instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Find the equation of the tangent line to the parabola y = x^2 + x + 1 that passes through the point (0,1). Express your answer as a linear equation in the form Ax + By + C = 0. | x - y + 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $P(AB)=\frac{3}{10}$ and $P(B|A)=\frac{1}{2}$, calculate the value of $P(A)$. Express your answer as a fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given \( \alpha, \beta \in (0, \pi) \), and \( \tan \alpha, \tan \beta \) are the roots of the equation \( x^{2} + 3x + 1 = 0 \), calculate the value of \( \cos(\alpha - \beta) \). Express your answer as a simplified fraction. | \frac{2}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities",
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If the lines $y=2x-1$ and $y=kx$ are perpendicular to each other, find the value of the real number $k$. Express your answer as a single real number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let the function f(x) = x^2 - 2x - 3. If a real number x_0 is randomly selected from the interval [-2, 4], then calculate the probability that the selected real number x_0 satisfies f(x_0) ≤ 0. Express your answer as a simplified fraction. | 2/3 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 4 different balls into 3 different boxes, with each box containing at least 1 ball. How many different ways are there to do this? Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
There are 66 cats in my street. I don't like 21 of them because they catch mice. Of the rest, 32 have stripes and 27 have one black ear. What is the smallest possible number of cats that have both stripes and one black ear? | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of $\sin {210^{0}}$. Express your answer as a simplified fraction. | -\frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given $sinα=\frac{4}{5}$, $α∈(\frac{π}{2},π)$, calculate the value of $\cos 2\alpha$. Express your answer as a fraction. | -\frac{7}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence $\{a_n\}$, let the sum of the first $n$ terms be $S_n$. If $S_{m-1} = -2$, $S_m = 0$, and $S_{m+1} = 3$, find the value of $m$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first 5 terms of an arithmetic sequence is 25, and the second term is 3, find the value of the seventh term of the sequence. Express your answer as a single numerical value. | 13 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Using the method of completing the square, transform the quadratic equation $x^{2}-4x=5$ into the form $(x+a)^{2}=b$. Then, calculate the value of $a+b$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola defined by x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0), and one of its asymptotes intersecting the circle x^2 + (y-2)^2 = 4 to form a chord of length 2*sqrt(3), calculate the eccentricity of the hyperbola. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with a common difference of -2 and its first term a_1 = 7, find the smallest natural number n that makes the sum of its first n terms, S_n, less than 0. Express your answer as a single natural number. | 9 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (2z+3)i = 3z, find the conjugate of z, denoted as \\overline{z}. Express your answer as a complex number in the form a + bi, using exact fractions. | -\\frac{6}{13} - \\frac{9}{13}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the real part and the imaginary part of the complex number $\frac{i}{1+i}-\frac{1}{2i}$. Express your answer as a single fraction or integer in the form $\boxed{[answer]}$. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \( (5^2 - 4^2)^3 \) and provide your answer as a single numerical value. | 729 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
For the inequality $-4x-k\leqslant 0$, if the negative integer solutions are $-1$ and $-2$, calculate the range of values for $k$. Express your answer in interval notation (e.g., $[a, b)$ or $(a, b]$). | [8, 12) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A circle passes through the vertices of a triangle with side-lengths $7\\tfrac{1}{2}, 10, 12\\tfrac{1}{2}.$ Find the radius of the circle. Express your answer as a fraction. | \\frac{25}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The packaging waste generated by the express delivery industry in China was about 4 million tons in 2015. The annual growth rate in 2016 was 50%. Experts predict that if no measures are taken, the packaging waste will continue to grow at this rate in the future. Calculate the year when the packaging waste generated by ... | 2021 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a}=(10,20)$, $\overrightarrow{b}=(5,5)$, $\overrightarrow{c}=\overrightarrow{a}+k\overrightarrow{b}$, if $\overrightarrow{b} \perp \overrightarrow{c}$, find the value of the real number $k$. Express your answer as a single real number. | -3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
The extensions of a telephone exchange have only 2 digits, from 00 to 99. Not all extensions are in use. By swapping the order of two digits of an extension in use, you either get the same number or the number of an extension not in use. What is the maximum possible number of extensions that can be in use? Provide your... | 55 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Select three different numbers from the set {1,2,3,...,10} such that these three numbers form a geometric sequence. Count the number of such geometric sequences. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A regular octagon has 8 vertices. If a triangle is formed by randomly choosing 3 vertices of the octagon, what is the probability that at least one side of the triangle is also a side of the octagon? Express your answer as a simplified fraction. | \frac{5}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $z$ is a complex number, $\overline{z}$ is the conjugate of $z$, and $\overline{z}=|z|-1+5i$, find the imaginary part of $z$. Express your answer as a single number. | -5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the perimeter p of an equilateral triangle inscribed in a circle, calculate the area of the circle in terms of p and pi (π). | \frac{\pi p^2}{27} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
How many three-digit numbers are divisible by 13? Express your answer as a whole number. | 69 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains 8 yellow marbles, 7 red marbles, and 5 black marbles. Without looking in the bag, Igor removes N marbles all at once. What is the maximum possible value of N such that, no matter which choice of N marbles he removes, there are at least 4 marbles of one color and at least 3 marbles of another color left i... | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The vertex of parabola C is at the origin, and its focus is on the x-axis. The line x-y=0 intersects parabola C at points A and B. If P(1,1) is the midpoint of segment AB, then what is the equation of parabola C? Express your answer in the standard form of a parabola equation. | y^2=2x | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Triangle $A B C$ has vertices $A(1,2), B(4,0)$ and $C(1,-4)$. Calculate the area of $\triangle A B C$. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the time duration in seconds that is equivalent to the product $60 \times 60 \times 24 \times 7$. Express your answer in seconds. | 604800 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit, and let $\bar{z}$ denote the complex conjugate of the complex number $z$. If $z = 1 + i$, then calculate the value of $-i \cdot z + i \cdot \bar{z}$. Provide your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $P=\{x\in \mathbb{N} | x(x-3)\leqslant 0\}$ and $Q=\{x | x\geqslant 2\}$, calculate the intersection $P \cap Q$. Express your answer as a set of numbers. | \{2,3\} | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers x and y satisfy 2^x + 2^y = 1, find the maximum value of x + y. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence, the sum of the first m terms is 30, and the sum of the first 2m terms is 100. Find the sum of the first 3m terms. | 210 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
There are 3 different mathematics books and 3 different Chinese books placed on the same shelf of a bookcase. Calculate the number of ways to arrange the books such that books of the same type are not adjacent. Express your answer as a single numerical value. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the unit imaginary number, and $z(1+i) = \left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i \right)^3$, determine the quadrant of the complex plane where the conjugate of $z$ is located. Express your answer using standard quadrant notation (I, II, III, IV). | I | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence ${a_n}$ is an arithmetic sequence, and the sequence ${b_n}$ is a geometric sequence, where $({a_{2017}}+{a_{2018}}=π )$ and $({{b_{20}}}^2=4 )$, find the value of $\tan \frac{{a_{2}}+{a_{4033}}}{{b_{1}}{b_{39}}}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
For any whole number $n$, what is the smallest number that always divides the expression $n^2(n^2 - 1)$? Provide your answer as a single integer. | 12 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^{2}=2px$ with the latus rectum line equation $x=-2$, find the value of $p$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
How many different ways are there to express 2016 as the sum of a natural number multiple of 2 and a natural number multiple of 3? Express your answer as a single integer. | 337 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 2034000 in scientific notation. Your answer should be in the form of a number between 1 and 10 multiplied by 10 raised to a power. | 2.034 \times 10^6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
How many different ways can 5 letters be placed into 3 mailboxes? Express your answer using exponential notation. | 3^5 \text{ ways} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A number \( N \), when expressed in base 7, is a three-digit number. When expressed in base 9, its digits are exactly reversed. Find the middle digit of \( N \). Express your answer as a single digit. | 0 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A particle moving in a straight line is subjected to a force F(x) = 1 - e^(-x) at any position x. Calculate the work done by the force F(x) as the particle moves from x_1 = 0 to x_2 = 1 along the x-axis. Express your answer as a single value. | \frac{1}{e} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Randomly split 2.5 into the sum of two non-negative numbers. Then, round each number to its nearest integer. What is the probability that the sum of the two resulting integers is 3? Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A list of six positive integers \( p, q, r, s, t, u \) satisfies \( p < q < r < s < t < u \). The sums of the 15 pairs of numbers that can be formed by choosing two different numbers from this list are:
\[ 25, 30, 38, 41, 49, 52, 54, 63, 68, 76, 79, 90, 95, 103, 117 \]
Calculate the sum \( r+s \) using the given info... | 54 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the number triangle shown, each disc is to be filled with a positive integer. Each disc in the top or middle row contains the number which is the product of the two numbers immediately below. What is the value of \( n \)? Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is symmetric about the line x = -2 and has a period of 2, and when x ∈ [-3, -2], f(x) = (x + 2)^2, find the value of f(5/2). Express your answer as a single numerical value. | \frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation: $1 - \frac{1}{1-x} = \frac{1}{1-x}$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the decimal representation of \(\frac{1}{7}\) as \(0.\overline{142857}\), what is the 100th digit to the right of the decimal point? Provide your answer as a single digit. | 8 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The general term formula of the sequence $\{a_n\}$ is $a_n = 2n - 49$. Find the value of $n$ when the sum of the first $n$ terms, $S_n$, reaches its minimum. Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{AB}=(2,3)$ and $\overrightarrow{AC}=(-1,2)$, calculate the vector $\overrightarrow{CB}$ using vector subtraction. Express your answer as an ordered pair (x, y). | (3,1) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
There are 4 teachers teaching mathematics in 4 different classes of the same grade. During a math test, each teacher is not allowed to proctor in their own class. Calculate the number of ways to arrange the proctoring. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Joe filled up his car's gas tank. After traveling 165 km, 3/8 of the gas in the tank was used. At this rate, approximately how much farther can the car travel before its fuel tank is completely empty? Express your answer in kilometers as a whole number. | 275 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
What is the reciprocal of 0.2? Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Analyze the given frequency distribution table of average weekly labor time spent at home by seventh-grade students. Identify the correct statement about the median of the average weekly household labor time based on the information provided in the table.
| Group | One | Two | Three | Four |
|-------|-----|-----|-----... | The median of the average weekly household labor time falls into Group Two. | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2^x, if a real number x is randomly chosen from the interval [-2,2], then find the probability that the inequality f(x) > 2 holds. Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
If the terminal side of angle α passes through the point P0(-3, -4), calculate the value of tan α. Express your answer as a fraction in simplest form. | \frac{4}{3} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Tangent"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the definite integrals S1 = ∫[1,2] x dx, S2 = ∫[1,2] e^x dx, and S3 = ∫[1,2] x^2 dx, find the ordering of S1, S2, and S3 from smallest to largest. Express your answer using the less than (<) symbol (e.g., S1 < S2 < S3). | S1 < S3 < S2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, point $D$ is the midpoint of side $BC$. Let $\overrightarrow{AB}=\overrightarrow{a}$ and $\overrightarrow{AC}=\overrightarrow{b}$. Find the vector $\overrightarrow{AD}$ in terms of $\overrightarrow{a}$ and $\overrightarrow{b}$. Express your answer as a vector equation. | \frac{1}{2}(\overrightarrow{a} + \overrightarrow{b}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of a circle with its center at (a, a) (where a ≠ 0) and passing through the origin. Express your answer in the standard form of the equation of a circle. | (x-a)^2 + (y-a)^2 = 2a^2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the constraints \(3 \leq p \leq 10\) and \(12 \leq q \leq 21\), calculate the difference between the largest and smallest possible values of \(\frac{p}{q}\). Express your answer as a simplified fraction. | \frac{29}{42} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
When a die is rolled twice in succession, what is the probability that the numbers facing up are consecutive natural numbers? Express your answer as a simplified fraction. | \dfrac{5}{18} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
If $k$, $-1$, and $b$ form an arithmetic sequence, find the fixed point that the line $y = kx + b$ must pass through. Express your answer as an ordered pair (x, y). | (1, -2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The cost of 1 piece of gum is 1 cent. What is the total cost of 1000 pieces of gum in dollars? Express your answer rounded to two decimal places. | 10.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overrightarrow{a}=(2,x)$ and $\overrightarrow{b}=(1,2)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of the real number $x$. Express your answer as a single real number. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
The following twelve integers are written in ascending order:
$$
1, x, x, x, y, y, y, y, y, 8, 9, 11
$$
The mean of these twelve integers is 7. Calculate the median of these twelve integers. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the parabola $x=4y^2$, find the coordinates of its focus. Express your answer as an ordered pair $(x, y)$. | \left(\frac{1}{16},0\right) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequences \( \{a_n\} \) and \( \{b_n\} \), let \( S_n \) and \( T_n \) be the sums of the first \( n \) terms of these sequences, respectively. It is given that
\[ \frac{S_n}{T_n} = \frac{2n + 6}{n + 1}. \]
If \( \frac{a_m}{b_m} \) is a prime number, find the value of the positive integer \( m \)... | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of \(2 \frac{1}{10} + 3 \frac{11}{100} + 4 \frac{111}{1000}\) and express your answer as a decimal with three places. | 9.321 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Approximate the product of $2.46\times 8.163\times (5.17+4.829)$ to the nearest whole number. | 200 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $a$, $b$, $c$, $d$ satisfying the equations
$$\frac{a - 2e^a}{b} = \frac{1 - c}{d - 1} = 1,$$
where $e$ is the base of the natural logarithm, find the minimum value of $(a-c)^2 + (b-d)^2$. | 8 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the definite integral $\int_{\frac{\pi}{4}}^{\frac{9\pi}{4}} \sqrt{2}\cos(2x + \frac{\pi}{4})dx$. Express your answer as a single number in the format $oxed{[answer]}$. | 0 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the greatest common divisor (GCD) of the products $360 \times 473$ and $172 \times 361$. Provide your answer as a single integer. | 172 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given angle α is in the fourth quadrant, and cos(α) = 3/5, calculate the value of the expression: (1 + √2 * cos(2α - π/4)) / sin(α + π/2). Express your answer as a fraction. | -\frac{2}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation \(\frac{1}{x-1}=\frac{2}{x-2}\). Express your answer as a single value of x in the format \(\boxed{\text{your answer here}}\). | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)$, defined on $\mathbb{R}$, have an inverse function $f^{-1}(x)$, and for any $x \in \mathbb{R}$, it always holds that $f(x) + f(-x) = 2$. Calculate the value of $f^{-1}(2008-x) + f^{-1}(x-2006)$. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $$\lim_{\Delta x \to 0} \frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}=k$$, find the value of $$\lim_{\Delta x \to 0} \frac{f(x_0+2\Delta x)-f(x_0)}{\Delta x}$$ in terms of k. | 2k | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence ${a_{n}}$ with its first $n$ terms sum $S_{n}$, where $a_{n} = \frac{1}{(2n-1)(2n+1)}$, find the value of $S_{5}$. Express your answer as a simplified fraction. | \frac{5}{11} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation (5 - 3x)^5 = -1. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance between two adjacent intersection points of the line y=a and the curve y=tan(x). Express your answer in terms of pi. | \pi | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
If \( 3x + 4 = x + 2 \), solve for the value of \( x \). Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the point P(2,5) and the line equation x+y=0, find the coordinates of the point symmetric to P with respect to the line. Express your answer as an ordered pair (x, y). | (-5, -2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic progression {x, a1, a2, a3,..., an, y} where n real numbers are inserted between x and y, find the common difference of this sequence. Express your answer as a fraction in terms of x, y, and n. | \frac{y-x}{n+1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the pattern in the squares of numbers consisting of consecutive 1's: $11^{2}=121$, $111^{2}=12321$, $1111^{2}=1234321$, find the result of $1111111^{2}$. Express your answer as a single numerical value. | 1234567654321 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that tan(θ) = -2, simplify the expression sin(θ)(1 + sin(2θ)) / (sin(θ) + cos(θ)). Express your answer as a simplified fraction. | \frac{2}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x, y$ are positive real numbers, and $x + y + \frac{1}{x} + \frac{1}{y} = 5$, find the maximum value of $x + y$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, 2)$, $\overrightarrow{b} = (2, -3)$, find the value of vector $\overrightarrow{c}$ that satisfies $\overrightarrow{c} \perp (\overrightarrow{a} + \overrightarrow{b})$ and $\overrightarrow{b} \parallel (\overrightarrow{a} - \overrightarrow{c})$. Express your answer as an ordered p... | (\frac {7}{9}, \frac {7}{3}) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = 2x^2 at the point P(1, 2). Express your answer in the standard form of a line (Ax + By + C = 0). | 4x - y - 2 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial expression (3x - 1)^7 = a_7 x^7 + a_6 x^6 +... + a_0, calculate the sum of the coefficients a_7 + a_6 +... + a_0. Express your answer as a single integer. | 128 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
In the following right-angled triangle \( ABC, \triangle C = BC = 1 \) and \( DEF \) is an arc of a circle with center \( A \). Suppose the shaded areas \( BDE \) and \( CEF \) are equal and \( AD = \frac{x}{\sqrt{\pi}} \). Find the value of \( x \). Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given \( x \in \left(0, \frac{\pi}{2}\right) \), calculate the integer part of the number \( m = 3^{\cos^2 x} + 3^{\sin^5 x} \). Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.3125 | 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 -> Linear Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given condition \(p: x > 2\), condition \(q: x > a\), and \(p\) is a necessary but not sufficient condition for \(q\), derive the range of values for \(a\). Express your answer as an inequality involving the variable \(a\). | a > 2 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The 26 letters of the English alphabet are listed in an infinite, repeating loop:
ABCDEFGHIJKLMNOPQRSTUVWXYZABC...
Find the 258th letter in this sequence. Express your answer as a single letter. | X | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Pat leaves Coe Hill at 1:00 p.m. and drives at a speed of 80 km/h for the first 60 km of the 150 km trip to Calabogie. If he must arrive at Calabogie by 3:00 p.m., how fast must he travel for the remainder of the trip? Express your answer in km/h, rounded to the nearest whole number. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = x / (x + 2) at the point (-1, -1). Express your answer in slope-intercept form (y = mx + b). | y = 2x + 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\sqrt{(x-3)^2}=x-3$, find the range of values for $x$ that satisfy this equation. Express your answer as an inequality in the format $x \geq a$ or $x \leq a$. | x \geq 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are two consecutive integers and that $a < \sqrt{7} < b$, calculate the sum of $a$ and $b$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.9375 | 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.