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 function $f(x)=\begin{cases} e^{x}+1,x < 1 \\ f(\ln x),x\geqslant 1 \end{cases}$, find the value of $f(e)$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
If $M \subseteq \{1, 2\}$, calculate the total number of sets $M$ that satisfy this condition. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Last Thursday, each of the students in M. Fermat's class brought one piece of fruit to school. Each brought an apple, a banana, or an orange. In total, 20% of the students brought an apple and 35% brought a banana. If 9 students brought oranges, how many students were in the class? Express your answer as a whole number... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
If the perimeter of a sector is 40cm, find the maximum area of the sector. Express your answer in square centimeters (cm²) as a whole number. | 100 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the vertex of the parabola given by the equation y = x^2 - 2. Express your answer as an ordered pair (h, k). | (0, -2) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given two parallel lines $l_1: 3x + 4y + c_1 = 0$ and $l_2: 6x + 8y + c_2 = 0$, calculate the distance between them. Express your answer as a formula in terms of $c_1$ and $c_2$, including the absolute value and the square root in the denominator, in the format $\boxed{d = \frac{|...|}{...}}$. | d = \frac{|2c_1 - c_2|}{10} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ratio a/b = 2/5, find the value of a/(a+b). Express your answer as a simplified fraction. | \frac{2}{7} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $1 + \binom{6}{1}i + \binom{6}{2}i^2 + \binom{6}{3}i^3 + \binom{6}{4}i^4 + \binom{6}{5}i^5 + \binom{6}{6}i^6$, where $i$ is the imaginary unit, meaning $i^2 = -1$. Express your answer in the form $\boxed{a+bi}$, where $a$ and $b$ are integers. | -8i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the length of the imaginary axis of the hyperbola $x^2 + my^2 = 1$ is twice the length of the real axis, find the value of the real number $m$. Express your answer as a real number, including the negative sign if applicable. | -\frac{1}{4} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $(1-x)(3+x) > 0$. Express your answer in interval notation. | (-3, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\\tan \\theta = 2$, find the value of $\\cos^2\\left(\\theta + \\frac{\\pi}{4}\\right)$. Express your answer as a simplified fraction. | \\frac{1}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Let y = f(x) be a function with the domain of (0, +∞), and f(xy) = f(x) + f(y). If f(8) = 6, calculate the value of f(√2). Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $$y=\log_{3}(-x^{2}-2x)$$ in interval notation. Express your answer in the format (a, b), using parentheses to denote the interval, and include any necessary negative or positive infinity notations. | (-2, 0) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
A broadcaster broadcasts a volleyball match, which adopts a best-of-five sets format, meaning the match ends as soon as one side wins three sets. It is known that the two teams are equally matched, and the outcome of each set is independent of the others. If the broadcaster can earn 200,000 yuan for each set broadcaste... | \dfrac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, find the angle of inclination of the line $\sqrt {3}x-y-3=0$. Express your answer in degrees as a whole number. | 60 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given a point $P$ on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ whose distance to one focus of the ellipse is $3$, find the distance from $P$ to the other focus. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Ellipses"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
A tangent line is drawn from point $A(-3,4)$ to the circle $(x-2)^{2}+(y-3)^{2}=1$. Find the distance from $A$ to the point of tangency. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If the curve f(x) = a*cos(x) and the curve g(x) = x^2 + b*x + 1 have a common tangent line at the intersection point (0, m), calculate the sum of a and b. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha \in \left( \frac{3}{2}\pi,2\pi\right)$, and it satisfies $\cos \left( \alpha +\frac{2017}{2}\pi \right)=\frac{3}{5}$, find the value of $\sin \alpha +\cos \alpha$. Express your answer as a single fraction or decimal value. | \frac{1}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given a monotonically decreasing geometric sequence {a_n} that satisfies a_1a_5 = 9 and a_2 + a_4 = 10, find the common ratio q of the sequence. Express your answer as a fraction or decimal. | 1/3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given an odd function f(x) that satisfies f(1-x) = f(1+x), calculate the value of f(2008). Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $(1+2i)i=a+bi$, where $a\in\mathbb{R}$ and $b\in\mathbb{R}$, calculate the value of $a+b$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos (\alpha+\beta)=-1$ and $\tan \alpha=2$, find the value of $\tan \beta$. Express your answer as a single value. | -2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
From the set {1,2,3,…,10}, select three different numbers such that at least two of them are adjacent. Find the total number of different ways to do this. Express your answer as a single integer. | 64 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence with a common ratio of $\sqrt{2}$, if $\sin(a_2a_3) = \frac{3}{5}$, calculate the value of $\cos(a_1a_6)$. Express your answer as a single fraction. | \frac{7}{25} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Rearrange the quadratic equation \(3x^{2} = -2x + 5\) into its general form \(ax^{2} + bx + c = 0\), and then list the coefficients of the quadratic term, linear term, and constant term in that order. | 3, 2, -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Mother Hubbard had a Giant Bear chocolate bar. She gave each of her children one-twelfth of the chocolate bar. One third of the bar was left. How many children did she have? Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In the triangle \( \triangle ABC \), \( \angle BAC = 50^\circ \), and \( BE = BD \), \( CF = CD \). Find the measure of \( \angle EDF \) in degrees. | 65^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A building has 4 units, and two people, A and B, live in this building. Calculate the probability that A and B live in the same unit. Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
The equation $\sqrt{(x-4)^2 + y^2} + \sqrt{(x+4)^2 + y^2} = 10$ represents the sum of the distances from the point $(x, y)$ to the points $(4, 0)$ and $(-4, 0)$, which is greater than the distance between the two points. Therefore, the trajectory of the point is an ellipse with foci at $(4, 0)$ and $(-4, 0)$. Find the ... | \frac{x^2}{25} + \frac{y^2}{9} = 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given $i$ is the imaginary unit, and $a\in\mathbb{R}$, find the value of $a$ such that the complex number $2i - \frac{a}{1-i}$ is real. Express your answer as a single real number. | 4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? Express your answer in degrees as a whole number. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the real number $\lambda$ such that the inequality $a^2+8b^2 \geq \lambda b(a+b)$ holds for any real numbers $a$ and $b$. Express your answer as a closed interval. | [-8,4] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the following sum, \( O \) represents the digit 0. \( A, B, X, \) and \( Y \) each represent distinct digits. How many possible digits can \( A \) be?
$$
\begin{array}{r}
A O O B A O O B \\
+\quad B O O A B O O A \\
\hline X X O X Y X O X X
\end{array}
$$
Express your answer as a single integer representing the coun... | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin(\pi + \alpha) = -\frac{1}{2}$, find the value of $\sin(5\pi - \alpha)$, where $\alpha$ is a real number. Express your answer as a single fraction or decimal value. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
The result of the calculation \(9 \times 11 \times 13 \times 15 \times 17\) is the six-digit number'\(3 n 8185\) '. Calculate the value of the missing digit \(n\). Express your answer as a single digit. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, calculate the product of $(-1+i)$ and $(2-i)$. Express your answer as a complex number in the form $a+bi$. | -1+3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A company's survey of employees' BMI values shows that the proportion of obese men is 1/5, and the proportion of obese women is 1/10. It is known that the ratio of male to female employees in the company is 3:2. Find the probability that a randomly selected obese employee from the company is male. Express your answer a... | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number z corresponds to the point (1, -1) on the complex plane, calculate the value of ((1+i)/z)^3. Express your answer as a complex number in the form a+bi or a single value if purely real or imaginary. | -i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In a box, there are 15 balls of the same size, including 9 red balls and 6 white balls. If person A and person B each draw one ball without replacement, and person A draws a red ball, what is the probability that person B draws a white ball? Express your answer as a simplified fraction. | \frac {3}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression \(1+2+3-4+5+6+7-8+9+10+11-12\). Provide your answer as a single integer. | 30 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\overrightarrow{OC}$ are three unit vectors with the angle between each pair being $60^{\circ}$, calculate the magnitude of the vector sum $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC}$. Express your answer as a single numerical value. | \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the number x such that 3 + x = -1. Express your answer as an integer. | -4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = e^x - x in the interval [-1, 1]. Express your answer as a mathematical expression. | e - 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Let \(R\) be the set of real numbers, \(M=\{-1,0,1,5\}\), \(N=\{x|x^{2}-x-2\geqslant 0\}\). Calculate the intersection of \(M\) and the complement of \(N\) in \(R\), denoted as \(M \cap \complement_{R} N\). Express your answer as a set of numbers. | \{0,1\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Select 3 laptops from 4 Type A laptops and 5 Type B laptops, ensuring that there is at least one laptop of each type among the three. Find the total number of different selections. Express your answer as a whole number. | 70 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z satisfies z*(1-i) = 2i, where i is the imaginary unit, find the value of z. Express your answer in the form a + bi, where a and b are integers. | -1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given $U=\{x\in\mathbb{N}|x < 6\}$, $A=\{1,2,3\}$, and $B=\{2,4,5\}$, find $(\complement_U A)\cup(\complement_U B)$. Express your answer as a set, including all elements of the resulting set. | \{0,1,3,4,5\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the algebraic expression $(x^{2}+mx)(x-3)$, find the value of $m$ such that the simplified expression does not contain the term $x^{2}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. Calculate the number of two-legged birds that Margie counted. Provide your answer as a whole number. | 139 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given a positive term arithmetic sequence $\{a_n\}$ that satisfies $a_1 + a_{2014} = 2$, find the minimum value of $\frac{1}{a_{2}} + \frac{1}{a_{2013}}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A car has five tires (four road tires and a full-sized spare) that were rotated so that each tire was used the same number of miles during the first 30,000 miles the car traveled. Calculate the number of miles each tire was used. Express your answer as a whole number of miles. | 24,000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let \( a \) represent the decimal part of \( \frac{1}{3-\sqrt{5}} \). Calculate \( \log_{2} a(2a+1) \) and provide your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A moving circle passes through the point (1, 0) and is tangent to the line x = -1. Derive the equation of the trajectory of the circle's center in the Cartesian coordinate system (x, y). | y^2=4x | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
The numbers \( x \) and \( y \) satisfy the equation \( 17x + 51y = 102 \). Find the value of \( 9x + 27y \). Express your answer as a single integer. | 54 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A store sells two calculators with different cost prices, both for $135. One is sold with a profit of 25%, while the other is sold at a loss of 25%. Calculate the overall profit or loss from this transaction. Express your answer in dollars, including a negative sign if the result is a loss. | -18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{BA}=(3,-2)$ and $\overrightarrow{AC}=(0,6)$, calculate the magnitude of $\overrightarrow{BC}$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, $\angle C=90^{\circ}$. Given $\overrightarrow{AB}=(t,1)$ and $\overrightarrow{AC}=(2,2)$, calculate the value of $t$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $P(a, a-2)$ is in the fourth quadrant, determine the range of $a$ that satisfies this condition. Express your answer as an inequality defining the range of $a$. | 0 < a < 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,0,-1)$ and $\overrightarrow{b}=(0,1,-2)$, calculate the result of the vector operation $2\overrightarrow{a}-\overrightarrow{b}$. Express your answer as a vector in the form $(x,y,z)$. | (4,-1,0) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
If all the sample points in a scatter plot fall on a straight line with a slope of 2, what is the value of $R^2$? Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the linear function y = -2x + 3 is moved up by 2 units. Find the coordinates of the point where the new function intersects the y-axis. Express your answer as an ordered pair (x, y). | (0, 5) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Functions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the analytical expression of the parabola obtained by shifting the parabola y = 2(x-3)^2 + 2 three units to the left and then two units down. Express your answer as a simplified algebraic equation. | y = 2x^2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of ordered pairs $(m,n)$ of positive integers that satisfy the equation $\frac{4}{m}+\frac{2}{n}=1$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Let the universal set $U = \{1, 3, 5, 7\}$, and let $M = \{1, 5\}$. Find the complement of $M$ in $U$. Express your answer as a set, with elements separated by commas and enclosed in curly brackets. | \{3, 7\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the even function $f(x)$ is differentiable on $\mathbb{R}$, and $f'(1) = -2$, $f(x+2) = f(x-2)$, calculate the slope of the tangent line to the curve $y=f(x)$ at $x=-5$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of sets A that satisfy {1} ⊆ A ⊊ {1,2,3}. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $a_1=2$ and $a_{n+1}=a_n+2n$, calculate the value of $a_{100}$. Express your answer as a single integer. | 9902 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 1/x, find the value of lim (Δx → 0) [f(2 + 3Δx) - f(2)]/Δx. Express your answer as a numerical value. | -3/4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \(15 - 3 - 15\). Provide your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Each edge of a cube is increased by 50%. Calculate the percent increase of the surface area of the cube. Express your answer as a percentage. | 125 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The surface area of a cube is 96 cm². Find the volume of the cube in cubic centimeters (cm³). Express your answer as a whole number. | 64 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Let set $A=\{1,2,3\}$, $B=\{4,5\}$, $M=\{x|x=a+b, a\in A, b\in B\}$. Calculate the number of unique elements in set $M$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
When the polynomial x^3 - 2 is divided by the polynomial x^2 - 2, what is the remainder? Express your answer as a polynomial. | 2x - 2 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function
$$
f(x) = x^2 - 53x + 196 + |x^2 - 53x + 196|,
$$
calculate the value of \( f(1) + f(2) + \cdots + f(50) \). Express your answer as a single integer. | 660 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Precalculus -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the definition of the "ideal number" of a sequence as T_n = (s_1 + s_2 +... + s_n) / n, where s_n is the sum of the first n terms of the sequence, and the "ideal number" of the sequence a_1, a_2,..., a_500 is 2004, find the "ideal number" of the sequence 8, a_1, a_2,..., a_500. Express your answer as a single int... | = 2008 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the real number $b$ and the pure imaginary number $z$ satisfy the equation $(2-i)z=4-bi$ (where $i$ is the imaginary unit), calculate the value of $b$. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a\in R$, if the complex number $\left(a-2i\right)\left(3+i\right)$ is a real number, find the value of $a$. Express your answer as a single value. | 6 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vector $\overrightarrow{a} = (2, 4)$ is collinear with the vector $\overrightarrow{b} = (x, 6)$, determine the value of the real number $x$. Express your answer as a single real number. | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $ax+y+a+1=0$, regardless of the value of $a$, find the fixed point through which this line always passes. Express your answer as an ordered pair $(x, y)$. | (-1,-1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
The equation x = 0 represents a vertical line that intersects the x-axis at the origin. What is the angle of inclination of this line with respect to the positive direction of the x-axis? Express your answer in degrees. | 90 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that vector $a=(2,-1,3)$, $b=(-1,4,-2)$, and $c=(7,5,λ)$, find the value of the real number $λ$ such that vectors $a$, $b$, and $c$ are coplanar. Express your answer as a fraction. | \frac{65}{7} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the average score for six students is 92.5, the highest score is 99, and the lowest score is 76, with all scores being unique integers, calculate the minimum possible score for the student who ranks 3rd from the highest. | 95 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let $M=\{x||x-1| < 2\}$ and $N=\{x|x < 2\}$. Calculate the intersection of sets M and N, and express your answer in interval notation. | (-1, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given two sets of real numbers \( A = \{a_{1}, a_{2}, \cdots, a_{100} \} \) and \( B = \{b_{1}, b_{2}, \cdots, b_{50} \} \), if a mapping \( f \) from \( A \) to \( B \) makes every element in \( B \) have a preimage, and
$$
f(a_{1}) \leq f(a_{2}) \leq \cdots \leq f(a_{100})
$$
calculate the number of such mappings. Ex... | \binom{99}{49} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
On the number line, the point A represents the number -3. If point A is moved 7 units to the right on the number line to point B, then what is the number represented by point B? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
There are five different products arranged in a row on a shelf, among which products A and B must be arranged together, while products C and D cannot be arranged next to each other. Calculate the total number of different arrangements. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Combinatorics -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence with common difference d > 0, the sum of the first n terms is denoted by S_n. If a_1 + a_{11} = 0, then find the value(s) of n when S_n reaches its minimum value. Express your answer as a numerical value or a range of values separated by commas. | 5, 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given two-dimensional vectors $\overrightarrow {a} = (1, -2)$ and $\overrightarrow {b} = (-2, x)$, find the value of $x$ such that $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression $i(1+i)^2$. Express your answer as a complex number in the form $a+bi$ or a real number if the imaginary part is zero. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression \( S = i^n + i^{-n} \), where \( i \) is the imaginary unit and \( n \) is an integer, find the number of distinct possible values of \( S \). Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation $x^2+y^2-2x+6y+6=0$, find the center coordinates (x, y) and the radius value. Express your answer in the format (x, y), radius = r. | (1, -3), radius = 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, if $\tan A \tan B = 1$ and $AB = \sqrt{3}$, calculate the size of angle $A$ in radians when the area of $\triangle ABC$ reaches its maximum value. Express your answer in radians, rounded to 4 decimal places if necessary. | \frac{\pi}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole 24 cm across at the top and 8 cm deep. Calculate the radius of the ball. Express your answer in centimeters. | 13 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given $a = \log_{3}2$, express $\log_{3}8 - 2\log_{3}6$ in terms of $a$ as a simplified expression. | a-2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A cylinder with radius \(2 \text{ cm}\) and height \(8 \text{ cm}\) is full of water. A second cylinder of radius \(4 \text{ cm}\) and height \(8 \text{ cm}\) is empty. If all of the water is poured from the first cylinder into the second cylinder, calculate the depth of the water in the second cylinder. Express your a... | 2 \text{ cm} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the angle of inclination of the line given by the parametric equations:
\[
\begin{cases}
x = -t\cos 20^{\circ} \\
y = 3 + t\sin 20^{\circ}
\end{cases}
\]
where $t$ is a parameter. Express your answer in degrees (°). | 160^{\circ} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Let \(f(x)\) be an odd function defined on \(\mathbb{R}\), and \(f(2)=0\). When \(x > 0\), it always holds that \(xf''(x)-f(x) < 0\). Find the solution set of the inequality \(x^{2}f(x) > 0\). Express your answer in interval notation. | (-\infty, -2) \cup (0, 2) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(n) = 1 + n(n+1)/2, which describes the number of parts a plane is divided into by n intersecting lines, find the algebraic expression representing the term added to f(k) to obtain f(k+1) during the induction step of a mathematical induction proof. Express your answer as a simple algebraic expressio... | k+1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Lorri took a 240 km trip to Waterloo. On her way there, her average speed was 120 km/h. She was stopped for speeding, so on her way home her average speed was 80 km/h. What was her average speed, in km/h, for the entire round-trip? Express your answer as a whole number. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
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