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 that $\tan \alpha = \sqrt{2}$, calculate the value of $\cos \alpha \sin \alpha$. Express your answer as a fraction with a square root. | \dfrac{\sqrt{2}}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the cube root of a number $a$ is a number $x$ such that $x^3 = a$, find the cube root of 64. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Mary had ten pieces of paper. Some of them were squares, and the rest were triangles. She cut three squares diagonally from corner to corner. She then found that the total number of vertices of the 13 pieces of paper was 42. Calculate the initial number of triangles Mary had before making the cuts. Express your answer ... | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the intersection points of the parabola $y = x^{2} - 2x + 1$ with the x-axis. Express your answer as coordinate(s) (x, y), noting that for intersections with the x-axis, y = 0. | (1, 0) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Suppose for an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is defined as $S_n$. Given that $S_{m-1}=-2$, $S_m=0$, and $S_{m+1}=3$, find the value of $m$. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f'"({x}_{0})=-3$, calculate the limit: $\lim\limits_{h\to0} \frac{f\left({x}_{0}+h\right)-f\left({x}_{0}-3h\right)}{h}$. Express your answer as a single integer. | -12 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In a grid of numbers and letters, starting from the letter A in the middle of the grid, calculate the total number of different paths that can spell out "AMC8". Each path must move to an adjacent (up, down, left, or right) letter or number only once (no diagonal moves allowed). Provide your answer as a single integer r... | 24 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x|x\leq 1\}$ and $P=\{x|x>t\}$, find the condition on the real number $t$ such that $\emptyset \subsetneq (M\cap P)$. Express your answer as an inequality involving $t$. | t<1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The school wants to select 3 out of 4 photography enthusiasts to participate in 3 sessions of an off-campus photography workshop (only 1 student per session). Due to scheduling conflicts, both students A and B cannot participate in the first session. Find the total number of different selection methods. Express your an... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality \(\left(\frac{\pi}{2}\right)^{(x-1)^{2}} \leq\left(\frac{2}{\pi}\right)^{x^{2}-5 x-5}\). Express your answer in interval notation. | [-\frac{1}{2}, 4] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n} that satisfies \frac {\ln a_1}{2} \cdot \frac {\ln a_2}{5} \cdot \frac {\ln a_3}{8} \cdot \ldots \cdot \frac {\ln a_n}{3n-1} = \frac {3n+2}{2} (n \in \mathbb{N}^*), calculate a_{10}. Express your answer as a power of e. | e^{32} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The first 3 terms of an arithmetic sequence {a_n} are -3, 1, 5. Find the value of S_10 - S_7, where S_n is the sum of the first n terms of the sequence. Express your answer as a single integer. | 87 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $\left(3-2i\right)\left(1-i\right)$. Express your answer as a numerical value. | -5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the quadrant in which the angle $-1120^\circ$ is located. Express your answer as the name of the quadrant (e.g., First quadrant, Second quadrant, etc.). | Fourth quadrant | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Equilateral triangle ABC has sides of length 4. The midpoint of BC is D, and the midpoint of AD is E. Calculate the value of EC^2. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
If $y = f(x)$ is differentiable on $(-\infty, +\infty)$, and $\lim\limits_{\Delta x \to 0} \frac{f(a+2\Delta x)-f(a)}{3\Delta x}=1$, calculate the value of $f'(a)$. Express your answer as a fraction. | \frac{3}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $S_n$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$, and $S_6=3$, $S_{11}=18$, calculate the value of the 9th term $a_9$. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Let the function y = f(x) be differentiable. Evaluate the limit $\lim_{\Delta x \to 0} \frac{f(1+3\Delta x)-f(1)}{3\Delta x}$. Express your answer as a mathematical expression. | f'(1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if a_2 = 3, and a_6 = 11, where S_n is the sum of the first n terms of the sequence {a_n}, find the value of S_7. Express your answer as a single numerical value. | 49 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
If the slope angle of the line $ax + (2a-3)y = 0$ is $45^\circ$, find the value of $a$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The sum of two natural numbers is 11. What is the maximum possible product that can be obtained with these numbers? Express your answer as a single numerical value. | 30 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
If the number of people in groups A, B, and C are 18, 24, and 30 respectively, and a stratified sampling method is used to draw a total of 12 people from these three groups, calculate the number of people drawn from group B. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that U is the universal set, and sets M, N ⊆ U, if M ∩ N = N, then what is the relationship between the complements of M and N, C_U M and C_U N? Express your answer using proper set notation. | C_U M ⊆ C_U N | big_math | [
"Mathematics -> Other -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Eight students from a university are preparing to carpool for a trip. There are two students from each of the four grades (first, second, third, and fourth grade). They will travel in two cars, named A and B, with each car carrying exactly four students. The twin sisters from the first grade need to ride in the same ca... | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $m$ that generates an extraneous root in the equation $\frac{x+7}{x-1}+2=\frac{m+5}{x-1}$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha$ is an obtuse angle, and $3\sin 2\alpha = \cos\alpha$, find the value of $\sin\alpha$. Express your answer as a numerical value. | \frac{1}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A regular dodecagon (12 sides) is inscribed in a circle with radius r inches. Find the area of the dodecagon in square inches. Express your answer as a formula in terms of r. | 3r^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that vector $\overrightarrow {a}$ = (2, 3) and vector $\overrightarrow {b}$ = (-1, 2), find the dot product of $\overrightarrow {a}$ and ($\overrightarrow {a}$ - $2\overrightarrow {b}$). Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
There are ten horses numbered from 1 to 10. The \( k \)-th horse (\( k = 1, 2, \cdots, 10 \)) takes exactly \( k \) minutes to run one lap on a circular track. Initially, all horses start at the starting point of the track at the same time, facing the same direction, and run at their own constant speeds along the circu... | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let the set \( A = \left\{ x \mid x^{2} + x - 6 < 0, x \in \mathbf{Z} \right\} \). Calculate the number of non-empty proper subsets of the set \( A \). Express your answer as a numerical value. | 14 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the line defined by the parametric equations x=(2-t sin 30°) and y=(-1+t sin 30°), and the circle defined by the equation x^2+y^2=8, the line intersects the circle at points B and C. Point O is the origin. Calculate the area of triangle BOC. Express your answer as a simplified radical expression. | \frac{\sqrt{15}}{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between two skew lines a and b is $50^{\circ}$, and P is a point in space, determine the number of unique lines passing through point P that make an angle of $30^{\circ}$ with both lines a and b. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum period of the function y=cos^4x-sin^4x+2. Express your answer in terms of pi (π). | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
A shopping mall, based on market information, adjusts the prices of two different models of air conditioners for sale. One air conditioner, after the price adjustment, sells for a 10% profit (relative to the cost price), while the other sells at a 10% loss. The selling prices of the two air conditioners after the price... | -1\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
If the temperature of $4^{\circ}\mathrm{C}$ above zero is denoted as $+4^{\circ}\mathrm{C}$, then how would you denote the temperature of $6^{\circ}\mathrm{C}$ below zero? Express your answer in degrees Celsius with the appropriate sign. | -6^{\circ}\mathrm{C} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence \(1, 3, 2, \cdots\), each term after the first two is equal to the difference of the two preceding terms (the previous term minus the term before it). Find the sum of the first 100 terms of this sequence. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Three volunteers, A, B, and C, are assigned to participate in volunteer activities in seven projects. Each volunteer can only participate in one project. Find the probability that exactly two people are assigned to the same project. Express your answer as a fraction. | \frac{18}{49} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $a$ in the equation $a^2 = a$. Provide all possible values of $a$, separating multiple values with commas if necessary. | 0, 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the dataset: $2$, $1$, $3$, $2$, $2$, calculate the variance of this data set. Express your answer as a decimal value. | 0.4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
How many ordered triples of integers $(a, b, c)$ satisfy the following system? $$ \begin{cases} ab + c &= 17 a + bc &= 19 \end{cases} $$ Provide your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given a convex n-gon, circles are drawn with each of its sides as diameters, ensuring that the n-gon can be completely covered by these n circles. Determine the maximum value of n for which this is possible. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} where the sum of the first n terms is denoted by S_n, and given that a_{1000} + a_{1018} = 2, find the value of S_{2017}. Express your answer as a single numerical value. | 2017 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the condition $M \cup \{1\} = \{1, 2\}$, how many sets $M$ satisfy this condition? Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A 300 gram container of sunflower seeds is given. If 15 seeds weigh about 1 gram, approximately how many sunflower seeds are in the container? Express your answer as a whole number. | 4500 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | 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"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $M = \{x | x^2 - x \leq 0\}$, and the function $f(x) = \log_2(1-|x|)$ have a domain of $N$. Calculate the intersection of $M$ and $N$, expressing your answer as an interval in standard interval notation. | [0, 1) | big_math | [
"Mathematics -> Algebra -> Inequalities",
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Molly has a set of cards numbered 1 to 12. She places eight of them at the vertices of an octagon so that the sum of every pair of numbers at opposite ends of an edge of the octagon is a multiple of 3. Find the four numbers that Molly did not place on the vertices of the octagon. Express your answer as a list of four n... | 3, 6, 9, 12 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Tegwen has the same number of brothers as she has sisters. Each one of her brothers has 50% more sisters than brothers. Using algebraic relationships, calculate the total number of children in Tegwen's family. Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A wooden rectangular prism with dimensions 4 by 5 by 6 is painted green and then cut into 1 by 1 by 1 cubes. Calculate the ratio of the number of cubes with exactly two green faces to the number of cubes with three green faces. Express your answer as a ratio in simplest form, using a colon (:) to separate the numbers. | 9:2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The normal vector of the line $3x+2y+5=0$ is $(a, a-2)$. Find the value of $a$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A ball is rolled on a rectangular table 5 units long and 2 units wide from point P at an angle of 45 degrees to PQ and bounces off SR. The ball continues to bounce off the sides at 45 degrees until it reaches point S. Calculate the number of bounces of the ball required to reach point S. Express your answer as a whole ... | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, if a_3 * a_5 * a_7 = (-sqrt(3))^3, then find the value of a_2 * a_8. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=f′( \frac {π}{4})\cos x+\sin x$, find the value of $f( \frac {π}{4})$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the point (1,0) and is parallel to the line x-2y-2=0. Express your answer in the standard form of a linear equation (Ax + By + C = 0). | x - 2y - 1 = 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In a circle with a radius of $3cm$, find the arc length corresponding to a central angle of $\frac{\pi}{7}$. Express your answer in centimeters (cm) and include π in your answer if necessary. | \frac{3\pi}{7} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the lateral area of a cone is twice the area of its base, find the central angle of the sector in the unfolded lateral surface of the cone. Express your answer in degrees as a whole number. | 180 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Find the abscissas (x-coordinates) of the intersection points between the parabola y = x^2 - 4x and the line y = 5. Express your answer as a pair of x-coordinates, separated by a comma. | -1, 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Jim drives 60 km south, 40 km west, 20 km north, and 10 km east. What is the straight-line distance from his starting point to his finishing point? Express your answer in kilometers as a whole number. | 50 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, it is known that a = 2, b = 3, and C = 120 degrees. Calculate the value of sin(A). Express your answer as a fraction with a square root if necessary. | \frac{\sqrt{57}}{19} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 + x^2 - ax + 1, find the range of real number 'a' for which f(x) is a monotonically increasing function on R. Express your answer in interval notation. | (-\infty, -\frac{1}{3}] | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer \( n \) that satisfies the inequality \( \sqrt{n} - \sqrt{n-1} < 0.01 \). Express your answer as a single positive integer. | 2501 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
The function $f(x)$ is defined as follows:
$$
f(x) =
\begin{cases}
x^{3} - 3x, & x \leqslant 0 \\
-2x + 1, & x > 0
\end{cases}
$$
Find the maximum value of $f(x)$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A cylinder has both a diameter and height of 4. Calculate the lateral area of the cylinder. Express your answer in terms of pi. | 16\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the median of the numbers in the list \( 19^{20}, \frac{20}{19}, 20^{19}, 2019, 20 \times 19 \). Express your answer as a single number. | 2019 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
A compact disc originally sells for $10.00. If the price of the compact disc is increased by 40% and this new price is later decreased by 30%, what is the final price? Express your answer as a dollar value rounded to two decimal places. | $9.80 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $6\tan45^\circ - 2\cos60^\circ$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Isabella had a week to read a book for a school assignment. She read an average of 36 pages per day for the first three days and an average of 44 pages per day for the next three days. She then finished the book by reading 10 pages on the last day. Calculate the total number of pages in the book. Express your answer as... | 250 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=
\begin{cases}
f(x+1), & x < 1\\
3^{x}, & x \geqslant 1
\end{cases}$, evaluate $f(-1+\log_{3}5)$. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
If $P = \{1, 2\}$, $Q = \{1, a^2\}$, and $P = Q$, then find the value of $a$. Express your answer as a mathematical expression, including the ± symbol if necessary. | ±\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Let $\{a_n\}$ be a geometric sequence with the first term as 1. If $\left\{ \frac {1}{2a_{n}+a_{n+1}}\right\}$ is an arithmetic sequence, then calculate the value of
$$( \frac {1}{2a_{1}}+ \frac {1}{a_{2}})+( \frac {1}{2a_{2}}+ \frac {1}{a_{3}})+…+( \frac {1}{2a_{2012}}+ \frac {1}{a_{2013}}).$$
Express your answer as ... | 3018 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, a_4 = 1 and a_6 = 1/3. Given that the sequence {1/a_n} is an arithmetic sequence, calculate the value of a_19. Express your answer as a simplified fraction. | \frac{1}{16} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Quality inspector Xiao Li tested a batch of telephones produced by his factory and found that among the first 50 units, 49 were of high quality. For every subsequent group of 8 units, 7 were of high quality. If the quality rate of this batch of telephones is not less than 90%, calculate the maximum number of telephones... | 210 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, AL bisects angle A, CM bisects angle C, and points L and M are on BC and AB respectively. The sides of triangle ABC are a, b, and c. Determine the value of k in the equation AM/MB = k * CL/LB. Express your answer as a ratio of side lengths in the form c/a. | \frac{c}{a} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function y=3sin(2x-π/6). Express your answer as a mathematical expression. | y′=6cos(2x-π/6) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane point sets
\[
\begin{array}{l}
A=\left\{(x, y)|(y-x) y-\frac{1}{x} \geqslant 0\right\}, \\
B=\left\{(x, y) \mid (x-1)^{2}+(y-1)^{2} \leqslant 1\right\},
\end{array}
\]
calculate the area of the plane figure represented by $A \cap B$. Express your answer as a single value. | \frac{\pi}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(\lambda,1,3)$ and $\overrightarrow{b}=(0,-3,3+\lambda)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, calculate the value of the real number $\lambda$. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The set $M=\{(x,y) \mid x^2+y^2 = 1\}$, and the set $N=\{(x,y) \mid x=1, y \in \mathbb{R}\}$. Find the intersection $M \cap N$. Express your answer as a set of ordered pairs (x, y). | \{(1,0)\} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
6 students are lined up in a row, and students A and B must be next to each other. Calculate the number of different arrangements of the students. Express your answer as a single integer. | 240 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the monotonic increasing interval of the function $f\left(x\right)=\left(3-x\right)e^{x}$. Express your answer as an interval in standard notation. | (-\infty, 2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=(\cos 2\alpha,\sin \alpha)$ and $b=(1,2\sin \alpha-1)$, where $\alpha\in\left( \frac{\pi}{2},\pi\right)$, if $a\cdot b= \frac{2}{5}$, calculate the value of $\tan \left(\alpha+ \frac{\pi}{4}\right)$. Express your answer as a numerical value. | \frac{1}{7} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the right focus of the hyperbola $\frac{x^2}{a^2} - y^2 = 1 (a > 0)$ coincides with the focus of the parabola $y^2 = 8x$, find the equation of the asymptotes of this hyperbola. Express your answer as an equation of a line in the form $y = mx$, where $m$ is a constant. | y = \pm \frac{\sqrt{3}}{3}x | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
There are two rectangles, A and B. The length and width of rectangle A are $x$ cm and $y$ cm, respectively. The length and width of rectangle B are each 1 cm less than those of rectangle A. Calculate the difference in area between rectangle A and rectangle B. Express your answer as an algebraic expression in terms of $... | x + y - 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\left\{\begin{array}{l}{x=1}\\{y=3}\end{array}\right.$ is a solution to the binary linear equation $x-2y=m$ with respect to $x$ and $y$, calculate the value of $m$. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \\(\\{a_n\\}\\) satisfies \\(a_{n+1}+a_n=n\\) and \\(a_1=2\\), calculate the value of \\(a_4-a_2\\). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the normal vectors of planes α and β are $\overrightarrow{a}=(-1,2,4)$ and $\overrightarrow{b}=(x,-1,-2)$ respectively, and that α is parallel to β, find the value of x. Express your answer as a single numerical value. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = cos(ωx + φ) where (ω > 0, 0 < φ < π) has a minimum positive period of T, and f(T) = 1/2, and x = 7π/3 is a critical point of the function f(x), find the minimum value of ω. Express your answer as a fraction. | \frac{2}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A cylinder has a rectangular side unfolding that is a square. Calculate the ratio of the surface area to the side area of this cylinder. Express your answer as a ratio in the form $\boxed{\dfrac{a+b\pi}{c\pi}}$. | \dfrac{1+2\pi}{2\pi} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The mean age of the members of a jazz band is 21. The saxophonist, singer, and trumpeter are 19, 20, and 21 years old respectively. The other three musicians are all the same age. Calculate the age of each of these three musicians. | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Six people are lined up from left to right. Find the total number of arrangements where at least one of the persons A or B is at one of the ends. Express your answer as a single integer. | 432 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If $\triangle A_{1} A_{2} A_{3}$ is an equilateral triangle and $A_{n+3}$ is the midpoint of segment $A_{n} A_{n+1}$ for all positive integers $n$, then find the degree measure of $\angle A_{44} A_{45} A_{43}$. Express your answer in degrees as a numerical value. | 120 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of the expression (1+tan 17°)(1+tan 28°)(1+tan 27°)(1+tan 18°). | 4 | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number $(1-i)(a+i)$ is real (where $i$ is the imaginary unit), find the value of the real number $a$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(m+2)x^{m^{2}-2}+2x+1=0$ is a quadratic equation in $x$, find the value of $m$ that satisfies this condition. Provide your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given \( a > 1 \) and \( b > 1 \) and \(\lg (a+b) = \lg a + \lg b\), calculate the value of \(\lg (a-1) + \lg (b-1)\). Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors \(a = (1, m)\) and \(b = (4, -2)\), and \(a \perp b\), find the value of \(m\). Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Two cylinders sit on a flat surface. Cylinder A, with a radius of 6 cm and a height of 50 cm, is empty. Cylinder B, with a radius of 8 cm and a height of 50 cm, is full of water. After pouring some water from Cylinder B into Cylinder A, the height of the water in both cylinders is the same. Calculate the height of the ... | 32.0 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
If the square roots $\sqrt{5b}$ and $\sqrt{3+2b}$ can be combined, solve for the value of $-b$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos \left( \frac {\pi}{2}+α \right)= \frac {3}{5}$, and $α∈\left( \frac {\pi}{2}, \frac {3π}{2}\right)$, calculate the value of $\tan α$. Express your answer as a simplified fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
The length, width, and height of a rectangular prism are three consecutive natural numbers. The volume of the prism is equal to twice the sum of the lengths of all its edges. Find the surface area of this rectangular prism. Express your answer as a whole number. | 148 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) with its right focus at F, if a line passing through point F with a slope angle of 60° intersects the right branch of the hyperbola at two points, calculate the range of the eccentricity of this hyperbola. Express your answer in interval notation. | (1,2) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
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