instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations, the u... | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $y= \sqrt {\log_{ \frac {1}{2}}(5x-2)}$. Express your answer in interval notation. | (\frac{2}{5}, \frac{3}{5}] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given \(\overrightarrow{a}=(\cos α,1,\sin α)\) and \(\overrightarrow{b}=(\sin α,1,\cos α)\), find the angle between the vectors \(\overrightarrow{a}+\overrightarrow{b}\) and \(\overrightarrow{a}-\overrightarrow{b}\). Express your answer in degrees. | 90^\circ | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the directrix of the parabola y = -x^2. Express your answer as an equation of a line (e.g., y = k or x = k). | y = \dfrac{1}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Let $M=\{x|x^2+2x=0, x\in \mathbb{R}\}$, $N=\{x|x^2-2x=0, x\in \mathbb{R}\}$. Find the union of sets M and N, expressing your answer as a set of numbers. | \{-2, 0, 2\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of ${(x+2+\frac{1}{x})}^{3}$. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a right triangle with sides a and b, and hypotenuse c, where a:b = 1:3, a perpendicular from the vertex divides c into segments r and s, adjacent respectively to a and b. Calculate the ratio of r to s, expressing your answer in the form a:b and simplifying to its simplest form. | 1:9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the three-dimensional Cartesian coordinate system, given points A(-1,0,1) and B(5,2,1), calculate the coordinates of the midpoint of the line segment AB. Express your answer as (x, y, z). | (2, 1, 1) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.75 | 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"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $f(x)$ satisfies $f(x-1)=x^2+1$, find the value of $f(-1)$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
If the equations 3x + by + c = 0 and cx - 2y + 12 = 0 represent the same graph, find the number of pairs (b, c) that satisfy the condition. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point P(x, y) is transformed into point P'(-y+1, x+2). We call point P'(-y+1, x+2) the terminal point of point P(x, y). Given that the terminal point of point P1 is P2, the terminal point of point P2 is P3, the terminal point of point P3 is P4, and so on until Pn, if the coordinates ... | (-3, 3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Nate is driving to see his grandmother. If he drives at a constant speed of 40 km/h, he will arrive 1 hour late. If he drives at a constant speed of 60 km/h, he will arrive 1 hour early. At what constant speed should he drive to arrive just in time? Express your answer in km/h as a whole number. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A certain middle school has 3500 high school students and 1500 junior high school students. To understand the students' learning situation, a stratified sampling method is used to draw a sample of size $n$ from the students of this school. It is known that 70 high school students are drawn. Calculate the sample size $n... | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $z$ satisfies the equation $(z-2)(1+i)=1-i$, where $i$ is the imaginary unit, calculate the value of $z$. Express your answer in the form $a+bi$. | 2-i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact sum of \(0.001 + 1.01 + 0.11\), ensuring to align the decimal points correctly. Express your answer as a decimal number, including all significant digits. | 1.121 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Round the number 0.040925 to the nearest hundredth and express your answer as a decimal, accurate to the hundredths place. | 0.04 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x\left|\right.-1\leqslant x \lt 3\}$ and $B=\{-1,0,2,3\}$, find the intersection $A\cap B$. Express your answer as a set of numbers. | \{-1, 0, 2\} | big_math | [
"Mathematics -> Set Theory -> Operations on Sets"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $y=kx+1$ is tangent to the curve $y=x^{3}+ax+b$ at the point $(1,3)$, find the value of $b$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 - x - 2 where x ∈ [-5, 5], calculate the probability that a randomly chosen point x_0 will satisfy f(x_0) > 0. Express your answer as a decimal value between 0 and 1. | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x\left|\right.-1 < x < 4\}$ and $B=\{-4,1,3,5\}$, find the intersection $A\cap B$. Express your answer in standard set notation. | \{1,3\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A group of girls stands in a circle. Florence is the fourth on the left from Jess and the seventh on the right from Jess. Find the total number of girls in the group. Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
What is the total number of ways to choose three non-consecutive numbers from the set {1, 2, 3,..., 10}? Express your answer as a single integer. | 56 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions that satisfy the inequality \(x - 1 < (x - 1)^2 < 3x + 7\). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A school has a total of 24 classes across three grades. To understand the psychological state of the students, each class is numbered from 1 to 24. Now, using systematic sampling, 4 classes are selected for the survey. If the smallest number drawn is 3, find the largest number drawn. Express your answer as a whole numb... | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A television station's animated program wants to reward its enthusiastic young viewers for the week by drawing 10 lucky young spectators from 10,000 with assigned numbers. The system sampling method is used for the draw. Calculate the sampling interval required to select 10 lucky viewers from the total population of 10... | 1000 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the linear function $y = ax - 3a + 1$ must pass through a certain point. Find the coordinates of this point. Express your answer as a coordinate pair (x, y). | (3, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \frac {1}{2}x^{2}+2xf'(2017)-2017\ln x$, find the value of $f'(2017)$. Express your answer as an integer. | -2016 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given a point P on the parabola $y^2 = 4x$, find the minimum value of the sum of the distance from point P to point Q (2, 1) and the distance from point P to the focus of the parabola. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the polynomial expansion of \((1 + 2x - x^2)^4\), what is the coefficient of \(x^7\)? Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given $x > y > 0$, and $x^2 - y^2 = 1$, find the minimum value of $2x^2 + 3y^2 - 4xy$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of 10 miles per hour. Jack walks to the pool at a constant speed of 4 miles per hour. Calculate the difference in minutes between their arrival times at th... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of all values of x that satisfy the equation (x^2 - 5x + 5)^(x^2 + 4x - 60) = 1. Consider the conditions for a^b = 1 and provide the exact sum of valid x values. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x^{2}-3x=2$, calculate the value of the algebraic expression $3x^{2}-9x-7$. Express your answer as a single numerical value in the format $\boxed{[answer]}$. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x||x-3| \lt 2\}$, $B=\left\{{x|{\frac{{x+1}}{{x-2}}≤0}}\right\}$, calculate the union $A\cup B$. Express your answer as an interval, including any relevant brackets or parentheses. | [-1, 5) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = x^4 that is perpendicular to the line x + 4y - 8 = 0. Express your answer in the standard form of a line equation (Ax + By + C = 0), including all coefficients and constants. | 4x - y - 3 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given a quadratic function f(x) = ax^2 + bx + c with its derivative f'(x), and f'(0) > 0, for any real number x, if f(x) >= 0, find the minimum value of f(1)/f''(0). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the opposite number of $(-2)^3$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(5-\sqrt{y^2-25})^2$. Provide your answer as a simplified algebraic expression. | y^2 - 10\sqrt{y^2-25} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Sacha places 25 counters into 14 boxes so that each box contains 1, 2, or 3 counters. No box is inside any other box. Seven boxes contain 1 counter. Find the number of boxes that contain 3 counters. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Two students, A and B, each choose 2 out of 6 extracurricular reading materials. Calculate the number of ways in which the two students can choose their materials such that they have exactly 1 material in common. Express your answer as a whole number, including the unit 'ways'. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1, where x is a real number, find the value of x such that f(x) = 1. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
The parabola y^2 = -4x opens to the left, and its directrix equation is x = 1. If the distance from a point P on the parabola to the y-axis is 5, find the distance from point P to the focus of this parabola. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $3 \sqrt {12} \div 3 \sqrt {\frac {1}{3}} - 2 \sqrt {3}$ and express your answer in simplest radical form. | 6 - 2 \sqrt {3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, if $3\sin B\cos C = \sin C(1-3\cos B)$, then find the ratio $\sin C : \sin A$. Express your answer as a ratio of integers, using a colon (:) to separate the two parts of the ratio. | 3:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation $x^2+y^2+2x-3=0$ and the parabola equation $y^2=2px$ ($p>0$), find the value of $p$ such that the length of the line segment cut by the circle on the directrix of the parabola is 4. Express your answer as a single positive number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (x-2)(x+3) = x^2 + ax + b, calculate the values of a and b. Express your answer as a pair of values, (a, b). | (1, -6) | big_math | [
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given $f(x)=2x+3$ and $g(x)=4x-5$, determine the expression for $h(x)$ such that $f(h(x))=g(x)$. Express your answer as a function of x. | h(x)=2x-4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors AB = (4, -2, 3), AD = (-4, 1, 0), and AP = (-6, 2, -8), calculate the height h from point P to the plane ABCD. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
If \( f(x) \) is an odd function periodic with a period of 5 on \( \mathbf{R} \), and \( f(7) = 9 \), calculate the value of \( f(2020) - f(2018) \). Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the probability distribution of the random variable $\xi$ as $P(\xi=k)= \frac{1}{2^{k}}$, where $k=1$, $2$, $...$, determine the value of $P(2 < \xi\leqslant 4)$. Express your answer as a simplified fraction. | \frac{3}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given two points O(0,0) and Q(a,b), point P1 is the midpoint of the line segment OQ, point P2 is the midpoint of the line segment QP1, P3 is the midpoint of the line segment P1P2, and so on, Pn+2 is the midpoint of the line segment PnPn+1. Calculate the limit position of point Pn as n approaches infinity. Express your ... | \left( \frac{2a}{3}, \frac{2b}{3} \right) | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the intersection point of the graph of the linear function y = 5x + 1 with the y-axis. Express your answer as an ordered pair (x, y) with integer coordinates. | (0, 1) | big_math | [
"Mathematics -> Algebra -> Algebra -> Linear Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) shares its foci with the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$, and that their eccentricities are reciprocal to each other, find the minimum value of $\frac{|PF_1|^2}{|PF_2|}$, where $F_1$ and $F_2$ are the left and right foci of th... | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, square ABCD has a side length of 4 units. Find the total area of the shaded regions. Express your answer in square units as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Nine lines parallel to the base of a triangle divide the other sides each into 10 equal segments and the area into 10 distinct parts. If the area of the largest of these parts is 38, then find the area of the original triangle. | 200 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
How many positive roots does the polynomial $x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0$ have, where $a_{2001} = 2002$ and $a_k = -k - 1$ for $0\leq k \leq 2000$? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Let $P(x)=x^2+bx+c$. Suppose $P(P(1))=P(P(-2))=0$ and $P(1)\neq P(-2)$. Calculate $P(0)$. Express your answer as a single fraction or integer, including a negative sign if applicable. | -\frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A certain number divided by 9 yields a quotient of 6 and a remainder of 4. What is the number? Express your answer as a whole number. | 58 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the dataset $1$, $2$, $3$, $4$, $3$, $5$, what is the mode? Provide your answer as the number that appears most frequently. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the lengths of two sides of a triangle are $3cm$ and $8cm$ respectively, calculate the range of possible lengths for the third side of this triangle using the triangle inequality theorem. Express your answer as a range in centimeters (cm). | (5, 11) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select three numbers $a$, $b$, and $c$ in the interval $[0,1]$. Calculate the probability that the inequality $a^{2}+b^{2}+c^{2}\leqslant 1$ holds. Express your answer as a numerical value. | \dfrac {\pi}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
If there are 4 different red envelopes in a WeChat group of 3 people, and each red envelope can only be grabbed once, and each person must grab at least one red envelope, calculate the total number of ways the red envelopes can be grabbed. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $M$ is a moving point on the circle $C: x^2+y^2=1$, and point $N(2,0)$, derive the equation of the trajectory of the midpoint $P$ of $MN$. Express your answer in the standard form of a circle equation, including the center coordinates and radius. | (x-1)^2+y^2= \frac {1}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain kingdom, numbers are expressed in base r. A person buys a car for 440 monetary units (m.u.) and gives the seller 1000 m.u., receiving 340 m.u. in change. What is the value of r? Express your answer as a positive integer. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a triangle with side lengths satisfying a = 3 and b + c = 5, find the maximum value of the area of this triangle using the Heron-Qin Jiushao formula. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sqrt{2m+1}$ is meaningful, find the smallest integer value that $m$ can take. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = x - \ln x$, find its derivative $f'(x)$. Express your answer as a mathematical expression for the derivative, using standard notation. | \frac{x-1}{x} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A man was x years old in the year x^2. If he was born in the first half of the 19th century, calculate his year of birth. Express your answer as a single year. | 1806 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Let $P$ be the increase in circumference of a circle resulting from an increase of $\pi$ units in the diameter. Calculate the value of $P$. | \pi^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
When two dice are rolled simultaneously, what is the probability that the sum of the points obtained is 5? Express your answer as a simplified fraction. | \dfrac{1}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x + 1/x = 4, find the value of x^3 + 1/x^3. Express your answer as a single number. | 52 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Convert 1200 minutes to degrees. Express your answer in degrees. | 20^{\circ} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, what are the coordinates of the point P with respect to the origin, given that P is located at (-2, -4)? Express your answer as an ordered pair (x, y). | (-2, -4) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Among the six numbers $(-3)^2$, $-2^2$, $|-2|$, $(-1)^3$, $-|-2|$, $(-1)^{2n-1}$ (where $n$ is a positive integer), how many are negative? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
In an 8x8 chessboard, which consists of 64 squares, calculate the total number of squares (of different sizes or positions) that are composed of an integer number of smaller squares. Express your answer as a whole number. | 204 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors OA = a, OB = b, OC = c, and the condition AC = 3BC, express c in terms of a and b. Provide your answer as a vector equation, including all necessary operations and coefficients. | -\frac{1}{2}a + \frac{3}{2}b | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a data set with a maximum value of 145, a minimum value of 50, and an interval of 10, calculate the number of groups the data can be divided into. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the variance of the data $a_1, a_2, a_3...a_n$ is $\sigma^{2}$, what is the variance of the data $2a_1, 2a_2, 2a_3...2a_n$? Express your answer in terms of the original variance $\sigma^{2}$. | 4\sigma^{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A slanted line segment in a plane is twice the length of its projection on the plane. Find the angle between the slanted line and the plane. Express your answer in degrees as a numerical value. | 60 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Select 3 people from a group of 5 people with different heights and arrange them in a line from left to right in descending order of height. Calculate the total number of ways to satisfy this condition. Express your answer as a single numerical value. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $S_{n}$ represents the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$, and $S_{7}=28$, find the value of $a_{4}$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} where a_1 = 3 and a_{n+1} = 1/(a_n - 1) + 1, calculate the value of a_{2014}. Express your answer as a fraction. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of real solutions to the equation x = sin(x) + 1993. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is a root of the function $f(x)=2^{x}-\log_{\frac{1}{2}}x$, if $0 < x_{0} < a$, determine the sign of $f(x_{0})$. Express your answer in the format 'f(x0) < 0', 'f(x0) > 0', or 'f(x0) = 0'. | f(x0) < 0 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the square root of 81. Provide both positive and negative roots, if applicable. Express your answer in the form of ±n, where n is the absolute value of the root. | ±9 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with common difference d and a geometric sequence {b_n} with common ratio q, where d is not equal to 0 and q is not equal to 1, it is known that the sum of the first n terms of {a_n} is equal to the sum of the first 2011-n terms of {a_n} for every positive integer n less than 2011, an... | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). Find the reflex angle at \( G \). Express your answer in degrees as a whole number. | 340 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let a and b be positive real numbers. Given the conditions:
1. 1/a + 1/b ≤ 2√2
2. (a-b)^2 = 4(ab)^3
Calculate the value of log_a b. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $m$ for which the equation $\frac{x-3}{x-2}=\frac{m}{2-x}$ has no solution. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.111111 | SynthLabsAI/Big-Math-RL-Verified |
The line $l$: $x-y+1=0$ is rotated counterclockwise by $90^{\circ}$ around point $A(2,3)$, obtaining line $l_{1}$. Find the equation of line $l_{1}$ in the form $Ax + By + C = 0$. | x+y-5=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Lines"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence {a_n} satisfies a_n + a_{n+1} = 1/2, and a_2 = 2. Find the sum of the first 21 terms of the sequence, S_{21}. Express your answer as a simplified fraction. | 7/2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of monomials is given by $a$, $3a^{2}$, $5a^{3}$, $7a^{4}$, $9a^{5}$, $\ldots$. Based on the pattern observed in the sequence, express the nth monomial as a formula in terms of n and a. | (2n - 1)a^{n} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $P=\{0,1\}$ and $Q=\{-1,0,1\}$, and a mapping $f$ from $P$ to $Q$, count the number of mappings that satisfy the condition $f(0) > f(1)$. Provide your answer as a single integer. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $M=\{x|x^{2}-1\leqslant 0\}$ and $N=\{x|x^{2}-3x \gt 0\}$, find the intersection of sets $M$ and $N$. Express your answer in interval notation. | [-1, 0) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A dolphin swims freely in a rectangular pool that is $30 m$ long and $20 m$ wide. What is the probability that the dolphin's snout is no more than $2 m$ away from the edge of the pool? Express your answer as a simplified fraction. | \frac{23}{75} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that a certain unit has 120 employees, of which 90 are male, a stratified sampling (stratified by gender) is used to draw a sample. If it is known that there are 18 male employees in the sample, then calculate the total size of the sample. Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A jar contains quarters (worth $0.25 each), nickels (worth $0.05 each), and pennies (worth $0.01 each). The value of the quarters is $10.00. The value of the nickels is $10.00. The value of the pennies is $10.00. If Judith randomly chooses one coin from the jar, what is the probability that it is a quarter? Express you... | \frac{1}{31} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $1101_2$ to its decimal equivalent. Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A certain brand of soap originally sells for 2 yuan per piece. For purchases of more than 2 pieces (including 2 pieces), the store offers two promotional sales methods. The first method: one piece of soap is sold at the original price, and the rest are sold at 70% of the original price; the second method: all are sold ... | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
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