instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Given that $\overline{z} = \frac{i}{1-i}$ is the complex conjugate of the complex number $z$, find $z$. Express your answer as a complex number in the form $a+bi$. | -\frac{1}{2} - \frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_3 + a_5 = 20 and a_4 = 8. Find the value of a_2 + a_6. Express your answer as a single integer. | 34 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the value of the algebraic expression $3x^2 - 2x - 1$ is $2$, find the value of the algebraic expression $-9x^2 + 6x - 1$. Express your answer as a single numerical value. | -10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation 3x - 4y = 0 and the circle equation (x-1)^2 + (y-2)^2 = 2, find the length of the chord intercepted by the line on the circle. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The surface area of a large cube is 5400 cm². This cube is cut into a number of identical smaller cubes. Each smaller cube has a volume of 216 cm³. Find the number of smaller cubes. Express your answer as a whole number. | 125 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, with an eccentricity of $\dfrac { \sqrt {3}}{2}$, calculate the value of $\dfrac {a}{b}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Use the completing the square method to solve the equation $x^{2}-4x-7=0$. What is the transformed equation in the exact format it would appear after completing the square? Express your answer in the format of a transformed equation, using the boxed notation for the final answer. | (x-2)^{2}=11 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $a_1=1$ and $a_n - a_{n-1} = 2$ (for $n \geq 2$), find the general term $a_n$. Express your answer as a formula using $n$ to represent the term number. | 2n - 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\overrightarrow{a} = 5\overrightarrow{e}, \overrightarrow{b} = -3\overrightarrow{e}, \overrightarrow{c} = 4\overrightarrow{e},$$ calculate the result of the operation $2\overrightarrow{a} - 3\overrightarrow{b} + \overrightarrow{c}$. Express your answer as a vector in terms of $\overrightarrow{e}$. | 23\overrightarrow{e} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=({2,4})$ and $\overrightarrow{b}=({1,x})$, and knowing that vector $\overrightarrow{a}$ is perpendicular to vector $\overrightarrow{b}$, find the value of the real number $x$. Express your answer as a single real number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of c such that the system of equations |x+y| = 2007 and |x-y| = c has exactly two solutions (x, y) in real numbers. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the functions $f(x)=2\cos^{2}\left( \frac {\pi}{4}+x\right)$ and $g(x)= \sqrt {3}\cos2x$, find the maximum value of $|MN|$ where M and N are the intersection points of the line $x=a$ with the graphs of $f(x)$ and $g(x)$ respectively. | 3 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The line $4x-3y=0$ intersects the circle $(x-1)^2+(y-3)^2=10$. Calculate the length of the chord formed by this intersection. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that is symmetrical about the x-axis to the line y = 2x + 1. Express your answer in the form y = mx + b. | y = -2x - 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Let $x$ and $y$ be positive numbers. Find the minimum value of $(x+y)\left(\frac{1}{x} + \frac{4}{y}\right)$. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f\left(x\right)=\cos 2x\cdot \ln x$, find the derivative of $f\left(x\right)$. Express your answer as a mathematical expression. | -2\sin 2x\cdot \ln x + \frac{\cos 2x}{x} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the real and imaginary parts of the complex number $\frac{1+3i}{1-i}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that point M is on the parabola x²=4y, find the minimum distance from point M to the line y=x-3. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given a real number $\lambda > 0$, find the minimum value of $\lambda$ such that the inequality ${e^{\lambda x}} - \frac{{\ln x}}{{2\lambda }} \ge 0$ holds for any $x \in (0, +\infty)$. Express your answer as a single value. | \frac{1}{2e} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of the real number a such that the inequality |x+3| + |x-1| >= a^2 - 3a holds true for any real number x. Express your answer using interval notation. | [-1, 4] | big_math | [
"Mathematics -> Algebra -> Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
One lap of a standard running track is 400 meters. Calculate the number of laps each athlete runs in a 5000 meter race. Express your answer as a mixed number or decimal, including units if necessary. | 12.5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
The first terms of both the arithmetic sequence {a<sub>n</sub>} and the geometric sequence {b<sub>n</sub>} are 1. The common difference d ≠ 0, and the common ratio q > 0 and q ≠ 1. Find the maximum number of elements in the set {n | a<sub>n</sub> = b<sub>n</sub>}. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\{a_n\}$ that satisfies $a_1 = -1$, and $S_n = 2a_n + n$ (where $S_n$ is the sum of the first $n$ terms of $\{a_n\}$), calculate the value of $a_6$. Express your answer as an integer. | -63 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of inequalities about x: $\left\{\begin{array}{l}{2x+3＞5}\\{x-a≤0}\end{array}\right.$ with exactly one integer solution, find the range of real numbers a that satisfy this condition. Express your answer as an interval (e.g., 2 ≤ a < 3). | 2 ≤ a < 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the result of subtracting 8 from the highest common divisor (HCD) of 6432 and 132. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that θ is a real number, if the complex number z = sin(2θ) - 1 + i(√2cosθ - 1) is a pure imaginary number, calculate the imaginary part of z. Express your answer as a numerical value. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the inequality ax^2 - bx - 1 >= 0 is [-1/2, -1/3], find the solution set of the inequality x^2 - bx - a < 0. Express your answer as an interval or union of intervals. | (2, 3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In △ABC, AC:CB = 3:4. The external angle bisector of ∠C intersects the extension of BA at P (with A between P and B). Find the ratio PA:AB. Express your answer as a ratio in the format a:b. | 3:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f'(x)$ is the derivative of $f(x)$, determine the value of $\lim_{t \to 0} \frac{f(3) - f(3 - t)}{t}$. Express your answer as a derivative. | f'(3) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given $m=2\sin 18^{\circ}$, and if $m^{2}+n=4$, calculate the value of $\dfrac {m \sqrt {n}}{2\cos ^{2}27 ^\circ -1}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let M and N be points on the sides BC and CD, respectively, of a square ABCD. If |BM| = 21, |DN| = 4, and |NC| = 24, what is the measure of angle MAN in degrees? | 45^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the progression $10^{\dfrac{1}{11}}, 10^{\dfrac{2}{11}}, 10^{\dfrac{3}{11}}, 10^{\dfrac{4}{11}},\dots, 10^{\dfrac{n}{11}}$, find the least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $100,000$. Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Two circles \(C_{1}\) and \(C_{2}\) touch each other externally and the line \(l\) is a common tangent. The line \(m\) is parallel to \(l\) and touches the two circles \(C_{1}\) and \(C_{3}\). The three circles are mutually tangent. If the radius of \(C_{2}\) is 9 and the radius of \(C_{3}\) is 4, calculate the radius ... | 12 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$ and a parabola $y^2=8x$ share a common focus $F$, and one point of intersection $P$. If $|PF|=5$, find the equation of the asymptotes of the hyperbola. Express your answer in the form $y = mx$. | y = \pm \sqrt{3}x | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the ratio of the length to the width of a rectangle is 4:3, and the length of the diagonal is d, express the area of the rectangle as k d^2 and find the value of k. Provide your answer as a fraction. | \frac{12}{25} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $\sqrt{64}$. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Ahmed is going to the store. One quarter of the way to the store, he stops to talk with Kee. He then continues for 12 km and reaches the store. What is the total distance travelled by Ahmed? Express your answer in kilometers as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(2\log _{4}3+\log _{8}3)(\log _{3}2+\log _{9}2)$ to a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The polynomial $x^{2}+mx+25$ is a perfect square trinomial. Find the value of $m$. Express your answer as a numerical value, including the possibility of a negative value. | \pm 10 | big_math | [
"Mathematics -> Algebra -> Algebra -> Quadratic Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of polynomials of the form $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$ that satisfy the condition $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|=3$, where $n$ is a non-negative integer, $a_0$ is a positive integer, and the remaining $a_i$ are integers or zero. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_{n}} where a_{1} = 2, and a_{n} = 2 - \\frac{1}{{a_{n-1}}} for n >= 2, calculate the value of a_{4}. Express your answer as a fraction. | \\frac{5}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive integer solutions (m, n) to the equation 4/m + 2/n = 1. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{x \mid x(x-3) < 0\}$ and $N=\{x \mid |x| < 2\}$, calculate the intersection $M \cap N$. Express your answer in interval notation. | (0, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if a_1 + a_3 + a_5 = π, then find the value of cos(a_3). Express your answer as a numerical value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the vertices of triangle ABC as A(1,0), B(2,-3), and C(3,3), find the equation of the median line on side AB. Express your answer in the form Ax + By + C = 0. | 3x - y - 6 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
There are a certain number of red balls, green balls, and blue balls in a bag. Of the balls in the bag, $\frac{1}{3}$ are red and $\frac{2}{7}$ are blue. The number of green balls in the bag is 8 less than twice the number of blue balls. Calculate the exact number of green balls in the bag. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a^{m}=4$ and $a^{n}=2$, calculate the value of $a^{m-2n}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose complex numbers z1 and z2 correspond to points A and B in the complex plane, and |z1| = 4, 4z1^2 - 2z1z2 + z2^2 = 0, with O being the origin. Calculate the area of triangle OAB. | 8\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers x and y satisfy (1+i)x + (1-i)y = 2, find the value of xy. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Consider the convex quadrilateral ABCD. Prolong the edges AD and BC through points D and C respectively to intersect at point E. Let s = ∠CDE + ∠DCE, and s' = ∠BAD + ∠ABC. If r = s/s', determine the value of r. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set R, set A = {x | 2^x ≤ 2√2}, and set B = {x | ln(2-x) < 0}, find the intersection of the complement of A with B, denoted as (∁_R A) ∩ B. Express your answer in interval notation. | (3/2, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, a_1 and a_{4025} are the extreme points of the function f(x) = (1/3)x^3 - 4x^2 + 6x - 1. Calculate the value of log_2(a_{2013}) as a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $x^{2} \leqslant 4$ and express the solution set in interval notation. | [-2, 2] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Define a function f(x) on R that satisfies f(x) + f''(x) < e, where f(0) = e + 2 and e is the base of the natural logarithm.
Find the solution set of the inequality e^x*f(x) > e^(x+1) + 2. Express your answer in interval notation. | (-\infty, 0) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $\frac{22}{7}$, $-\sqrt{9}$, $\frac{π}{2}$, $1.414$, $3$, $0.1010010001\ldots$ (the number of zeros between every two ones increases by $1$ each time), how many are irrational? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Tommy Thomas's tankard holds 480 ml when it is one quarter empty. How much does it hold when it is one quarter full? Express your answer in milliliters as a whole number. | 160 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the normal vectors of planes α and β are $\overrightarrow{a}=(1,2,-2)$ and $\overrightarrow{b}=(-2,1,m)$, respectively, and that α and β are perpendicular, find the value of m that satisfies this condition. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given $a\in R$, find the value of $a$ such that the complex number $\left(a-2i\right)\left(3+i\right)$ is a real number. Express your answer as a single value. | 6 | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x) = ax^5 + bx^3 - cx + 2$. Given that $f(-3) = 9$, calculate the value of $f(3)$. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given two points O(0,0) and Q(a,b), point P1 is the midpoint of the line segment OQ, point P2 is the midpoint of the line segment QP1, P3 is the midpoint of the line segment P1P2, and so on, Pn+2 is the midpoint of the line segment PnPn+1. Find the limit position of point Pn as n approaches infinity. Express your answe... | \left( \frac{2a}{3}, \frac{2b}{3} \right) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Francis spends \(\frac{1}{3}\) of his day sleeping, \(\frac{1}{4}\) of his day studying, and \(\frac{1}{8}\) of his day eating. Calculate the number of hours he has left in the day. Express your answer in hours as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given two fixed points A(0,-2) and B(0,2), point P is on the ellipse x^2/12 + y^2/16 = 1 and satisfies |AP| - |BP| = 2. Calculate the dot product of vectors AP and BP. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the square root of 16. Express your answer in the form \\pm n, where n is a whole number. | \\pm 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The area of a trapezoidal field is 1400 square yards, and its altitude is 50 yards. Find the number of possible pairs of bases if each base is an integer divisible by 8. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Integers x and y with x > y > 0 satisfy the equation x + y + xy = 80. What is the value of x? Express your answer as a single integer. | 26 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, S_n is the sum of the first n terms. If a_2 + 2a_8 + a_{14} = 8, then find the value of S_{15}. Express your answer as a single numerical value. | 30 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse C: x^2/2 + y^2 = 1 with its right focus at F, and the line l: x = 2. If point A lies on line l, and segment AF intersects ellipse C at point B, and if Vector FA = 3 * Vector FB, then calculate the magnitude of Vector AF, |AF|. Express your answer as a numerical value, including the square root symbol ... | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $x$ and $y$ that 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 -> Intermediate Algebra -> Exponential Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Four high school sophomores plan to volunteer at three local tourist attractions. If each attraction must have at least one student, and students A and B cannot be at the same attraction, calculate the total number of possible arrangements for these four students. Express your answer as a single integer. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
What is the negation of the proposition \(\exists x \in \mathbb{R}, x^3 - 2x + 1 = 0\)? Express your answer in the correct logical form. | \forall x \in \mathbb{R}, x^3 - 2x + 1 \neq 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The equation of a circle is given as $x^2+y^2-4x-2y-5=0$. Find the coordinates of the center of the circle. Express your answer as a coordinate pair (x, y). | (2, 1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the first three terms of a harmonic sequence are 3, 4, and 6, calculate the sum of the first four terms (S4) of this sequence. | 25 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A pagoda with seven levels has twinkling red lanterns that double in number with each higher level. In total, there are 381 lanterns. Find the number of lanterns on the very top level. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Sharon drives from her home to her office, a total distance of 59 km. Her drive includes 20 minutes on a highway and 40 minutes on city roads. If her average speed when she is on city roads is 45 km/h, calculate her average speed, in km/h, at which she drives on the highway. Express your answer as a whole number. | 87 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
During the Nanchang Flower Expo, 6 volunteers are arranged to provide services in 4 exhibition areas. It is required that each of the exhibition areas A and B should have one person, and the remaining two exhibition areas should have two people each. Additionally, Xiao Li and Xiao Wang cannot be in the same group. Calc... | 156 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a = (-1, 1)$ and $b = (2, x)$, if vector $a$ is perpendicular to $(a + b)$, then find the value of the real number $x$. Express your answer as a single real number. | 0 | big_math | [
"Mathematics -> Algebra -> Vector Algebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, AB = 3, BC = sqrt(13), AC = 4. Using the cosine theorem, calculate the size of angle A in degrees. Express your answer in the format "A = X degrees", where X is the calculated angle. | A = 60 degrees | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $\left( {{x}^{2}}+2 \right){{\left( \frac{1}{{{x}^{2}}}-1 \right)}^{5}}$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x$ and $y$ are positive real numbers satisfying $2x + y = 1$, find the maximum value of $xy$. Express your answer as a numerical value. | \frac{1}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A particle moves along the positive direction of the x-axis. At any position x meters, the force exerted on it is $F(x) = 3x^2$ Newtons. Calculate the work done by the force $F(x)$ as the particle moves from the origin to 4 meters. Express your answer in Joules. | 64 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence {a_n}(n ≥ 1) that satisfies a_{n+2} = a_{n+1} - a_{n} and a_2 = 1, if the sum of the first 1999 terms of this sequence is 2000, calculate the sum of the first 2000 terms. | 2001 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of $a$ such that the graphs of the lines $2y+x+3=0$ and $3y+ax+2=0$ meet at right angles. | -6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 - ax + 1, find the range of the real number a such that f(x) takes negative values. Express your answer as an interval or inequality, including any necessary absolute value signs or inequality symbols. | (-\infty, -2) \cup (2, \infty) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
In the polar coordinate system, find the distance from point $(2, \frac{\pi}{3})$ to line $\rho\cos \theta=2$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the real numbers $a$, $b$, and $c$ satisfy $a^2 + b^2 + c^2 = 9$, find the maximum value of the algebraic expression $(a-b)^2 + (b-c)^2 + (c-a)^2$. Express your answer as a single number. | 27 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
From 9 pieces of paper labeled 1, 2, 3,..., 9, two are drawn at random. Find the probability that the product of the numbers on the two pieces of paper is even. Express your answer as a simplified fraction. | \frac{13}{18} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, find the imaginary part of the complex number $z= \frac {3-4i}{1-i}$. Express your answer as a single number. | -\frac {1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the line $\sqrt{3}x + y - 1 = 0$ and the line $2\sqrt{3}x + my + 3 = 0$ are parallel, find the distance between them. Express your answer as a fraction or decimal. | \frac{5}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Among the eight vertices of a cube, four of them are precisely the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron. Express your answer as a simplified ratio. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Compute the product of $\sqrt {2}$ and $\sqrt {6}$ and simplify the resulting expression. Express your answer in simplified radical form. | 2 \sqrt {3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given an odd function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(x) = 2^x - 4$ for $x > 0$, find the set $\{x|f(x-1) > 0\}$. Express your answer as a set of intervals, e.g., $\boxed{\{x|-1 < x < 1 \text{ or } x > 3\}}$. | \{x|-1 < x < 1 \text{ or } x > 3\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (sinθ + cosθ)/(sinθ - cosθ) = 2, find the value of sinθ cosθ. Express your answer as a fraction. | \frac{3}{10} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given a nonzero real number triplet \((a, b, c)\), consider the expression:
\[
\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{abc}{|abc|}.
\]
Determine the set of all possible values that this expression can take. Provide your answer as a set of numbers, describing all possible outcomes based on the signs of... | \{-4,0,4\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the general term of the arithmetic sequence $\{a_n\}$ is $a_n = 3n - 5$, find the term number of this sequence that corresponds to the coefficient of the $x^4$ term in the expansion of $(1+x)^5 + (1+x)^6 + (1+x)^7$. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Algebra -> Sequences and Series",
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles triangle, one side has a length of 2, and the lengths of the other two sides are the two real roots of the quadratic equation x^2 - 6x + k = 0 with respect to x. Find the value of k that results in a valid isosceles triangle. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
The mean score of the students who took a mathematics test was 6. Exactly 60% of the students passed the test with a mean score of 8. Calculate the mean score of the students who failed the test. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line on which the common chord of the circles $(x^{2}+y^{2}-2x-8=0)$ and $(x^{2}+y^{2}+2x-4y-4=0)$ lies. Express your answer as an equation of a line in standard form (e.g., Ax + By + C = 0). | x - y + 1 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M={x|x^{2}-2x-3 < 0}$ and $N={x|x > 1}$, find the intersection $M∩N$. Express your answer as an interval, using parentheses or brackets as necessary to denote the type of interval. | (1,3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overset{→}{a}$ and $\overset{→}{b}$ are unit vectors, and $\overset{→}{a}+ \overset{→}{b}+ \overset{→}{c}= \overset{→}{0}$, determine the maximum value of $|\overset{→}{c}|$. Provide your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
An aluminum can in the shape of a cylinder is closed at both ends. Its surface area is 300 cm². If the radius of the can were doubled, its surface area would be 900 cm². If instead the height of the can were doubled, what would its surface area be? Express your answer in square centimeters (cm²) as a whole number. | 450 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
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