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 equation $x^{2}+mx+6=0$ with one root being $-2$, calculate the value of the other root. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Convert the quinary number $444_{(5)}$ to an octal number. Express your answer in octal format with a subscript (e.g., 174_{(8)}). | 174_{(8)} | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Divide 996 jin of cotton among eight sons for their expenses. Distribute it in such a way that each son receives 17 jin more than his elder brother. How much cotton does the eighth son receive? Express your answer in jin as a whole number. | 184 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $abc^{2}$ in the expansion of $(a+2b-3c)^{4}$. Express your answer as a single integer. | 216 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
If $\left(3b+a\right)\left(\ \ \right)=9b^{2}-a^{2}$, find the algebraic expression that fills in the parentheses. Express your answer as an algebraic expression involving variables a and b. | 3b-a | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with all positive terms, where the sum of the first 20 terms is 100, find the maximum value of $a_3 \cdot a_{18}$. Express your answer as a numerical value. | 25 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a function $y=f(x)$ defined on the interval $[-1,1]$ with a range of $[-2,0]$, determine the range of the function $y=f(\cos 2x)$. Express your answer as an interval. | [-2,0] | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given two random variables X and Y, where X follows a binomial distribution with parameters n=8 and p=1/2, and Y follows a normal distribution with mean μ and variance σ^2. If μ = E(X) and P(Y < 0) = 0.2, calculate the probability P(4 ≤ Y ≤ 8). Express your answer as a decimal value between 0 and 1, rounded to 1 decima... | 0.3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function μ = √(2x + 1) + √(3y + 4) + √(5z + 6) and the constraint 2x + 3y + 5z = 29, find the maximum value of μ. Express your answer as a boxed numerical value. | 2\sqrt{30} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the definite integral $\int_{1}^{2}{4xdx}$ and express your answer as a single number. | 6 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{bx(a^2x^2 + 2a^2y^2 + b^2y^2) + ay(a^2x^2 + 2b^2x^2 + b^2y^2)}{bx + ay}$. Express your answer as a simplified algebraic expression. | (ax + by)^2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A middle school radio station randomly selects two festivals from the five traditional Chinese festivals: Spring Festival, Lantern Festival, Qingming Festival, Dragon Boat Festival, and Mid-Autumn Festival. Calculate the probability that at least one of the Spring Festival and Dragon Boat Festival is selected. Express ... | 0.7 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, m+2)$ and $\overrightarrow{b} = (m, -1)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, find the magnitude of $\overrightarrow{b}$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of real roots of the equation
$$
\sqrt{x^{2}+2x-63}+\sqrt{x+9}-\sqrt{7-x}+x+13=0.
$$
Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given x = 2, y = x^2 - 5, and z = y^2 - 5, calculate the value of z. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, if the complex number $z$ satisfies $(1+i)z=1-i$, calculate the conjugate of $z$, denoted as $\overline{z}$. Express your answer as a complex number in the form $a+bi$. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of real roots of the equation $x + \sqrt{x-2} = 4$. Provide your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Applied to a bill for $\textdollar{10,000}$, calculate the difference in dollars between a single discount of $40$% and two successive discounts of $36$% and $4$%. Express your answer in dollars, including the dollar sign. | \textdollar{144} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of f(x) is R, y=f(2x-1) is an odd function, and y=f(x+1) is an even function, and knowing that f(x)=e^x when x∈(-1,1), calculate the value of f(194). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The hands of a clock moved from 0 hours 45 minutes to 3 hours 30 minutes. Calculate the minimum time that has passed. Express your answer in hours and minutes (e.g., X hours Y minutes). | 2 hours 45 minutes | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A die is rolled twice. What is the probability that the number of points on the second roll is twice that of the first roll? Express your answer as a simplified fraction. | \frac{1}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A piece of paper with a thickness of 0.1mm is continuously folded, with each fold doubling its thickness. What is the minimum number of folds needed for the total thickness of the folded paper to exceed 12mm? Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the line $kx-y+1=3k$, where $k$ is a variable parameter, find the fixed point through which all lines pass. Express your answer as an ordered pair (x, y). | (3, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, if $(1+i)^2 - \frac{a}{i}$ is a real number, calculate the value of the real number $a$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve f(x) = 2x - e^x at the point (0, f(0)). Express your answer as a linear equation in standard form, including all necessary variables (x and y). | x - y - 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
For the expression \(\left[\left(\frac{\sqrt{3}+i}{2}\right)^{8}+1\right]^{n}\), where n takes on values from 1 to 100, calculate the number of distinct values this expression can take. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Complex Numbers -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the cubic equation $x^{3}-ax^{2}+1=0$ and the condition $a > 3$, find the number of real roots of the equation in the interval $(0,2)$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = (x+1)(x+2)(x+3)(x+4) + 35. Express your answer as a single number. | 34 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the zero of the function y = 4x - 2. Express your answer as a single number. | 1/2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the contrapositive of the proposition "If $a > b$, then $a - 8 > b - 8$". Express your answer as a logical statement in the form "If..., then...". | If $a - 8 \leq b - 8$, then $a \leq b$. | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
There are six classes in the second grade of a certain school. Four students are transferred from other places and need to be arranged into two of these classes, with two students in each class. Calculate the total number of different arrangement plans possible. Express your answer as a single integer. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
A clothing merchant sells two sets of clothes at the same time, with a selling price of 168 yuan per set. One set makes a 20% profit, while the other set incurs a 20% loss. Calculate the overall profit or loss in yuan. If the result is a loss, include a negative sign in your answer. | -14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In the right triangle ABC, side AC is divided into 8 equal parts, and from each division point, a segment parallel to BC is drawn, resulting in a total of 7 segments. Given that BC = 10, find the sum of the lengths of these 7 segments. Express your answer as a whole number. | 35 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $A=\{x\in \mathbb{N} | -2 \lt x \lt 2\}$ and $B=\{-1,0,1,2\}$, find the intersection $A\cap B$. Express your answer in standard set notation. | \{0,1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, 2)$ and $\overrightarrow{b} = (-3, 0)$, if $(2\overrightarrow{a} + \overrightarrow{b}) \parallel (\overrightarrow{a} - m\overrightarrow{b})$, calculate the value of $m$. Express your answer as a single numerical value, including the negative sign if applicable. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $2x-8=0$ for $x$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangle is divided into three smaller congruent rectangles. Each smaller rectangle is similar to the large rectangle. Find the ratio of the length of a longer side to that of a shorter side in each of the four rectangles. Express your answer as a simplified ratio (e.g., x:y). | \sqrt{3} : 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x\left|\right.-2 \lt x \lt 1\}$ and $B=\{x\left|\right.-1\leqslant x\leqslant 2\}$, find the intersection $A\cap B$. Express your answer in interval notation. | \left[-1,1\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
The point M(x, y) lies on the line x+y-10=0, and x, y satisfy -5 ≤ x-y ≤ 5. Calculate the range of values for √(x² + y²). Express your answer as an interval [min, max]. | [5\sqrt{2}, \frac{5\sqrt{10}}{2}] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let $a$ be a real number, and let the function $f(x) = x^3 + ax^2 + (a-3)x$ have the derivative $f'(x)$. Given that $f'(x)$ is an even function, find the equation of the tangent line to the curve $y = f(x)$ at the origin. Express your answer in the form $y = mx + b$, where $m$ is the slope of the tangent line and $b$ i... | y = -3x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number \[Z = \frac {4+2i}{(1+i)^{2}}\] (where \(i\) is the imaginary unit), the point corresponding to \(Z\) in the complex plane lies on the line \(x-2y+m=0\). Calculate the value of \(m\). Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2^x + 2^y = 1$, calculate the range of values for $x + y$. Express your answer as an interval. | (-\infty, -2] | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Xiamen No.1 Middle School plans to set up a display board showcasing the achievements of an outstanding young alumni at each of its two campuses. Five student volunteers, including A and B, will assist in setting up the display boards. Each person will participate in and only participate in setting up one display board... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Six athletes are preparing to compete on 6 tracks, where athlete A cannot stand on the first or second track, and athlete B must stand on either the fifth or sixth track. Calculate the total number of different arrangements. Express your answer as a single integer. | 144 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Matilda and Ellie divide a white wall in their bedroom in half, each taking half of the wall. Matilda paints half of her section red. Ellie paints one third of her section red. Find the fraction of the entire wall that is painted red. Express your answer as a simplified fraction. | \frac{5}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given an infinite geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, if $S_{6}=3S_{3}$, calculate the value of $\frac{a_{7} + a_{9}}{a_{1} + a_{3}}$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The perimeter of a rectangle is 30. If one side of the rectangle is represented by the letter x, express the length of the other side in terms of x. | 15 - x | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $\frac {2i}{1-i}+2$. Express your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In 1998, the population of Canada was 30.3 million. Express 30.3 million as a single integer. Provide your answer in the form \boxed{ ext{your answer here}}. | 30300000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse x^2/a^2 + y^2 = 1 and the point P(1, sqrt(6)/3) that lies on the ellipse, calculate the eccentricity e of the ellipse. Express your answer as a fraction or decimal and include the necessary calculations. | \dfrac{\sqrt{6}}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains 6 balls, among which there are 3 red balls and 3 white balls. Two balls are drawn successively without replacement. Given that the first ball drawn is red, find the probability of drawing a red ball again on the second draw. Express your answer as a simplified fraction. | \dfrac {2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given $x \in \left(-\frac{\pi}{2}, 0\right)$ and $\cos x = \frac{4}{5}$, find the value of $\tan 2x$. Express your answer as a fraction or decimal. | -\frac{24}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
For the function f(x) = (x^2 - 2x + 2) / (2x - 2) and the interval -4 < x < 1, find the maximum value of f(x). Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Let $ABC$ be a triangle with circumradius $1$. If the center of the circle passing through $A$, $C$, and the orthocenter of $\triangle ABC$ lies on the circumcircle of $\triangle ABC$, what is the length of $AC$? Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In a Huangpu River transportation and sightseeing map with a scale of 1:38000, a certain tunnel is about 7 cm long. Calculate the actual length of the tunnel in kilometers. Express your answer in kilometers (km), rounded to 2 decimal places. | 2.66 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, points R and S lie on PT and PQ, respectively. If ∠PQR = 90°, ∠QRT = 158°, and ∠PRS = ∠QRS, what is the measure of ∠QSR? Express your answer in degrees as a whole number. | 79^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A right triangle has a hypotenuse of 10 inches and an inscribed circle with a radius of 1 inch. Find the perimeter of the triangle. Express your answer in inches as a whole number. | 24 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 3 interest groups, and students A and B each join one of them, with an equal probability of choosing any of the groups. Find the number of ways that A and B can join different groups. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\left\{{\begin{array}{l}{x\sin x, x\geq 0}\\{f(x+\pi), x<0}\end{array}}\right.$, find the derivative $f'(x)$ and evaluate it at $x = -\frac{3\pi}{2}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_3 + a_8 > 0, and S_9 < 0. Find the smallest among S_1, S_2,..., S_9. Express your answer in the form S_n (e.g. S_5). | S_5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of $f(x)$ is $\left[1,3\right]$, determine the domain of the function $g(x)=\frac{{f(3x-2)}}{{2x-3}}$. Express your answer in interval notation, including any necessary union or intersection symbols. | [1,\frac{3}{2}) \cup (\frac{3}{2},\frac{5}{3}] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given positive integers a and b such that 4a + b = 30, find the pair (a, b) that minimizes the value of 1/a + 1/b. Express your answer as a pair of integers (a, b). | (5,10) | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition "$\exists x > 0$, $|x| + x \geqslant 0$", what is its negation? Express your answer as a logical statement using proper quantification and mathematical notation. | $\forall x > 0$, $|x| + x < 0$ | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of two numbers and their quotient are both 6, calculate the value of the product of these two numbers minus their difference (greater minus smaller). Express your answer as a fraction in simplest form. | \frac{6}{49} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence \( \{a_n\} \), if \( a_5 + a_6 = 10 \), then find the sum of the first 10 terms, \( S_{10} \). Express your answer as a single numerical value. | 50 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A star player gives all 5 identical small gifts to 3 different fans, ensuring each fan receives at least one gift. Find the total number of different ways to distribute the gifts. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer pairs (x, y) that satisfy the equation xy - 3x + 5y = 0. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
If $x^2 - (m+1)x + 1$ is a perfect square, find the value(s) of $m$. Express your answer as a numerical value or a list of numerical values if more than one solution exists. | 1, -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, a complex number z satisfies the equation (1-i) * z = 2. Solve for z. Express your answer as a complex number in the form a + bi. | 1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x + 3 = 10, find the value of 5x + 15. Express your answer as a single integer. | 50 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function $f(x) = 2\sin x\cos x\cos 2x$. Express your answer in terms of $\pi$. | \dfrac{\pi}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Let $a \in \mathbb{R}$, and $z = (a+1) + (1+a^2)i$. If the complex number $z$ is purely imaginary, find the value of $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
What 2D plane figure is most analogous to a parallelepiped in space, considering the geometric characteristics of a parallelepiped? Express your answer as the name of a specific 2D shape. | Parallelogram | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{1,2,3\}$ and $\complement_{U}A=\{2\}$, find the number of proper subsets of set $A$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Determine whether the condition q = 0 is necessary and sufficient for the sequence with sum formula S_n = An^2 + Bn + q to be an arithmetic sequence. Provide your answer as a direct assessment of the condition. | The condition q = 0 is necessary and sufficient. | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In right triangle ABC with angle C = 90 degrees, side AC is divided into 8 equal parts. Seven line segments are drawn parallel to BC from these points of division to AB. Given BC = 10, find the sum of the lengths of the seven line segments. Express your answer as a whole number. | 35 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains red, black, and white balls. The probability of drawing a red ball is 0.4, and the probability of drawing a black ball is 0.25. Find the probability of drawing a white ball. Express your answer as a decimal value between 0 and 1. | 0.35 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of the expression $-1÷(-5)÷(-\frac{1}{5})$. Provide your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \dfrac {a+i}{i}$, and $z\in \mathbb{R}$, find the real value of $a$. Express your answer as a single real number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^{3}y^{3}$ in the expansion of $(x+\\frac{{y}^{2}}{x})(x+y)^{5}$. Express your answer as a numerical value. | 15 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^3+x^2+x=-1$, find the value of $x^{-28}+x^{-27}+\ldots+x^{-2}+x^{-1}+1+x^{1}+x^{2}+\ldots+x^{27}+x^{28}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
If the function $f(x) = 1 + |x| + x^3$, calculate the value of $$f(\log_2{2}) + f(\log_2{\frac{1}{2}}) + f(\log_2{5}) + f(\log_2{\frac{1}{5}})$$ and express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\cos \frac {πx}{4}$, and the set $A={2,3,4,5,6}$, two different numbers $m$ and $n$ are randomly selected from set $A$. Calculate the probability that $f(m)⋅f(n)≠0$. Express your answer as a simplified fraction. | \frac {3}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^{2}+y^{2}-x+y+m=0$, find the range of values for $m$ such that the equation represents a circle. Express your answer as an inequality involving $m$. | m < \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given $0 \leqslant \alpha \leqslant \frac{\pi}{2}, 0<\beta \leqslant \frac{\pi}{2}$, find the minimum value of $\cos ^{2} \alpha \sin \beta+\frac{1}{\sin \beta}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number (1+i)(1-ai) is real, find the value of the real number a. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation x(x(x+1)+2)+3 = x^3 + x^2 + x - 6. Express your answer as a single integer. | -9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given a right-angled triangle with sides a, a+d, and a+2d, where a and d are both positive numbers, find the ratio of a to d. Express your answer as a ratio a:d in simplest form. | 3:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
The equation of a circle is given by $x^2+y^2-4x+6y=0$. Find the coordinates of the center of the circle. Express your answer as a coordinate pair (x, y). | (2,-3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The fixed telephone local call charge rule is: the first three minutes cost 0.22 yuan (if less than three minutes, it is charged as three minutes), and thereafter, each minute costs 0.11 yuan (if less than one minute, it is charged as one minute). Calculate the charge for a local call that lasts 550 seconds. Express yo... | 0.99 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\log_x 8 = 3$, find the value of $x$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
There are two types of products, with pass rates of 0.90 and 0.95, respectively. One item of each type is taken for inspection. Calculate the probability that exactly one item fails the inspection. Express your answer as a decimal value. | 0.14 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance between the two parallel lines given by the equations $3x-4y-5=0$ and $3x-4y+5=0$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In a class with 21 students, at least two of any three students are friends. What is the largest possible value of k such that there exists a student with at least k friends? Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the internal angle that guarantees the similarity of two isosceles triangles. Express your answer as a single angle measurement in degrees. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a} = (-3, 1)$ and $\overrightarrow{b} = (-1, 2)$, solve for the real numbers $m$ and $n$ in the equation $m\overrightarrow{a} - n\overrightarrow{b} = (10, 0)$. Express your answer as a pair of real numbers $(m, n)$. | (-4, -2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression 2.017 × 2016 - 10.16 × 201.7. Provide your answer as a single number. | 2017 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If $i$ is the imaginary unit, calculate the value of the complex number $(3+2i)i$. Express your answer as a complex number in the form $a+bi$. | -2+3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Snacks are purchased for 17 soccer players. Juice boxes come in packs of 3 and cost $2.00 per pack. Apples come in bags of 5 and cost $4.00 per bag. Calculate the minimum amount of money that Danny spends to ensure every player gets a juice box and an apple. Express your answer as a dollar amount, rounded to two decima... | 28.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 2023 as a sum of numbers using only the digit 7, using a 7 a total of 19 times. Calculate the number of times the number 77 occurs in the sum. Provide your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
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