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 complex number $$z= \frac {2-ai}{i}$$ (where $i$ is the imaginary unit and $a \in \mathbb{R}$) is a pure imaginary number, find the value of $|z|$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
At the same moment, Xiaoqiang, who is 1.6 meters tall, casts a shadow of 0.8 meters under the sunlight. A big tree casts a shadow of 4.8 meters. What is the height of the tree? Express your answer in meters as a decimal number. | 9.6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the fractional part of the common logarithm of a positive number M is 1/a, find the fractional part of the common logarithm of 1/M. Express your answer as a fraction or an equivalent expression. | \frac{a-1}{a} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
An extracurricular interest group has 5 members, including 3 female members and 2 male members. Now, 2 members are randomly selected for a study report. Let X denote the number of female members selected. Calculate the mean of X, expressing your answer as a fraction or decimal. Include all necessary calculations in you... | 6/5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of line $l$ is $2x-5y+10=0$, find its intercepts on the x-axis ($a$) and on the y-axis ($b$), and then calculate $|a+b|$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, where the sum of the first n terms is S_n, if a_3 + a_9 = 6, calculate the value of S_{11}. Express your answer as a single integer. | 33 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular ceiling PQRS measures 6m by 4m and is to be completely covered using 12 rectangular tiles, each measuring 1m by 2m. A beam TU is positioned so that PT = SU = 2m, and no tile can cross the beam. Calculate the total number of possible arrangements of tiles. Express your answer as a single integer. | 180 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the general term formula for a sequence, $a_n = (-1)^n \frac{n}{n+1}$, find the value of $a_3$. Express your answer as a fraction. | -\frac{3}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $12a^{4}b^{3}c\\div (-4a^{3}b^{2})$. Express your answer as a simplified algebraic expression. | -3abc | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given $x^3 = 8$, evaluate the function $f(x) = (x-1)(x+1)(x^2+x+1)$. Express your answer as a single numerical value. | 21 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the power function f(x) = x^α and that the point (1/2, √2/2) lies on its graph, calculate f(4). | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of zeros of the function f(x) = sin 2x - sqrt(3)cos 2x + 1 in the interval [0, pi]. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
A certain school has 100 teachers, 600 male students, and 500 female students. A stratified sampling method is used to draw a sample of size n from all the teachers and students. If it is known that a total of 40 female students were sampled, calculate the value of n. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence ${a_n}$ where $a_n > 0$, and $a_1 + a_2 +... + a_{10} = 30$, find the maximum value of $a_5 \cdot a_6$. Express your answer as a numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given seven integers with a sum of $-1$, what is the maximum number of these integers that can be greater than 13? | 6 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
On the plane region {(x, y) | |x| ≤ 1 and |y| ≤ 1}, the inequality ax - 2by ≤ 2 always holds. Calculate the area of the plane region formed by the moving point P(a, b). | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)= \begin{cases} \left( \frac {1}{2}\right)^{x}-7 & (x < 0) \\ \sqrt {x} & (x\geqslant 0)\end{cases}$. If $f(a) < 1$, then find the range of the real number $a$ as an interval. | (-3, 1) | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $-\sqrt{5}$, $0.\dot 42\dot 1$, $3.14$, $0$, $\frac{π}{2}$, $\frac{{22}}{7}$, $\sqrt{81}$, $0.1616616661\ldots$ (with one more $6$ between each pair of $1$), how many are irrational numbers? Provide your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle α intersects the unit circle at point (-√5/5, 2√5/5), find the value of cos 2α. Express your answer as a simplified fraction. | -3/5 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( f(n) \) be the sum of the squares of the digits of positive integer \( n \) (in decimal). For example, \( f(123) = 1^{2} + 2^{2} + 3^{2} = 14 \). Define \( f_{1}(n) = f(n) \), and \( f_{k+1}(n) = f\left(f_{k}(n)\right) \) for \( k = 1, 2, 3, \ldots \). Calculate the value of \( f_{2005}(2006) \). Express your an... | 145 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \( 1 - 4(3 - 1)^{-1} \) and provide the result as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In a mathematics competition, 1000 students are numbered as follows: 0001, 0002, 0003, …, 1000. A sample of size 50 is to be drawn using systematic sampling, divided into 50 parts. If a number 0015 is randomly selected from the first part, calculate the 20th sample number. Express your answer as a four-digit number wit... | 0395 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^2-7|x|+6=0$, calculate the sum of its roots. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A line $l$ bisects the circle $x^{2}+y^{2}-2x-4y=0$ and is parallel to the line $x+2y=0$. Determine the equation of line $l$ in the standard form $Ax + By + C = 0$. | x+2y-5=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = 1 + |x| + x^3$, calculate the value of $$f(\log_2{2}) + f(\log_2{\frac{1}{2}}) + f(\log_2{5}) + f(\log_2{\frac{1}{5}}).$$ Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence \\(\\{a_n\\}\\), if \\(a_5 = 2\\), find the value of the product \\(a_1a_2a_3a_7a_8a_9\\). Express your answer as a single numerical value. | 64 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation \sqrt{4+9+x^{2}}=7. Express your answer as a single number, including the \pm symbol if there are multiple solutions. | \pm 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} -x^{2} & ;x < 0 \\ g(x) & ;x > 0 \end{cases}$, where $f(x)$ is an odd function, calculate the value of $g(2)$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the following division operations: 7.86÷9, 34.2÷15, 48.3÷6, 34.78÷37, how many of these divisions result in a quotient less than 1? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $| \overrightarrow {a}|=1，| \overrightarrow {b}|=2$, and $\overrightarrow {a}⊥( \overrightarrow {a}- \overrightarrow {b})$, find the projection of vector $\overrightarrow {a}$ on the direction of $\overrightarrow {b}$. Express your answer as a fraction or decimal. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
How many fractions of the form $\frac{n}{n+1}$ are less than $\frac{7}{9}$, given that $n$ is a positive integer? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {zn}, if z1 = 1, z2 = a + bi, and z3 = b + ai, where a and b are real numbers and a > 0, find the smallest value of n such that z1 + z2 +... + zn = 0. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $x^2-2y^2=1$ with focus $F$, calculate the distance from focus $F$ to one of its asymptotes using the point-to-line distance formula. Express your answer as a simplified fraction or decimal value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)= \frac {\ln (x-1)}{ \sqrt {4-x^{2}}}$. Express your answer as an interval. | (1,2) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
What is the square root of $9$? Express your answer in the form $\pm x$, where $x$ is the positive root. | \pm 3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A cuboid has three faces meeting at one vertex with areas of $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{6}$, respectively. Calculate the length of the diagonal of this cuboid. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The shape in the diagram is made up of a rectangle, a square, and an equilateral triangle, all of which have the same perimeter. The length of the side of the square is \( 9 \mathrm{~cm} \). Calculate the length of the shorter sides of the rectangle. Express your answer in centimeters (cm). | 6 \text{ cm} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a$ and $b$ that satisfy $|a|=2$, $|b|=1$, and $a \cdot b = 1$, find the angle $\theta$ between vector $a$ and vector $(a-b)$. Express your answer in radians. | \dfrac{\pi}{6} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. Find the perimeter of square $III$. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$ with a non-zero common difference, some terms of the sequence ($a_{k_1}, a_{k_2}, a_{k_3}, \ldots$) form a geometric sequence $\{a_{k_n}\}$. If $k_1 = 1$, $k_2 = 2$, and $k_3 = 6$, find the value of $k_4$. Express your answer as a whole number. | 22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 98 and 63. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the expression $(x+1)(x+2)(x+3)(x+4) + 2019$ for $x \in \mathbf{R}$. Express your answer as a single integer. | 2018 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Place 6 balls, labeled from 1 to 6, into 3 different boxes. Each box is to contain 2 balls, and the balls labeled 1 and 2 are to be placed in the same box. Calculate the total number of different ways to do this. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.171875 | 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"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given $f(x) = 2x + 3$ and $g(x + 2) = f(x)$, find the expression for $g(x)$. Simplify your answer fully and express it as a function of $x$. | 2x - 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Two people, A and B, depart simultaneously from the same location in opposite directions. After one hour, they reach their respective destinations, A and B. If they start again from the original location but swap destinations, person A reaches B 35 minutes after person B reaches A. Find the ratio of person A's speed to... | 3/4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a group of 5 people, choose 4 persons to visit four different cities: Paris, London, Washington, and Sydney, with the requirement that each city is visited by one person, and each person visits only one city. Among these 5 people, individual A does not go to Paris. Calculate the total number of different selectio... | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the odd function f(x) that satisfies f(x+2) = -f(x) and f(x) = 3^x + 1/2 for x in (0, 1), find the exact value of f(log_3 54). Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2,3)$ and $\overrightarrow{b}=(-1,2)$, find the coordinates of $2\overrightarrow{a}-\overrightarrow{b}$. Express your answer as a coordinate pair (x, y), enclosed in parentheses. | (5,4) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a^{m}=9$ and $a^{n}=3$, calculate the value of $a^{m-n}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Exponential Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of the first n terms denoted as S_n, if a_4 = 9 and a_6 = 11, calculate the value of S_9. | 90 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ln(x) + x^2, find the minimum value of the slope of the tangent line to the curve y = f(x) at any point P. Express your answer as a single numerical value. | 2\sqrt{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
When a ship is sailing, the fuel cost per unit time $u$ is directly proportional to the cube of its speed $v$. If the speed of the ship is 10 km/h, the fuel cost is 35 yuan per hour, and the other costs are 560 yuan per hour, which do not vary with speed. Given that the maximum speed of the ship is 25 km/h, calculate t... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Let $F$ be the focus of the parabola $y^2=2x$, and let $A$, $B$, $C$ be three points on the parabola. If $F$ is the centroid of $\triangle ABC$, then find the value of $|\overrightarrow{FA}|+|\overrightarrow{FB}|+|\overrightarrow{FC}|$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, $\angle C=50^{\circ}$. With point $C$ as the center, draw an arc with a suitable length as the radius, intersecting $AC$ at point $M$ and $BC$ at point $N$. With $M$ and $N$ as centers, draw arcs with lengths greater than $\frac{1}{2}MN$ as the radius. These two arcs intersect at point $D$.... | 25 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation |-x| = 5. Express your answer using the '±' notation for combined solutions. | \pm 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
In a cube $A C_{1}$, $M$ and $N$ are the midpoints of edges $C_{1} D_{1}$ and $A B$ respectively. The points $A_{1}, M, C,$ and $N$ lie in the same plane. Calculate the sine of the angle between $C D$ and the plane $A_{1} M C N$. Express your answer as a simplified fraction or expression involving square roots. | \frac{\sqrt{2}}{4} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^2=12x$ and that the distance from point P on the parabola to its focus is 8, find the distance from point P to the directrix. Express your answer as a numerical value. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given an increasing sequence of positive integers {a_n} with the property a_{n+2} = a_{n+1} + a_n, and given a_7 = 120, calculate the value of a_8. Express your answer as a single positive integer. | 194 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Bev is driving from Waterloo, ON to Marathon, ON. She has driven 312 km. She has 858 km still to drive. Calculate the distance Bev must drive to be halfway from Waterloo to Marathon. Express your answer in kilometers (km). | 273 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sqrt(a^2 - x^2) / (|x + a| - a), find the necessary and sufficient condition on 'a' for f(x) to be an odd function. Express your answer as an inequality or equality condition on 'a', including all necessary mathematical symbols and expressions. | a > 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line $ax+y-1-a=0$ is parallel to the line $x-\frac{1}{2}y=0$, find the value of $a$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $y=f(x)$ is an increasing function on $\mathbb{R}$, and $f(2m) < f(9-m)$, find the range of the real number $m$. Express your answer as an interval in interval notation. | (-\infty, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the point that is symmetric to point $P(5,-3)$ with respect to the x-axis. Express your answer as a coordinate pair (x, y). | (5, 3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
If the volume of a cylinder is exactly equal to its surface area, what is the minimum possible value for the volume of the cylinder? Express your answer as a multiple of pi, in the format \\boxed{N \\pi}. | 54 \\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the condition on c such that the solution set of the inequality \(x^{2}+x+c > 0\) is all real numbers. Express your answer as an inequality involving c. | c > \dfrac {1}{4} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A real number x is randomly selected from the interval (10,20). Find the probability that the inequality {{x}^{2}}-14x < 0 holds. Express your answer as a simplified fraction. | \dfrac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
There are 9 different Chinese books, 7 different math books, and 5 different English books. How many different ways are there to select 2 books that do not belong to the same subject? Express your answer as a whole number. | 143 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex numbers $z_1=-1+2i$ and $z_2=2+i$, where $i$ is the imaginary unit, calculate the product $z_1 \cdot z_2$. Express your answer in the form $a+bi$. | -4+3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A plane perpendicular to the radius of a sphere at the midpoint of the radius cuts the sphere. Find the ratio of the area of the resulting circular section to the surface area of the sphere. Express your answer as a simplified fraction. | \dfrac{3}{16} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
What is the maximum number of Mondays that can occur in the first 45 days of a year? Provide your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the recurrence relation F(n+1) = (2F(n) + 1)/2 for n = 1, 2,..., and F(1) = 2, calculate the value of F(101). Express your answer as a single integer. | 52 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle $x^{2}+(y-1)^{2}=4$ and the line $3x+2y+1=0$, find the equation of the line that passes through the center of the circle and is perpendicular to the given line. Express your answer in the standard form $Ax+By+C=0$. | 2x-3y+3=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=1-i$ ($i$ is the imaginary unit), calculate the value of $\bar{z}+\frac{2i}{z}$. Express your answer in the simplest form of a complex number, using the boxed notation. | 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Among the four-digit numbers composed of the digits 0, 1, 2, and 5 without any repetition, how many numbers cannot be divided by 5? Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
On a stormy night, four people on an isolated island must cross a narrow bridge to reach the mainland for refuge. To cross the bridge, they need to use a lamp for illumination, but they only have one lamp, and the bridge can only accommodate two people at a time. If there are still people on the island, the people who ... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(4a^{2}+kab+9b^{2}\) is a perfect square, find the value of the constant \(k\). Express your answer as a numerical value, including the ± symbol if applicable. | ±12 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given $0 < a < 1$ and $c > 1$, determine the relationship between $ac+1$ and $a+c$. Express your answer as an inequality or equality statement. | ac+1 < a+c | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(-1, 1) and B(0, 2), if vector $\overrightarrow {AC}$ = (-2, 3), find vector $\overrightarrow {BC}$. Express your answer as a vector in the form (x, y). | (-3, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given natural numbers \( a, b, \) and \( c \) with the sum \( S \), satisfying \( a + b = 1014 \), \( c - b = 497 \), and \( a > b \). Calculate the maximum value of \( S \). Express your answer as a single integer. | 2017 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The function $f(x)= \frac {1}{3}x^{3}- \frac {1}{2}(2b+1)x^{2}+b(b+1)x$ has a local minimum in the interval $(0,2)$. Find the range of values for $b$ that satisfy this condition. Express your answer as an inequality in standard notation. | -1 < b < 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Alex drives from point \( P \) to point \( Q \), then immediately returns to \( P \). Bob drives from point \( Q \) to point \( P \), then immediately returns to \( Q \). They travel on the same road, start at the same time, and each travels at a constant speed. Alex's speed is three times Bob's speed. They meet each o... | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
One morning, the temperature was -2°C. It rose by 6°C at noon and then dropped by 8°C at midnight. What was the temperature at midnight? Express your answer in degrees Celsius (°C). | -4°C | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, points B, C, and D have coordinates B(3, 9), C(3, -3), and D(-2, -3). Using the Pythagorean Theorem, calculate the length of side BD. Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
If $a > 1$, find the minimum value of $\frac{a^2 - a + 1}{a - 1}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Express 1.5 million in scientific notation. Provide your answer in the format a × 10^b, where a is the coefficient and b is the exponent. | 1.5 × 10^6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function y=f(x) is [1,2], find the domain of the function y=f(sqrt(x)). Express your answer in interval notation. | [1,4] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A train, originally of mass M, is traveling on a frictionless straight horizontal track with constant speed v. Snow starts to fall vertically and sticks to the train at a rate of ρ, where ρ has units of kilograms per second. The train's engine keeps the train moving at constant speed v as snow accumulates on the train.... | \frac{1}{2} \rho v^2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Using only $2 coins and $5 bills, in how many different ways can you make $207? Express your answer as a single integer. | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $$f(x)= \begin{cases} x(x+3) & (x\geq 0) \\ x(x-3) & (x<0) \end{cases}$$, evaluate $f(-2)$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Numbers \( m \) and \( n \) are on the number line, as shown. The distance between 0 and 30 is 6 spaces. Each space represents 5 units. \( n \) is 2 spaces to the right of 60, and \( m \) is 3 spaces to the left of 30. Calculate the value of \( n - m \). Express your answer as a single integer. | 55 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the quadrant (I, II, III, or IV) of the point corresponding to the complex number $\frac{i}{1+i}$ in the complex plane. Express your answer as the standard quadrant label. | I | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{a+2b}{a}=4$, find the value of $\frac{a}{b-a}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence below, each △ represents a number, and starting from the third term, each number is the sum of the two preceding numbers. Calculate the sum of all 9 numbers listed: △, △, △, △, 7, △, △, △, 47. Provide your answer as a single integer. | 122 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function $f(x)=\sqrt{2}\sin(\frac{x}{2}+\frac{π}{3})$. Express your answer in radians as an exact value. | 4π | big_math | [
"Mathematics -> Precalculus -> Functions -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the algebraic expression $49m^{2}+km+1$, find the value(s) of $k$ for which this expression is a perfect square trinomial. Provide all possible values of $k$ that satisfy this condition. | \pm 14 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {an}, it is known that a3=5, a2+a5=12, and an=29. Calculate the value of n. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a\in (\frac{\pi}{2}, \pi)$ and $\sin \alpha = \frac{3}{5}$, find $\tan \left(\alpha + \frac{\pi}{4}\right)$. Express your answer as a single fraction or number. | \frac{1}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The length of the chord intercepted on the circle $x^{2}+y^{2}=9$ by the line $egin{cases} x=2t-1 \ y=t+1 \end{cases} (t ext{ is a parameter})$ is to be calculated. Find the exact length of the chord. | \frac{12\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a\in\left( \frac{\pi}{2},\pi\right)$ and $\sin \alpha= \frac{3}{5}$, calculate $\tan \left(\alpha+ \frac{\pi}{4}\right)$. Express your answer as a single value. | \frac{1}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.