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 |
|---|---|---|---|---|---|
Find the maximum value of the function $f(x)=x^{3}-3x^{2}-9x+2$ on the interval $[-2,2]$. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(\\sin \\alpha + \\cos \\alpha) = \\sin \\alpha \\cos \\alpha$, find the value of $f(\\sin \\frac{\\pi}{6})$. Express your answer as a single fraction. | -\\frac{3}{8} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Starting with some gold coins and some empty treasure chests, I tried to put 9 gold coins in each treasure chest, but that left 2 treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had 3 gold coins left over. Calculate the total number of gold coins I have. Express your answer as a ... | 45 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In a dark box, there are $a$ balls with all the same characteristics except for color, and among these $a$ balls, only $3$ are red. After stirring the balls evenly, one ball is randomly picked, its color is noted, and then it is put back into the dark box. After conducting a large number of ball-picking experiments, it... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given a quadratic equation in x, x^2 + 2x + k = 0, with two distinct real roots, find the range of k. Express your answer as an inequality. | k < 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Let the real number $a \in (0,1)$. Find the probability that the function $f(x)=x^{2}-(2a+1)x+a^{2}+1$ has a root. Express your answer as a fraction. | \dfrac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(-2,1)$ and $\overrightarrow{b}=(-2,-3)$, calculate the projection vector of $\overrightarrow{b}$ onto $\overrightarrow{a}$. Express your answer as an ordered pair (x, y). | (-\frac{2}{5}, \frac{1}{5}) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the radius of circle $C$ is $3$, and a point $D$ on the diameter $AB$ satisfies $\overrightarrow{AB} = 3\overrightarrow{AD}$, $E$ and $F$ are the endpoints of another diameter. Calculate the dot product of vectors $\overrightarrow{DE}$ and $\overrightarrow{DF}$. Express your answer as a single numerical valu... | -8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $|\overrightarrow{a}|=2$, and $|\overrightarrow{b}|=1$, find the magnitude of $\overrightarrow{a}-2\overrightarrow{b}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Applied Mathematics -> Other"
] | 0.359375 | 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 -> Statistics -> Probability"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{AB}=(3,-4)$, calculate the magnitude of this vector, denoted as $|\overrightarrow{AB}|$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Three archers shoot two arrows each and record scores as follows:
First archer: 1 arrow in C and 1 arrow in A for a score of 15 points
Second archer: 1 arrow in C and 1 arrow in B for a score of 18 points
Third archer: 1 arrow in B and 1 arrow in A for a score of 13 points
If a fourth archer shoots 2 arrows into ... | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $$(2a^{m} \cdot b^{m+n})^{3} = 8a^{9}b^{15}$$, find the values of m and n. Express your answer as values for m and n, separated by a comma. | 3,2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{1, 2, 3\}$ and $B=\{y\ |\ y=2x-1, x\in A\}$, calculate the intersection $A \cap B$. Express your answer as a set of numbers using standard set notation. | \{1, 3\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
In the rectangular coordinate system xOy, points A and B lie on the curve y=1/x in the first quadrant. If triangle AOB is an equilateral triangle, what is its area S? Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence {a_n} is an arithmetic sequence, and S_n is the sum of the first n terms, and a_2 = 3a_4 - 6, find the value of S_9. Express your answer as a single numerical value. | 27 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the smallest positive period of the function $y=\sin\left(2x-\frac{\pi}{6}\right)$. Express your answer in terms of pi. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation defined as \\( \\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = ad - bc \\), let \\( z = \\begin{vmatrix} 1 & 2 \\\\ i & i^{4} \\end{vmatrix} \\) (where \\( i \\) is the imaginary unit). Determine the quadrant of the point corresponding to the complex number \\( \\overset{.}{z} \\) in the complex ... | IV | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = √3sin2x - 2cos^2x on the interval [0, π/2]. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the XOY plane of the Cartesian coordinate system, the parallel lines $(x=n)$, $(n=0,1,2…5)$ and the parallel lines $(y=n)$, $(n=0,1,2…5)$ form rectangles. Count the total number of rectangles formed. Express your answer as a single integer. | 225 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Counting Methods"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Starting from a point on one of the edges of a polygon (not a vertex), connecting all the vertices results in 2022 triangles. Find the number of sides of this polygon. Express your answer as a whole number. | 2023 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $I=\{x|x^{2} < 9,x\in \mathbb{Z}\}$, $A=\{1,2\}$, and $B=\{-2,-1,2\}$, find $A\cup(\complement_{I} B)$. Express your answer as a set of integers, including all elements of the resulting set. | \{0,1,2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} satisfying a_1 = 1, a_2 = 2, a_{n+2} = (1 + cos^2(n*pi/2))*a_n + sin^2(n*pi/2), find the sum of the first 12 terms of this sequence. Express your answer as a single integer. | 147 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)$ ($x \in \mathbb{R}$) satisfies $f(1)=1$, and the derivative of $f(x)$, $f'(x) < \frac{1}{3}$, find the solution set of the inequality $f(x^2) > \frac{x^2}{3}+\frac{2}{3}$. Express your answer as an interval or set of intervals. | (-1,1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, P is on RS so that QP bisects ∠SQR. Also, PQ = PR, ∠RSQ = 2y°, and ∠RPQ = 3y°. Find the measure of ∠RPQ in degrees. | 108 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the coefficients of the odd powers of x in the expansion of (a+x)(1+x)^4 is 32. Calculate the value of a. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given two planar vectors $a$ and $b$ with an angle of $\dfrac{\pi}{3}$ between them, and $|a|=1$, $|b|=\dfrac{1}{2}$, calculate the magnitude of the vector $a-2b$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
A tire company's quality inspection department wants to inspect the width (unit: mm) of a batch of tires. If at least 2 out of 3 tires randomly selected from this batch have a width within 195±3, then this batch of tires is considered basically qualified. The widths of this batch of tires are 195, 196, 190, 194, 200, r... | \frac{7}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If $A=\{(2,-2), (2,2)\}$, then how many elements are in set $A$? Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of $a$ such that the inequality $x^2+ax+1\geq0$ holds for all $x \in (0, \frac{1}{2})$. Express your answer as a single number. | -\frac{5}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Scientist Konstantin Tsiolkovsky proposed in 1903 that the maximum velocity $v$ of a single-stage rocket in an ideal situation without considering air resistance and Earth's gravity satisfies the formula: $v={v_0}\ln\frac{{m_1}+{m_2}}{{m_1}}$, where $m_{1}$ and $m_{2}$ are the structural mass of the rocket and the mass... | 3.6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} where a_1 = -1 and a_{n+1} = a_n - 3, calculate the value of a_4. Express your answer as an integer in a boxed format. | -10 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Sequences and Series"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, angle A = 45 degrees, angle B = 60 degrees, and side a = 10 units. Using the Law of Sines, calculate the length of side b. Express your answer as a simplified radical. | 5 \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the regression equation $\overset{\wedge}{y} = 2x + 1$, and a set of data obtained from an experiment is $(2, 5.1), (3, 6.9), (4, 9.1)$, calculate the sum of squared residuals. Express your answer to two decimal places. | 0.03 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $\frac{1}{\sin 10^{\circ}} - \frac{\sqrt{3}}{\sin 80^{\circ}}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A hyperbola passes through the point (m, n) where m > n > 0, and its asymptote equations are y = ±x. Determine the axis on which the foci of the hyperbola are located. | The foci of the hyperbola are located on the x-axis. | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of four-digit positive integers that are divisible by both 12 and 20, but are not divisible by 16. Express your answer as a whole number. | 113 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, the measure of angle ABC is 50 degrees, AD bisects angle BAC, and DC bisects angle BCA. Find the measure of angle ADC. Express your answer in degrees as a whole number or decimal if necessary. | 115^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} that satisfies the equation a_3 + a_5 + a_11 + a_13 = 80, find the value of a_8. Express your answer as a single numerical value. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
If $\alpha$ is an angle in the fourth quadrant and $\tan \alpha =-\frac{5}{12}$, find the value of $\sin \alpha$. Express your answer as a fraction, including the sign (positive or negative). | -\frac{5}{13} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In the rectangular coordinate system $O-xyz$, find the symmetric point of point $(1, 2, -2)$ about point $(-1, 0, 1)$. Express your answer as a coordinate (x, y, z). | (-3, -2, 4) | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations of two lines, x + (m+1)y - 1 = 0 and mx + 2y - 1 = 0, find the value of m that makes these lines parallel. Express your answer as a single value. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Let $x$ be a real number, $P=e^x+e^{-x}$, $Q=(\sin x+\cos x)^2$. Determine the relationship between $P$ and $Q$, expressing your answer as an inequality or equality relationship between $P$ and $Q$. | P \geq Q | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$ satisfies $a_1=2$, $a_{n+1}= \frac{a_n-1}{a_n+1} (n\in\mathbb{N}^*)$, calculate the value of $a_{2017}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of integers greater than -2.8 but not greater than π. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that a positive integer N divided by 60 leaves a remainder of 49, find the remainder when N is divided by 15. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.796875 | 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. How many times will the number 77 occur 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 |
Solve the system of equations y = (x+1)^2 and xy + y = 1. Determine the number of real and imaginary solutions (x, y) that satisfy both equations. Express your answer in the format: [number] real and [number] imaginary solutions. | 1 real and 2 imaginary solutions | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the large square has an area of 49 square units, the medium square has an area of 25 square units, and the small square has an area of 9 square units. The region inside the small square is shaded. The region between the large and medium squares is also shaded. Calculate the total area of the shaded regi... | 33 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the coordinates of point P are (2,1). Find the coordinates of the point symmetric to point P with respect to the y-axis. Express your answer as an ordered pair (x, y). | (-2,1) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
The Gauss Candy Company has 8362 lollipops to package. They place exactly 12 lollipops in each package. Calculate the number of lollipops that remain after the maximum possible number of packages are filled. Provide your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z = \frac{2\mathrm{i}}{1-\mathrm{i}}$, calculate the value of $\overline{z} \cdot z$, where $\overline{z}$ is the conjugate of $z$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A population of 3,204 students is to be sampled using a systematic sampling method to draw a sample of size 80. To ensure that the population size is divisible by the sample size, how many individuals should be randomly removed from the population? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
A certain product decreases its price by 25% every three years. If its current price is 640 yuan, calculate the price of this product after 9 years. Express your answer as a whole number in yuan. | 270 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be a differentiable function, satisfying $$\lim_{x \to 0} \frac{f(1)-f(1-x)}{2x} = -1.$$ Find the slope of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of linear equations $\left\{\begin{array}{l}{x+2y=8,}\\{2x+y=-5,}\end{array}\right.$, calculate the value of $x+y$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation x + 2 = 7 for x. Express your answer as a single integer. | x = 5 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the combinatorial expression $$C_n^0 - 2C_n^1 + 4C_n^2 - 8C_n^3 + \ldots + (-2)^n C_n^n.$$ Express your answer as a mathematical expression. | (-1)^n | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with 10 terms, the sum of its even terms is 30, and the sum of its odd terms is 25. Calculate the common difference d of the sequence. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Ming goes home at noon to cook noodles by himself, which involves the following steps: ①Wash the pot and fill it with water for 2 minutes; ②Wash the vegetables for 6 minutes; ③Prepare the noodles and seasonings for 2 minutes; ④Boil the water in the pot for 10 minutes (this step can be done concurrently with other ... | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1=1$, $a_{n+1}=a_n+2$ ($n\in\mathbb{N}^*$), calculate the value of $a_8$. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Sharon drove from her home to her mother's home at a constant speed, taking exactly 180 minutes. One day, she drove to her mother's home at her usual speed, and after covering 1/3 of the total distance, she encountered a snowstorm. She reduced her speed by 20 miles per hour and took a total of 276 minutes for the entir... | 135 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The sale ad read: "Buy three tires at the regular price and get the fourth tire for 3 dollars." Sam paid 240 dollars for a set of four tires at the sale. What was the regular price of one tire? Express your answer in dollars, rounded to two decimal places if necessary. | 79 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle C: $x^2+y^2-4x-4y+7=0$, and a line passing through point P(-2, 5) that is tangent to circle C at point Q, find the length of $|PQ|$. Express your answer as a numerical value, including the square root symbol if necessary. | 2 \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse with its focus on the x-axis, described by the equation $\dfrac{x^{2}}{m}+\dfrac{y^{2}}{4}=1$, and an eccentricity of $\dfrac{\sqrt {2}}{2}$, calculate the real number $m$. Express your answer as a single real number. | 8 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 79000 in scientific notation. Include the base number and the power of 10 in your answer, in the format $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer. | $7.9 \times 10^4$ | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Claire measures the interior angles of a convex polygon and calculates their sum to be 2017 degrees. However, she realizes that she forgot to include one of the angles in her calculation. Find the measure of the angle she forgot. Express your answer in degrees as a whole number. | 143 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 * e^x + 2x + 1, find the equation of the tangent line to the graph of f(x) at x = 0. Express your answer in the form Ax + By + C = 0. | 2x - y + 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 4 different balls into 3 different boxes, with each box containing at least 1 ball. Find the number of different ways to do this. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
The mean (average) of five consecutive even numbers is 12. Calculate the mean of the smallest and largest of these numbers. Express your answer as a single whole number. | 12 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition $p$: $\exists x_{0} > 0$, $2^{x_{0}} \geqslant 3$, what is the negation of $p$? Express your answer as a logical statement using proper notation for quantifiers. | \forall x > 0, 2^{x} < 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A number x is randomly selected from the interval [-1, 1]. Calculate the probability that the value of 2x^2 falls between 0 and 1/2. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition P: ∀x ∈ ℝ, x² + 2 > 2x, what is its negation? Express your answer as a logical statement with proper quantifiers and mathematical notation. | ∃x ∈ ℝ, x² + 2 ≤ 2x | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_{n}}$, where $a_{1}=a_{2}=1$, and $a_{n+2}-a_{n}=1$, find the sum of the first $100$ terms of the sequence. Express your answer as a numerical value. | 2550 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\mathbb{R}$, and the set $A=\{x|x^2+2x\geq0\}$, calculate the complement of $A$ in $U$, denoted as $\complement_U A$. Express your answer in interval notation. | (-2, 0) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(-2,1), B(1,4), and C(4,-3), find the coordinates of point E, which extends line segment DC such that CE/ED = 1/4, where point D lies on AB such that AD/BD = 1/2. Express your answer as a coordinate pair (x, y). | \left(-\frac{8}{3}, \frac{11}{3}\right) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
One morning, the temperature was $-3^{\circ}\mathrm{C}$. By noon, it rose by $6^{\circ}\mathrm{C}$, and then dropped by $7^{\circ}\mathrm{C}$ in the middle of the night. What was the temperature in the middle of the night? Express your answer in degrees Celsius (°C). | -4^{\circ}\mathrm{C} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the coordinates of the vertex of the parabola given by the quadratic function y = -x^2 + 2x + 3. Express your answer as an ordered pair (x, y) in the format (h, k) where (h, k) is the vertex. | (1, 4) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function $y = -2x^2 + 4x + 1$ needs to be translated to obtain the graph of $y = -2x^2$. Calculate the translation required to obtain the graph of $y = -2x^2$ from the graph of $y = -2x^2 + 4x + 1$. Express your answer as a translation (e.g., x units left/right, y units up/down). | 1\text{ unit to the left, 3 units down} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an even function defined on $\mathbb{R}$ and is monotonically increasing on $(-\infty,0)$, find the range of values for the real number $a$ that satisfies $f({2}^{|a-1|}) > f(4)$. Express your answer in interval notation. | (-1,3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The three views of a solid are shown in the figure. The solid is a right triangular prism with a base that is a right triangle. The lengths of the legs of the base triangle are 3 and 4, and the length of the hypotenuse is 5. The height of the prism is 5. Calculate the surface area of this solid. Express your answer as ... | 72 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sin^2(2\pi - \alpha) + \cos(\pi + \alpha) \cdot \cos(\pi - \alpha) + 1$ to a single numerical value. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of the first n terms as S_n, if a_1 = -14 and a_5 + a_6 = -4, find the value of n when S_n reaches its minimum value. Express your answer as a natural number. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains eight yellow marbles, seven red marbles, and five black marbles. Without looking in the bag, Igor removes N marbles all at once. If he is to be sure that, no matter which choice of N marbles he removes, there are at least four marbles of one color and at least three marbles of another color left in the b... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $f(x)=5\cos(wx+\varphi)$ satisfies $f\left(\frac{\pi}{3}+x\right)=f\left(\frac{\pi}{3}-x\right)$ for any real number $x$, and the function $g(x)=4\sin(wx+\varphi)+1$, calculate the value of $g\left(\frac{\pi}{3}\right)$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that one of the foci of a hyperbola coincides with the focus of the parabola x^2 = 20y, and the equations of its asymptotes are 3x ± 4y = 0, derive the standard equation of this hyperbola. | \dfrac{y^2}{9} - \dfrac{x^2}{16} = 1 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the largest number in the set {0.109, 0.2, 0.111, 0.114, 0.19}. Express your answer as a decimal number. | 0.2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Using the numbers 2, 0, and 5, what is the maximum number of two-digit decimals that can be formed? Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Counting Methods -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
What is the units digit of 33 * 219^89 + 89^19? Express your answer as a single digit. | 8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the variance of $x_1, x_2, x_3, \ldots, x_{2013}$ is 3, find the variance of $3(x_1-2), 3(x_2-2), 3(x_3-2), \ldots, 3(x_{2013}-2)$. Express your answer as a single number. | 27 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $4x^2+5y^2=1$, find the perimeter of the triangle $\triangle MNF$ formed by the left and right focal points $F$ and $F'$ and the points $M$ and $N$ where a line passing through $F'$ intersects the ellipse. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A student mistakenly multiplied $-3x$ by a polynomial instead of adding it, resulting in $3x^{3}-3x^{2}+3x$. What is the correct calculation result that should have been obtained by adding $-3x$ to the polynomial? Express your answer as a simplified polynomial. | -x^{2} - 2x - 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A set of data includes $12$, $16$, $20$, $23$, $20$, $15$, $28$, $23$. Find the median of this data set. Express your answer as a single number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If a rise in temperature by $2^{\circ}\mathrm{C}$ is denoted as $+2^{\circ}\mathrm{C}$, how would you denote a temperature drop by $4^{\circ}\mathrm{C}$? Provide your answer in the format of the temperature change notation. | -4^{\circ}\mathrm{C} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $\frac {1}{a}x^2+bx+c<0$ (where $ab>1$) with an empty solution set, find the minimum value of the expression $T= \frac {1}{2(ab-1)}+ \frac {a(b+2c)}{ab-1}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the real part of the complex number $Z= \frac{1-3i}{1+i}$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Two cones, A and B, have equal slant heights. The sum of the central angles of their unfolded lateral surfaces is $2\pi$. The lateral surface areas of the cones are $S_A$ and $S_B$, with $\frac{S_A}{S_B}=2$. Find the ratio of the heights of the two cones, $\frac{h_A}{h_B}$. Express your answer as a simplified ratio. | \frac{\sqrt{10}}{4} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
What is the square root of 4? Provide your answer in the form of a mathematical expression, including both positive and negative roots if applicable. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_7 = 35$, calculate the value of $a_4$, the 4th term of the sequence. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = g(x) + x^2 and the equation of the tangent line to the curve y = g(x) at the point (1, g(1)) is y = 2x + 1, find the slope of the tangent line to the curve y = f(x) at the point (1, f(1)). | 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
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