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 set A = {1, 2, 3} and set B = {x | x(x+1)(x-2) < 0, x ∈ Z}, compute A ∩ B. Express your answer as a set of integers. | \{1\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the integral $$∫_{ 0 }^{ 1 }(-x^2 - 1)dx.$$ Express your answer as a single number. | - \frac {4}{3} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, find the imaginary part of the complex number $z=\frac{1-i}{3-i}$. Express your answer as a fraction. | -\frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the modulus of the complex number $1 + \cos\alpha + i\sin\alpha$, given that $\pi < \alpha < 2\pi$. Express your answer as a function of $\alpha$. | 2|\cos\frac{\alpha}{2}| | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $C: \\dfrac{y^{2}}{a^{2}}-\\dfrac{x^{2}}{b^{2}}=1$ where the length of the real axis is $\\sqrt{2}$ times the length of the imaginary axis, find the equation of the asymptotes of hyperbola $C$. Express your answer in the form $y = \\pm kx$. | y = \\pm \\sqrt{2}x | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {1, 2, 3} and B = {-1, 3}, find the union of sets A and B. Express your answer as a set of unique elements, using set notation with curly brackets {}, and separate elements with commas. | {-1, 1, 2, 3} | big_math | [
"Mathematics -> Other -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
If \( k=2 \), calculate the value of \( (k^{3}-8)(k+1) \). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{\cos^{2} 5^{\circ} - \sin^{2} 5^{\circ}}{\sin 40^{\circ} \cos 40^{\circ}}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of lines that pass through the fixed point (2,3) and have equal intercepts on both coordinate axes. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of $60^{{∘}}$ between them, and knowing that $|\overrightarrow{b}|=1$ and $|2\overrightarrow{a}-\overrightarrow{b}|=1$, calculate the magnitude of vector $\overrightarrow{a}$. Express your answer as a single numerical value. | \dfrac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Rectangle PQRS is divided into three identical squares. If PQRS has a perimeter of 120 cm, what is its area in cm^2? Express your answer as a whole number. | 675 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) = ae^x - sin(x) has an extreme value at x = 0. Determine the value of a. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the constraints \( 10 \leq x \leq 20 \) and \( 40 \leq y \leq 60 \), calculate the largest possible value of the expression \( \frac{x^{2}}{2y} \). Provide your answer as a single number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given point A(-2,1) is symmetric to point B with respect to the origin, find the coordinates of point B. Express your answer as an ordered pair (x, y). | (2,-1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, let S_n denote the sum of the first n terms. Given that S_3 = 8 and S_6 = 7, find the value of a_7 + a_8 + a_9. Express your answer as a fraction. | 1/8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the functional equation \( f(x+y) = f(x) \cdot f(y) \) and \( f(1) = 2 \), calculate the value of the expression:
\[
\frac{f(2)}{f(1)} + \frac{f(4)}{f(3)} + \frac{f(6)}{f(5)} + \cdots + \frac{f(2000)}{f(1999)}
\]
Express your answer as a single integer. | 2000 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence with a non-zero first term and a non-zero common ratio, and an arithmetic sequence with a first term equal to 0, a third sequence is formed by adding the corresponding terms of the first two sequences. The first few terms of the third sequence are 1, 1, 2,.... Find the sum of the first ten te... | 978 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 people standing in a row. Calculate the number of arrangements where there is exactly 1 person between A and B. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
How many perfect squares are there between 2 and 20? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = -x^2 - cos(x), find the solution set of the inequality f(x-1) > f(-1). Express your answer in interval notation. | (0,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that each interior angle of a polygon is $120^{\circ}$, determine the number of sides of the polygon. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality series \( 1-x > (1-x)^{2} > (1-x)^{3} > (1-x)^{4} > \cdots \), determine the range of values for \( x \). Express your answer as an interval or inequality. | 0 < x < 1 | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Convert the number 37,600 into scientific notation. Express your answer in the form a × 10^n, where 1 ≤ a < 10. | 3.76 × 10^4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
An unequal-sided triangle has sides that are integers, and its perimeter is less than 13. Count the number of such triangles. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $$\frac {4+mi}{1+2i}$$ is a real number, find the magnitude of $$|m+6i|$$. Express your answer as a single number. | 10 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
If the decimal number 26 is equal to the number 32 in base k, find the value of k. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $2^{\log_{3} x} = \frac{1}{4}$. Express your answer as a simplified fraction. | \frac{1}{9} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = (ax)/(x^2 + 1) with a > 0, find the interval where f(x) is increasing. Express your answer as an interval, using interval notation (e.g., (-1, 1)). | (-1, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\left\{\begin{array}{l}x=1\\ y=-1\end{array}\right.$ is a solution of the equation $2x-ay=3$, find the value of $a$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The endpoints of segment AB are A(-1, 4) and B(-4, 1), respectively. Now, it is translated 4 units to the left to obtain segment A1B1. Find the coordinates of A1 and B1 after the translation. Express your answer in the format (x1, y1), (x2, y2). | (-5, 4), (-8, 1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
As shown in the figure, each small circle has a diameter equal to the radius of the large circle. If the combined area of the two small circles is 1, what is the area of the shaded region? Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $a^{2}-4a-12=0$, calculate the value of $-2a^{2}+8a+40$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Let z = (1+i)/sqrt(2). Evaluate the product of the two summations:
(sum from k=1 to 12 of z^(k^2)) * (sum from k=1 to 12 of 1/z^(k^2))
Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given a line passing through points A (-2, m) and B (m, 4) is parallel to the line $2x+y-1=0$, find the value of $m$. Express your answer as a single integer. | -8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Let {a_n} be an arithmetic sequence with a common difference d = -2, and let S_n be the sum of its first n terms. If S_{11} = S_{10}, then find the value of the first term a_1. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a1 = (1,0), a2 = (1,-1), a3 = (2,2) that are linearly dependent, find the value of k1 + 4k3 based on the linear dependence condition k1*a1 + k2*a2 + k3*a3 = 0. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
The equation $x^{2}-4x+m=0$ has two distinct real roots for $x$. Find the range of values for $m$ that satisfies this condition. Express your answer as an inequality. | m < 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given $3\sin\theta+\cos\theta=\sqrt{10}$, calculate the value of $\tan(\theta+\frac{\pi}{8})-\frac{1}{\tan(\theta+\frac{\pi}{8})}$. Express your answer as a single integer. | -14 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
How many perfect squares are there between 2 and 20? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
What is the smallest prime number that divides the sum $3^{11}+5^{13}$? Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The numbers x, y, p, and q are all integers, with p and q being positive constants. The four integers are related by the equation xy = px + qy. When y takes its maximum possible value, calculate the expression equal to y - x in terms of p and q. | (p-1)(q+1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with the sum of the first n terms denoted as s_n, and S_2 = 10, S_5 = 55, calculate the slope of the line passing through points P(n, a_n) and Q(n+2, a_{n+2}) (n ∈ ℕ*). Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x-1)=x^2+4x-5$, calculate the value of $f(x+1)$. Express your answer as a polynomial expression. | x^2+8x+7 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = e^x, find the value of the derivative f'(x) at x = -2. Express your answer as a single mathematical expression. | e^{-2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given three points P1 (1, 1, 0), P2 (0, 1, 1), and P3 (1, 0, 1), with O as the origin, find the magnitude of the vector $\overrightarrow{OP_1} + \overrightarrow{OP_2} + \overrightarrow{OP_3}$. Express your answer as a numerical value with a radical if necessary. | 2\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Four vertices of a quadrilateral are located at \((7,6), (-5,1), (-2,-3)\), and \((10,2)\). Calculate the area of the quadrilateral in square units. | 63 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
In parallelogram $ABCD$, the lengths of the diagonals $AC$ and $BD$ are $16$ and $12$ respectively. Using the properties of parallelograms and the triangle inequality theorem, calculate the range of the length of side $AB$. Express your answer as an inequality. | 2 < AB < 14 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function $f(x) = \sqrt{3}\sin(\frac{x}{2} - \frac{\pi}{4})$. Express your answer in terms of $\pi$. | 4\pi | big_math | [
"Mathematics -> Precalculus -> Functions -> Trigonometric Functions"
] | 0.671875 | 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, including all elements of the intersection. | \{2,3\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{1,3,7,8\}$ and $B=\{1,5,8\}$, determine the value of $A\cup B$. Express your answer as a set. | \{1,3,5,7,8\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a, b, c,$ and $d$ are positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$, find the prime number that must be a divisor of $a$. Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a, b \in \mathbb{R}$, and $a^2 + 2b^2 = 6$, find the minimum value of $a + b$. Express your answer as a single real number, including a negative sign if applicable. | -3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} that satisfies a_1 + a_2 = 3 and a_2 + a_3 = 6, find the value of a_7. Express your answer as a single numerical value. | 64 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit, and $\overset{.}{z}$ be the conjugate of the complex number $z$. If $z= \frac {2}{-1+i}$, calculate the value of $\overset{.}{z}$. Express your answer as a complex number in the form $a + bi$. | -1+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Insert a number into the sequence 9, 1, 15, 21, 4 so that the median becomes 8. What is the value of the inserted number? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Calculate the minimum value of $a$. Express your answer as a single value. | 3\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the product of 13 and 50. Express your answer in tens. | 65 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
An equilateral triangle is drawn with a side of length $a$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. Calculate the limit of the sum of the perimeters of... | 6a | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that one root of the quadratic equation $x^{2}-ax+6=0$ is $2$, find the value of $a$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The sequence {a_n} satisfies a_1=1, a_2=2, 2a_{n+1}=a_n+a_{n+2}, and b_n = 1/(a_n * a_{n+1}). Calculate the sum of the first 5 terms of the sequence {b_n}. Express your answer as a simplified fraction. | \frac{5}{6} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the initial function y = f(x) with graph c, the graph c1 is obtained by reflecting c about the line x = 1, and the graph c2 is obtained by shifting c1 one unit to the left. Find the function corresponding to the graph c2. Express your answer as a function of x, using function notation (e.g., f(x)). | y = f(1 - x) | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial equation x^4 - 7x^3 + 14x^2 - 14x + 4 = 0, which can be factored as (x^2 - 5x + 2)(x^2 - 2x + 2) = 0, find the sum of the real roots. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The curve of the equation $x^2+y^2-2x+6y+k=0$ is a circle. Find the range of values for $k$ that satisfies the inequality for real circles. Express your answer as an inequality involving $k$. | k<10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n} with a_n > 0, given that a_3 * a_7 = 64, find the value of a_5. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic progression where the sum of the first ten terms is four times the sum of the first five terms, find the ratio of the first term to the common difference. Express your answer in the form a:b. | 1:2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Person A can complete an engineering project alone in m hours. When working together with person B, they can complete the project in 20 hours. Calculate the time needed for person B to complete the project alone. Express your answer in hours, using the variable m in your formula. | \frac{20m}{m-20} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Two spinners are spun once, and each lands on one of the numbered sectors. The first spinner has sectors numbered 1, 3, and 5, while the second spinner has sectors numbered 2, 4, and 6. Calculate the probability that the sum of the numbers in the two sectors is prime. Express your answer as a simplified fraction. | \frac{8}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A subway train arrives every 10 minutes and stops at the station for 1 minute. What is the probability that a passenger arrives at the platform and boards the train immediately? Express your answer as a simplified fraction. | \dfrac{1}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation y^2 = 4x, a line passes through the focus of the parabola and intersects it at points A and B. If the x-coordinate of the midpoint of segment AB is 2, calculate the length of chord AB. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)=x^2-4x+3$, $g(x)=3x-2$, and the sets $M=\{x\in\mathbb{R}|f(g(x))>0\}$, $N=\{x\in\mathbb{R}|g(x)<2\}$. Describe the intersection set $M\cap N$ in interval notation. | (-\infty,1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all real roots of the equation \(\left|x^{2}-12 x+34\right|=2\). Express your answer as a single numerical value. | 18 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In the concave quadrilateral ABCD shown, ∠BCD = 90°, AB = 12, BC = 4, CD = 3, and AD = 13. Find the area S of the concave quadrilateral ABCD. Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = 2x^2 that is perpendicular to the line x + 4y - 8 = 0. Express your answer as a linear equation in the form Ax + By + C = 0. | 4x - y - 2 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Negate the proposition "For all $x$ in $\mathbb{R}$, $x \geq 0$". Express your answer as an existential statement. | \exists x \in \mathbb{R}, x < 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the circle with the focus of the parabola y^2 = -8x as its center, and tangent to the directrix of this parabola. Express your answer in standard form, including the center coordinates and the radius squared. | (x+2)^2 + y^2 = 16 | big_math | [
"Mathematics -> Geometry -> Coordinate Geometry -> Circles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the length, width, and height of a rectangular prism are 5, 4, and 3 respectively, calculate the surface area of the rectangular prism. Express your answer in square units as a whole number. | 94 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given a point P is 9 units away from the center of a circle with a radius of 15 units, how many chords passing through point P have integer lengths? Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a$ and $b$ that satisfy $|a|=|b|=2$, and $a\cdot(b-a)=-2$, calculate $|2a-b|$. Express your answer as a simplified radical expression if necessary. | 2\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the center of the circle given by the equation $(x+2)^2+y^2=5$. Express your answer as a coordinate pair in the format $\boxed{(x, y)}$. | (-2, 0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the center of a circle at (1, -1) and a radius of 2, calculate the equation of the circle in the format (x-h)^2 + (y-k)^2 = r^2, where h, k, and r are the given values. Express your answer as an equation. | (x-1)^2 + (y+1)^2 = 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = a^(x - 2) + 1, where a > 0 and a!= 1, find a point (x, y) that the graph of the function necessarily passes through. Express your answer as an ordered pair (x, y). | (2,2) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, a point corresponding to the complex number $z$ is symmetric with respect to the imaginary axis as $\frac{2}{{1-i}}$. Calculate the value of $z$ after applying this symmetry. Express your answer as a complex number in the form $a+bi$. | -1+i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} \sqrt{x},x\geqslant 4 \\ 2^{x},x < 4 \end{cases}$, find the exact value of $f[f(2)]$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $a_{n+2} = a_{n+1} - a_n$, and $a_1 = 2$, $a_2 = 3$, let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$. Calculate the value of $S_{2016}$. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are real numbers, and $$\frac {a+bi}{2-i}=3+i,$$ calculate the value of $a-b$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $c\cos B+b\cos C=2a\cos A$, $a=2$, and the area of $\triangle ABC$ is $\sqrt{3}$, calculate the perimeter of $\triangle ABC$. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1=1$, $a_{n+1}=a_n+1$. Calculate the value of $a_{2014}$. Express your answer as a single integer. | 2014 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Let set $A=\{1,2,3,5,7\}$, $B=\{x\in\mathbb{Z}|1<x\leq6\}$, and the universal set $U=A\cup B$. Calculate the intersection of $A$ and the complement of $B$ with respect to $U$, denoted as $A\cap C_{U}B$. Express your answer as a set of integers, including all elements of the resulting set. | \{1,7\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Let point $P$ be on circle $C_1: x^2 + y^2 + 4x + 2y + 1 = 0$, and point $Q$ be on circle $C_2: x^2 + y^2 - 4x - 4y + 6 = 0$. Calculate the minimum value of $|PQ|$. | 3 - \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given a line $l$ with an inclination angle of $\theta$ is perpendicular to the line $m: x-2y+3=0$, find the value of $\sin 2\theta$. Express your answer as a numerical value. | -\dfrac{4}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $$ \frac{2005^{2} + 2 \times 2005 \times 1995 + 1995^{2}}{800} $$ Express your answer as a single integer. | 20000 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\frac{x}{1-i}=1-yi$, where $x$, $y$ are real numbers, and $i$ is the imaginary unit, calculate the conjugate of $x+yi$. Express your answer in the form $a+bi$, where $a$ and $b$ are real numbers. | 2+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Siobhan received marks of 69, 53, 69, 71, 78, x, and y on seven tests, each out of 100 marks. If her average mark on the seven tests is 66, find the minimum possible value of x. Express your answer as a single numerical value. | 22 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of the expression $(\sqrt{11}+\sqrt{5})^{8} + (\sqrt{11}-\sqrt{5})^{8}$. Provide your answer as a single numerical value. | 903712 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $\left(m-1\right)x^{2}+3x-1=0$, find the condition on $m$ such that the equation remains quadratic. Express your answer as an inequality or equality condition on $m$. | m \neq 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(2x + π/6), find the new function expression when the graph is moved to the right by π/6 units. Express your answer as a function of x in the format y =.... | y = sin(2x - π/6) | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a^{\frac{2}{3}} = \frac{4}{9} (a > 0)$, calculate the value of $\log_{a} \frac{3}{2}$. Express your answer as a fraction, including a negative sign if necessary. | -\frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin(2x) = \frac{1}{7}$, find the numerical value of $\sin^4(x) + \cos^4(x)$. Express your answer as a fraction. | \frac{97}{98} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
At the Gaussian Store, puzzles cost $10 each or $50 for a box of 6 puzzles. If a customer would like exactly 25 puzzles, what is the minimum possible cost in dollars? | 210 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex numbers \( z_{1} \) and \( z_{2} \) correspond to the points \( A \) and \( B \) respectively on the complex plane, where \( \left| z_{1} \right| = 4 \) and \( 4 z_{1}^{2} - 2 z_{1} z_{2} + z_{2}^{2} = 0 \). If \( O \) is the origin, determine the area of \( \triangle OAB \). Express your answer as a mu... | 8 \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
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