them is broken and a number is missing. Given that the distance from Atown to Betown is 4 km and the distance from Atown to the second signpost is 6 km, calculate the distance that was written on the broken sign... | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Let {a_n} be a geometric sequence, and let S_n denote the sum of the first n terms. If S_4 = 2 and S_8 = 8, then find S_12. Express your answer as a single integer. | 26 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the total number of lattice points (including those on the boundary) in the region bounded by the x-axis, the line x=4, and the parabola y=x^2. Express your answer as a single integer. | 35 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
An object moves in the left and right directions. It is defined that moving $3m$ to the right is denoted as $+3m$. What is the notation for moving $3m$ to the left? Express your answer as a notation (e.g. $+3m$ or $-3m$). | -3m | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
For a real number $a$, if its reciprocal does not exist, then what is the value of $a$? Express your answer as a single real number. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
After an exam, 50 out of 1000 exam papers numbered from 1 to 1000 were selected for evaluation using a systematic sampling method. Calculate the number of papers selected from the interval [850, 949]. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $P=\{x|y= \sqrt {-x^{2}+x+2}, x \in \mathbb{N}\}$, $Q=\{x|\ln x < 1\}$, calculate the intersection $P \cap Q$. Express your answer as a set of numbers, using set notation. | \{1, 2\} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Applied Mathematics -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
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