instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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In a $7 \times 7$ grid, some squares are painted black and the rest are painted white, with at least one square of each color. A pattern is called "symmetric" if it looks the same when rotated $90^{\circ}$ and if it can be folded along either diagonal such that the two halves match. Find the total number of symmetric p... | 1022 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence ${a_n}$, $a_8 = 1$, and the common ratio $q = \frac{1}{2}$. Find the sum of the first 8 terms, $S_8$. Express your answer as a whole number. | 255 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given a > 0, b > 0, and two lines l1: (a-1)x + y - 1 = 0 and l2: x + 2by + 1 = 0, where l1⊥l2, find the minimum value of 2/a + 1/b. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Calculus -> Optimization"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Points A(11, 9) and B(2, -3) are vertices of triangle ABC with AB = AC. The altitude from A meets the opposite side at D(-1, 3). Calculate the coordinates of point C. Express your answer as a coordinate pair (x, y). | (-4, 9) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of $999 + 999$? Express your answer as a single integer. | 1998 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence ${a_n}$ with sum of the first $n$ terms $S_n$, and the equations $a_3=2S_2+5$ and $a_4=2S_3+5$, find the common ratio $q$ of this sequence. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}$=($\sqrt {3}，1$) and $\overrightarrow {b}$=(-3, $\sqrt {3}$), calculate the projection of vector $\overrightarrow {b}$ onto vector $\overrightarrow {a}$. Express your answer as a single value, possibly with a square root. | -\sqrt {3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if a_5 = 1 and a_1 + a_7 + a_{10} = a_4 + a_6, then find the sum of the first 10 terms, S_{10}. Express your answer as a fraction. | \frac{25}{3} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A gumball machine contains 9 red, 7 white, and 8 blue gumballs. What is the least number of gumballs a person must buy to be sure of getting four gumballs of the same color? Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-collinear vectors, $\overrightarrow{AB}=\lambda \overrightarrow{a}+ \overrightarrow{b}$, $\overrightarrow{AC}= \overrightarrow{a}+\mu \overrightarrow{b}$ ($\lambda,\mu\in\mathbb{R}$), find the condition under which points A, B, and C are collinear. Expres... | \lambda\mu=1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(ab^{3})^{3}$ by applying the rules of exponents. | a^{3}b^{9} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given vector $\overrightarrow{a}$, calculate the result of the vector addition $\overrightarrow{a}+2\overrightarrow{a}$. Express your answer as a vector in terms of $\overrightarrow{a}$. | 3\overrightarrow{a} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the side lengths of four squares are shown. The area of the fifth square is denoted by k. Using the given relationships between the heights of the squares on the left and right sides, find the value of k. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies the equation $z \cdot i = 3 - 4i$ ($i$ is the imaginary unit), find $z$ in standard form, $a + bi$. Express your answer in the form $\boxed{a + bi}$. | -4 - 3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, if 2a_7 = a_9 + 7, then find the sum of the first 9 terms of the sequence (S_9). Express your answer as a single integer. | 63 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \( 10^{\frac{1}{11}}, 10^{\frac{2}{11}}, 10^{\frac{3}{11}}, \ldots, 10^{11} \), find the smallest positive integer value of \( n \) such that the product of the first \( n \) terms of this sequence exceeds 100,000. | 11 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given tan(alpha) = 1/2 and pi < alpha < 3*pi/2, calculate the exact value of cos(alpha) - sin(alpha). Express your answer as a simplified fraction, including the negative sign if applicable. | -\frac{\sqrt{5}}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A sphere is intersected by a plane that is perpendicular to one of its radii at the midpoint. Determine the ratio of the intersecting circle's area to the sphere's surface area. Express your answer as a simplified fraction. | \frac{3}{16} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Three classes each choose one of five scenic spots to visit. What is the total number of different selection methods? Express your answer as a numerical value. | 125 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
There are 4 normal school graduates being allocated to teach at 3 different middle schools, with each school receiving at least 1 teacher. Calculate the total number of different allocation schemes possible. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $(1+x-2x^2)(1+x)^5$, find the coefficient of the term containing $x^3$. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A town's population increased by 1,200 people, and then this new population decreased by 11%. The town now has 32 less people than it did before the 1,200 increase. Calculate the original population of the town. Express your answer as a whole number. | 10,000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, find the distance from the point (-2, -√5) to the x-axis. Express your answer as a single numerical value. | √5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $D$ that satisfies $\left( z-2i \right)\left( 1-i \right)=-2$, find the value of $|z|$. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the values of $a$, $b$, and $c$ such that $(ax^2 - 2xy + y^2) - (-x^2 + bxy + 2y^2) = 5x^2 - 9xy + cy^2$ holds true. Express your answer as a comma-separated list of values for $a$, $b$, and $c$. | 4, 7, -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the actual distance between two points A and B (AB) is 5 km, and the distance between A' and B' drawn on the map (A'B') is 2 cm, calculate the ratio of the distance on the map to the actual distance. Express your answer as a ratio (a:b) in simplest form. | 1:250,000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given points M(-3, 3) and N(-5, -1), calculate the vector MN. Express your answer as an ordered pair (x, y). | (-2, -4) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (8, \frac{1}{2}x)$ and $\overrightarrow{b} = (x, 1)$, with $x > 0$, find the value of $x$ such that $\overrightarrow{a} - 2\overrightarrow{b}$ is collinear with $2\overrightarrow{a} + \overrightarrow{b}$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = x^4 that is perpendicular to the line x + 4y - 8 = 0. Express your answer in the standard form of a line equation (Ax + By + C = 0), including all necessary coefficients and constants. | 4x - y - 3 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Find the necessary and sufficient condition on a and b for the inequality a|a+b| < |a|(a+b) to hold, where a, b are real numbers. Express your answer as a condition on a and b using standard mathematical notation and formatting. | a < 0 \text{ and } b > -a | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given $ab=-3$, calculate the value of $a\sqrt{-\frac{b}{a}}+b\sqrt{-\frac{a}{b}}$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. Calculate the total time it took Joe to get from home to school. Express your answer in minutes as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Point P (4, a) lies on the curve defined by x = t/2 and y = 2 sqrt(t), and point F (2, 0). Find the distance |PF| between these two points. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality x / (2x - 1) > 1. Express your answer in interval notation. | (1/2, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $b$ and $c$ are in the interval $[-1, 1]$, find the probability that the equation $x^{2} + 2bx + c^{2} = 0$ has real roots. Express your answer as a fraction or decimal between 0 and 1. | \dfrac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that proposition $p$ is true, and proposition $q$ is false, then among the compound propositions formed by "p∨q", "p∧q", "￢p", "￢q", how many are true propositions? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a\_n\}$ satisfies $a\_1=1$, and $a_{n+1}-a_{n}=n+1 (n∈N_+)$. Calculate the sum of the first 10 terms of the sequence $\{ \frac{1}{a\_n}\}$. Express your answer as a simplified fraction. | \frac{20}{11} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Let \( M = \left\{(x, y) \mid \tan(\pi y) + \sin^2(\pi x) = 0 \right\} \) and \( N = \left\{(x, y) \mid x^2 + y^2 \leq 2 \right\} \). Calculate the number of points in the intersection \( M \cap N \). Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are both positive real numbers, find the minimum value of $(a+ \frac {1}{b})(b+ \frac {4}{a})$. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \( x_{1}, x_{2}, \cdots \) defined by the initial condition \( x_{1}=1 \), and for \( k \geqslant 2 \),
\[
x_{k} = x_{k-1} + 1 - 4 \left( \left\lfloor \frac{k-1}{4} \right\rfloor - \left\lfloor \frac{k-2}{4} \right\rfloor \right),
\]
calculate the value of \( x_{2010} \). Express your answer as a sin... | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\begin{cases}
-x^{2}+2x-2 & \text{for } x \leqslant 1 \\
2^{-|1-x|}-2 & \text{for } x > 1
\end{cases}$, and $g(x)=|a-1|\cos x$ for $x \in \mathbb{R}$, find the range of the real number $a$ such that $f(x_{1}) \leqslant g(x_{2})$ for any $x_{1}, x_{2} \in \mathbb{R}$. Express your answer as an i... | [0,2] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Convert the binary number $111_{(2)}$ to a decimal number. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Other -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Use the Euclidean algorithm to find the greatest common divisor of 456 and 357. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $m$ in the equation $$\frac {2}{1+i}=1+mi \quad (m \in \mathbb{R}, i \text{ denotes the imaginary unit}).$$ Express your answer as a real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola described by the equation $y^2 = 4x$, find the coordinates of its focus. Express your answer as an ordered pair $(x, y)$. | (1, 0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = $$\frac {(1+2i)^{2}}{-i+2}$$, calculate the value of |z|. Express your answer as a mathematical expression, possibly involving square roots. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Among the eight vertices of the cube ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, a plane passes through four of them, and the remaining four vertices are all at equal distances from the plane. Find the number of such planes. | 12 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If f(x) is an even function, and when x > 0, f(x) = 10^x, then find f(x) when x < 0. Express your answer as an exponential function of x. | 10^{-x} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Use systematic sampling to select 50 students from a group numbered 1 to 700. If student number 20 from the second segment is selected, calculate the student number that would be selected from the fifth segment. Express your answer as a single integer. | 62 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $(2-x)^5 = a_0 + a_1x + a_2x^2 + \ldots + a_5x^5$, calculate the value of $\frac {a_{0}+a_{2}+a_{4}}{a_{1}+a_{3}+a_{5}}$. Express your answer as a simplified fraction. | -\frac{122}{121} | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the distance from the center of the circle $x^2+y^2-2x-4y=0$ to the line $x-y+a=0$ is $\frac{\sqrt{2}}{2}$, calculate the possible values of $a$. Express your answer as whole numbers or fractions, listing all possible values. | a = 2 \text{ or } a = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
An opaque box contains 20 balls, each of the same size and shape, numbered from 1 to 20. Two balls are randomly chosen from the box. What is the probability that the sum of the numbers on these two balls is divisible by 3? Express your answer as a simplified fraction. | \frac{32}{95} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\int_{1}^{2}{(x+\frac{1}{x})dx}$. Express your answer as an exact value, including any logarithmic terms. | \frac{3}{2} + \ln 2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of extreme points of the function f(x) = x^3 - 3x^2 + 3x. Express your answer as a whole number. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
From the set {1, 2, 3,..., 20}, randomly select 4 different numbers such that these 4 numbers form an arithmetic sequence. Calculate the number of such arithmetic sequences. Express your answer as a whole number. | 114 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Assuming that car license plates are 5-digit numbers (which cannot all be 0), calculate the probability that the license plate number of an oncoming car is a multiple of 3. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations of two lines are $x+y+a=0$ and $x+y+b=0$, respectively, where $a$ and $b$ are the two real roots of the equation $x^{2}+x+c=0$, and $0 \leqslant c \leqslant \frac{1}{8}$, find the maximum distance between these two lines. Express your answer as a numerical value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
The radius of a certain globe is 20 cm. Calculate the length of the parallel of latitude at 60°N on this globe. Express your answer in terms of π and include units (cm). | 20\pi \text{ cm} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
If $m < 0$, find the absolute value of the difference between $m$ and its opposite number. Express your answer as an algebraic expression in terms of $m$. | -2m | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given a point $(1,-2)$ on the terminal side of angle $\alpha$, determine the value of $\sin \alpha$. Express your answer as a simplified fraction or exact decimal, including any necessary square roots or absolute value signs. | -\dfrac{2 \sqrt{5}}{5} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\ln (1-x)$. Express your answer in interval notation. | (-\infty,1) | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \\frac{\\sqrt{3}+i}{(1- \\sqrt{3}i)^{2}}$, calculate the value of $z \\cdot \\bar{z}$, where $\\bar{z}$ is the conjugate of $z$. Express your answer as a single numerical value. | \\frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Two positive numbers x and y are in the ratio a:b, where 0 < a < b. Given that x+y = c, calculate the value of the smaller of x and y in terms of a, b, and c. | \frac{ac}{a+b} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
A class is preparing for Christmas. On the first day, they work for 15 minutes. For the next five days, each day's work time is double the previous day's. Calculate the work time on the sixth day in hours. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the sum of the first and ninth terms is given as a_1 + a_9 = 10. Find the value of the fifth term, a_5. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Let $z=\frac{{2(3+5i)}}{{{{(1-i)}^2}}}$, then find $\overline{z}$. Express your answer as a complex number in the form a + bi. | -5-3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $E$: $\\dfrac{x^{2}}{a^{2}} - \\dfrac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) with an eccentricity of $\\dfrac{\\sqrt{7}}{2}$, find the equation of the asymptotes of $E$. Express your answer in the form $y = mx$, where $m$ is the slope. | y = \\pm \\dfrac{\\sqrt{3}}{2}x | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Four students randomly select one of two universities, A and B, to visit. Calculate the probability that both universities A and B are visited by the students. Express your answer as a simplified fraction. | \frac{7}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A class meeting is to be hosted by one person selected from a group of 3 female students and 2 male students. How many different ways are there to select the host? Express your answer as a single whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $\frac{2\sin 80{}^\circ -\sin 20{}^\circ }{\cos 20{}^\circ }$. Express your answer as a single value, using radicals if necessary. | \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality -1 < x < 0, simplify the expression sqrt(x^2) - sqrt((x+1)^2). Express your answer as a simplified algebraic expression. | -2x-1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^2=4x$ with focus $F$, the directrix $l$ intersects the x-axis at point $K$, and a point $P$ on the parabola. If $|PF|=5$, calculate the area of $\Delta PFK$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Mr. Chen took a taxi ride that cost him 15 yuan, and the meter showed a waiting time of 11 minutes and 30 seconds. The taxi fare is standardized as follows: The initial fare is 6 yuan for boarding, and this fare applies to trips not exceeding 2km. For distances beyond 2km, an additional charge of 1.5 yuan/km is applied... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations of two circles: $x^2+y^2-4x+6y=0$ and $x^2+y^2-6x=0$, find the equation of the line passing through their intersection points A and B. Express your answer in the standard form of a linear equation. | x+3y=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Express the value of 351.37 billion yuan in scientific notation. | 3.5137\times 10^{11} | big_math | [
"Mathematics -> Other -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola C: $$\\frac{x^{2}}{a^{2}}$$ - $$\\frac{y^{2}}{b^{2}}$$ = 1 (a > 0, b > 0), one of its asymptotic line equations is y = - $$\\frac{\\sqrt{5}}{2}$$x. Calculate the eccentricity of the hyperbola C. Express your answer as a fraction or decimal value. | $$\\frac{3}{2}$$ | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the following expression:
\[\frac{\left(3^{2008}\right)^2-\left(3^{2006}\right)^2}{\left(3^{2007}\right)^2-\left(3^{2005}\right)^2}\]
Express your answer as a simplified numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, and the set $$M=\{z|z=( \frac {i-1}{i+1})^{n}，n\in\mathbb{N}^{*}\}$$, calculate the number of non-empty subsets of set $M$. Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given points $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, respectively. A line passing through $F_{1}$ and perpendicular to the x-axis intersects the ellipse at points $A$ and $B$. If triangle $ABF_{2}$ is an equilateral triangle, calculate the eccentricity $e$... | \frac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Cindy leaves school at the same time every day. If she cycles at 20 km/h, she arrives home at 4:30 in the afternoon. If she cycles at 10 km/h, she arrives home at 5:15 in the afternoon. At what speed, in km/h, must she cycle to arrive home at 5:00 in the afternoon? Express your answer as a single number in km/h. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $\overset{⇀}{a} =(-2,3)$ and $\overset{⇀}{b} =(1,2)$, if the vector $λ\overset{⇀}{a} +\overset{⇀}{b}$ is perpendicular to $\overset{⇀}{b}$, find the value of the real number $λ$. Express your answer as a single real number. | -\dfrac{5}{4} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Convert the decimal number 2015 into binary. Express your answer in binary (base 2), including the base in your answer (e.g., 11111011111_{(2)}). | 11111011111_{(2)} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given a triangle ABC with points P, Q, R lying on the sides AB, BC, CA, respectively, and BP = PQ = QR = RC = 1, find the maximum possible area of triangle ABC. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_n\}$, where $a_1=3$ and $a_{n+1}=a_n+2 (n\in \mathbb{N}^*)$, calculate the sum of the first 10 terms, $S_{10}$. Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
A tower with 7 floors has a total of 381 lights. The number of lights on the next lower floor in each adjacent pair of floors is twice that of the upper floor. Calculate the number of lights at the top of the tower. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that line $l$ passes through the right focus $F$ of the hyperbola $\frac{x^{2}}{12} - \frac{y^{2}}{4} = 1$ and is perpendicular to the asymptotes of the hyperbola in the first and third quadrants, determine the equation of line $l$. Express your answer in the form $y = mx + b$. | y = -\sqrt{3}x + 4\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
The circle x^2+y^2+2x-6y+1=0 is symmetric with respect to the line ax-by+3=0 (a>0, b>0). Calculate the minimum value of 1/a + 3/b. | \frac{16}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A line passes through point M(-2, m) and point N(m, 4) with a slope of 1. Find the value of m. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_1 = -2 and a_5 = -8. Find the value of a_3. Express your answer as a single numerical value. | -4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Four middle school students are sent to three different communities to conduct social surveys, with at least one person going to each community. Calculate the total number of different dispatch plans possible. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Four students each choose one day to participate in a charity event, either on Saturday or Sunday. Calculate the probability that there are students participating in the charity event on both Saturday and Sunday. Express your answer as a simplified fraction. | \frac {7}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the numbers $49, 29, 9, 40, 22, 15, 53, 33, 13, 47$, group them in pairs so that the sum of each pair is the same. What number is paired with 15? Express your answer as a single number from the original list. | 47 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A rectangle is transformed into a square by doubling its width and halving its length. The perimeter of the resulting square is P. Find the perimeter of the original rectangle. Express your answer in terms of P. | \frac{5}{4}P | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
How many of the following five expressions give answers which are not prime numbers?
$$
1^{2}+2^{2}
$$
$2^{2}+3^{2}$
$3^{2}+4^{2}$
$4^{2}+5^{2}$
$5^{2}+6^{2}$
Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Find the direction vector of the line given by the equation x - 3y + 1 = 0. Express your answer as an ordered pair (x, y). | (3, 1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of non-negative integer solutions $(x, y)$ to the equation $3x^2 + y^2 = 3x - 2y$. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of elements in the set $\{x|C_{10}^{x} \leq 20\}$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the geometric sequence {a_n}, where a_1 + a_2 = 2 and a_4 + a_5 = 4, find the value of a_10 + a_11. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
If we split a cube with an edge of $1 \mathrm{~m}$ into small cubes with an edge of $1 \mathrm{~mm}$, what will be the height of a column formed by stacking all the small cubes one on top of another? Express your answer in kilometers (km) as a whole number. | 1000 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $C: x^2 = 4y$ and a fixed point A(0, 1), if the distance from a point P on the parabola to point A is 2, calculate the distance from P to the x-axis. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
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