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 minimum value of ω > 0 such that the graph of the function f(x) = sin(ωx) moved to the right by π/4 units is symmetric about the point (3π/4, 0). Express your answer as a single number. | ω = 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x=2$ is a root of the quadratic equation $x^{2}+mx+2=0$, find the value of $m$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Rory uses four identical standard dice to build a solid. Whenever two dice touch, the numbers on the touching faces are the same. The numbers on some faces of the solid are shown. Given that the numbers on opposite faces of a standard die add up to 7, what number is written on the face marked with a question mark? Expr... | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\sin \frac{\pi }{12}-\sqrt{3}\cos \frac{\pi }{12}$. Express your answer as a simplified numerical value. | - \sqrt{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^n, and its derivative f'(2) = 12, find the value of n. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_2 = 9 and a_5 = 243. Find the sum of the first 4 terms of the sequence. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Three circles \( P, Q \) and \( R \) are all tangent to each other. Their centers all lie on a diameter of \( P \), as shown in the figure. Find the value of \(\frac{\text{circumference of } Q + \text{circumference of } R}{\text{circumference of } P}\). Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Among all two-digit numbers consisting of two different digits, calculate the smallest quotient obtained when each two-digit number is divided by the sum of its digits. Express your answer as a decimal value rounded to 1 decimal place. | 1.9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that S_n is the sum of the first n terms of the sequence {a_n} and satisfies the equation 3a_n = 2S_n + λn (n ∈ N^*), and the sequence {a_n + 2} is a geometric sequence, calculate the value of λ. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $C$: $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{4}=1$ has a focus at $(2,0)$, calculate the eccentricity of $C$. Express your answer as a fraction or decimal, including any necessary square roots, and box the final answer. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
The breeding quantity $y$ (in individuals) of a certain animal species is related to time $x$ (in years) by the equation $y = a\log_3(x+1)$. It is given that in the second year, there are 100 individuals of this species. Calculate the population of this species in the eighth year. Express your answer as a numerical val... | 200 | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
What is $0.015 \%$ of 60 million? Express your answer as a whole number. | 9000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. The lengths of sides $AC$ and $BC$ are 6 and 7 respectively. Find the length of side $AB$. Express your answer as a square root if necessary. | \sqrt{17} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Xiao Ming participated in a math test. The average score of the first two times was 85, and the total score of the last three times was 270. Calculate the average score of Xiao Ming's five tests. Express your answer as a whole number. | 88 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Express the decimal number 0.00077 in scientific notation. Include the coefficient and the exponent in your answer. | 7.7 \times 10^{-4} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given \(θ\in\left(\dfrac{π}{2}, π\right)\) and \(\sin θ= \dfrac{3}{5}\), find the value of \(\tan\left(θ+ \dfrac{π}{4}\right)\). Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In a class, there are four subjects to be scheduled in the morning: Chinese, Mathematics, Physical Education, and Foreign Language. The Physical Education teacher is unavailable for the first and second periods. Calculate the number of different scheduling methods. Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence of letters KANGAROOKANGAROOKANG... the word KANGAROO is repeated indefinitely. What is the 2017th letter in this sequence? Provide your answer as a single letter. | K | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $$\\frac {x^{2}}{a^{2}}- \\frac {y^{2}}{b^{2}}=1$$ (where $a>0$, $b>0$) with eccentricity $e=2$, and a line through a point $M$ on the hyperbola intersects the hyperbola at points $A$ and $B$, with slopes $k_1$ and $k_2$ respectively. If the line $AB$ passes through the origin, calculate the value o... | 3 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Four coins are given, one of which is counterfeit and differs in weight from the real ones. Using a balance scale without weights, what is the minimum number of weighings required to determine which coin is counterfeit? | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
The distance from the right focus of the hyperbola x^2/3 - y^2 = 1 to the line x - sqrt(3)y = 0 is to be calculated. Find this distance. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the algebraic expression $x^{2}+2\left(k+1\right)x+16$, find the value or values of $k$ that make this expression a perfect square trinomial. Express your answer as a numerical value or a set of values. | 3 \text{ or } -5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $-x^{2}+3x+10 > 0$. Express your answer using interval notation (e.g., $(-2,5)$ or $(-∞,-2)∪(5,+∞)$). | (-2,5) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let $a \in \mathbb{R}$, and $z = (a+1) + (1+a^2)i$. If the complex number $z$ is purely imaginary, find the value of $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In 4 independent trials, the probability of event A occurring remains the same. If the probability of event A occurring at least once is $\frac{65}{81}$, then what is the probability of event A occurring in a single trial? Express your answer as a fraction or decimal between 0 and 1. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the minimum value of the function $f(x) = \frac{x^2 - 2x + 1}{x}$ on the interval $\left[\frac{1}{2}, 3\right]$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
If the number of sides of a convex polygon increases from 3 to n, what is the sum of its exterior angles? Express your answer in degrees. | 360 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Using 100 boxes to pack cups, with each box containing a different number of cups and none of the boxes being empty, calculate the minimum number of cups required. Express your answer as a single integer. | 5050 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
An isosceles triangle PQR, in which PQ = PR, is split into three separate isosceles triangles, such that PS = SQ, RT = RS, and QT = RT. Find the size, in degrees, of angle QPR. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player? Express your answer as a whole number. | 400,000 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x) = \begin{cases} 2^{-x} & (x < 1) \\ \log_{81}x & (x \geq 1) \end{cases}$, find all values of $x$ that satisfy $f(x) = \frac{1}{4}$. Express your answer as a single value or a list of values. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In $\Delta ABC$, $AB=3, BC= \sqrt{13}, AC=4$. Calculate the height to side $AC$. Express your answer as a fraction or decimal value. | \dfrac{3 \sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $x^3 - x + 1 = 0$, find the sum of the integers $a$ and $b$ ($a, b \in \mathbb{Z}$, and $b-a=1$) such that the equation has a root in the interval $(a, b)$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A bookshelf has 6 different English books and 2 different math books. If you randomly pick 1 book, what is the total number of different ways to pick the book? Express your answer as a single whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable X ~ N(1, σ^2), if P(0 < X < 3) = 0.5, and P(0 < X < 1) = 0.2, calculate the probability P(X < 3). Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place. | 0.8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the solution set of the inequality $\frac{3x-1}{2-x} \geqslant 0$. Express your answer in interval notation. | [\frac{1}{3}, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \((5 \times 5) + (5 \times 5) + (5 \times 5) + (5 \times 5) + (5 \times 5)\). Provide your answer as a single number. | 125 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $x^{2}-3x+1=0$, where its roots are denoted as $m$ and $n$, calculate the value of $-m-n-mn$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a^{2n} = 3$, calculate the value of $2a^{6n} - 1$. Express your answer as a single integer. | 53 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given six balls in a box, consisting of two red balls, two green balls, and two yellow balls, what is the probability that if Cheryl were to draw two balls, they would be of the same color? Express your answer as a simplified fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A certain type of cell divides from one into two every 30 minutes. After 3 hours, how many cells will there be in total? Express your answer as a numerical value. | 64 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^{3}$ in the expansion of $(2-x)(1-x)^{4}$. Express your answer as a single integer. | -14 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
A class of 30 students was asked what they did on their winter holiday. 20 students said that they went skating. 9 students said that they went skiing. Exactly 5 students said that they went skating and went skiing. Calculate the number of students who did not go skating and did not go skiing. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \\frac{20}{19} = 1 + \\frac{1}{1 + \\frac{a}{b}}, where a and b are positive integers, find the least possible value of a + b. Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\left\{a_{n}\right\}$, $a_{2}=3$, $a_{3}=6$, and the recursive relation $a_{n+2}=a_{n+1}-\frac{1}{a_{n}}$ holds. Calculate the value of $a_{5}$. Express your answer as a simplified fraction. | \frac{11}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is the smallest positive integer, $b$ is the largest negative integer, $c$ is the rational number with the smallest absolute value, and $d$ is the number whose reciprocal is equal to itself, calculate the value of the expression $a-b+c^2-|d|$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles triangle \( \triangle ABC \), the altitude to one of the legs is 1. This altitude forms a \( 45^\circ \) angle with the base. Find the area of \( \triangle ABC \). Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations y+4=(x-2)^2 and x+4=(y-2)^2, and x is not equal to y, calculate the value of x^2+y^2. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $\frac{x}{2}-1=3$ for $x$. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of an asymptote of the hyperbola given by $\frac{{y^2}}{8}-\frac{{x^2}}{6}=1$. Express your answer as a linear equation in the form of $Ax + By = 0$. | 2x - \sqrt{3}y = 0 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A "Long March" 2 series rocket carrying "Shenzhou VII" covers a distance of 2 km in the first second after ignition, and the distance it covers increases by 2 km per second. How many seconds does it take for the rocket to reach an altitude of 240 km from the ground? Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{a, b, c\}$ and $N=\{d, e\}$, calculate the total number of different mappings that can be established from set $M$ to set $N$. Provide your answer as a single integer. | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given that α ∈ (0, π) and cos(π+α) = 3/5, find the exact value of sin(α). Express your answer as a fraction. | \frac{4}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle α passes through the fixed point P(4,2) on the function y = log_a(x-3) + 2, find the value of sin(2α) + cos(2α). | \frac{7}{5} | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(2x + π/6), find the range of real number t such that f(x) is monotonically increasing on the interval [-t, t]. Express your answer as an interval of real numbers. | (0, π/6] | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If a mapping $f: A \rightarrow B$ is such that every element in set $B$ has a pre-image in $A$, it is called a "surjection". If set $A$ has 3 elements and set $B$ has 2 elements, then how many different surjections are there from $A$ to $B$? Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is a function defined on the real numbers, and for any real number x, it satisfies f(x+4) = -f(x) + 2√2. If the graph of the function y = f(x-1) is symmetric about the line x=1, find the value of f(2014). Express your answer as a single value, possibly involving square roots. | √2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $\arcsin(x)^2 + \arcsin(x) + x^6 + x^3 > 0$. Express your answer in interval notation, including any relevant endpoints or bounds. | (0, 1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A wheel of radius 8 rolls along the diameter of a semicircle of radius 25 until it bumps into this semicircle. Find the length of the portion of the diameter that cannot be touched by the wheel. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest three-digit number N that is the sum of three different two-digit palindromic numbers and is not itself a palindromic number. Calculate the sum of the digits of N. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
If each exterior angle of a polygon is $30^{\circ}$, find the number of diagonals starting from a vertex of this polygon. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
A middle school uses systematic sampling to select 50 students from a total of 800 students for a dental health check. The students are numbered from 001 to 800, and the interval number k is determined to be 16, meaning every 16th student is selected. A random number between 001 and 016 is generated, resulting in 007. ... | 039 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
For real numbers $x$ and $y$, if $|x-1| \leqslant 2$ and $|y-1| \leqslant 2$, find the maximum value of $|x-2y+1|$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given x = a/b, where a is not equal to b and b is not equal to 0, calculate the value of (a + b)/(a - b). Express your answer as a simplified fraction or algebraic expression. | \frac{x+1}{x-1} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is a root of the equation $x^{2}+2x-1=0$, calculate the value of $2a^{2}+4a-2024$. Express your answer as an integer in the form $\boxed{[answer]}$. | -2022 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations p = 2, p * q = 20, p * q * r = 202, and p * q * r * s = 2020, calculate the sum p + q + r + s. Express your answer as a single number, rounded to one decimal place if necessary. | 32.1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular prism has each of its corners cut off. The cuts do not intersect anywhere in or on the prism. Calculate the total number of edges in the new figure. Express your answer as a single integer. | 36 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the slope of line $l_1$ is $1$ and $l_1 \perp l_2$, find the angle of inclination of $l_2$. Express your answer in degrees as a whole number. | 135 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y = x^2 - 2x + 1 and points A(0, y1), B(1, y2), and C(-2, y3) on the parabola, determine the relationship among y1, y2, and y3. Express your answer using inequality symbols in the format yX > yY > yZ, where X, Y, and Z are the corresponding point indices. | y3 > y1 > y2 | big_math | [
"Mathematics -> Algebra -> Quadratic Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A pack of monosodium glutamate weighs 80g, a pack of salt weighs 500g, and a pack of laundry detergent weighs 420g. What is the total weight of these three items? Express your answer in grams (g) or kilograms (kg). | 1kg | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
From a bag containing 3 white balls and 2 red balls, 3 balls are randomly drawn. Calculate the probability that at most 1 red ball is among the 3 drawn balls. Express your answer as a simplified fraction. | \frac{7}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A residential community has installed a water recycling device. According to calculations, if originally $a$ days required $b$ tons of water, now these same amount of water can last for an additional $4$ days. Calculate the difference in the amount of water used per day now compared to before. Express your answer as a ... | \frac{4b}{a(a+4)} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vertex of angle $\alpha$ is at the origin of coordinates, the initial side is the positive half-axis of the x-axis, and the terminal side falls in the second quadrant. Point A $(x, y)$ is a point on its terminal side, vector $\overrightarrow{m} = (3, 4)$. If $\overrightarrow{m} \perp \overrightarrow{OA}$... | \frac{1}{7} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that four times the reciprocal of the circumference of a circle equals the diameter of the circle, find the area of the circle. Express your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \(x^{2} + y^{2} - 6x = 0\) for real numbers \(x\) and \(y\), determine the range of \(\sqrt{2x^{2} + y^{2} - 4x + 5}\). Express your answer as an interval, using square roots if necessary. | [\sqrt{5}, \sqrt{53}] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the space rectangular coordinate system Oxyz, given point D(2,1,0) and vector m=(4,1,2), where m is perpendicular to plane DEF, calculate the distance from point O to plane DEF. Express your answer as a fraction with a square root in the numerator. | \frac{3\sqrt{21}}{7} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\sin {18}^{\circ } \cdot \sin {78}^{\circ } - \cos {162}^{\circ } \cdot \cos {78}^{\circ }$. Express your answer as a simplified fraction. | \dfrac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Several different positive integers are written on a blackboard. The product of the smallest two of them is 16. The product of the largest two of them is 225. What is the sum of all the integers written on the blackboard? Express your answer as a single integer. | 44 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Toothpicks are used to form squares in a pattern: four toothpicks are used to form one square, seven to form two squares, and so on. If this pattern continues, how many toothpicks will be used to form 10 squares in a row? Express your answer as a whole number. | 31 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the real number $m$ such that the domain of the function $y= \sqrt {mx^{2}-6mx+m+8}$ is $\mathbb{R}$. Express your answer as an interval. | [0,1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\left(m-3\right){x}^{{m}^{2}-7}-4x-8=0$, find the value of $m$ for which this equation is a quadratic equation in one variable. Provide your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
A track and field team consists of 48 male athletes and 36 female athletes. If a stratified sampling method is used to draw a sample of 21 athletes, calculate the expected number of male athletes in the sample, ensuring that the proportion of male athletes in the sample is the same as in the total population. Provide y... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n} with its sum of the first n terms denoted as S_n, if S_4 = -4 and S_6 = 6, then find the value of S_5. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the shape enclosed by the curve $\rho = 4\cos\theta$. Express your answer in terms of $\pi$. | 4\pi | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Catherine's car gas tank is initially 1/8 full. When 30 litres of gas are added, the tank becomes 3/4 full. If the gas costs $1.38 per litre, how much will it cost her to fill the remaining quarter of the tank? Express your answer in dollars, rounded to two decimal places. | 16.56 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the terms 4x^(2m+2)y^(n-1) and -3x^(3m+1)y^(3n-5) are like terms, calculate the value of m-n. Express your answer as an integer. | -1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
For the reaction: $2X + 3Y \rightarrow 3Z$, the combination of $2.00$ moles of $X$ with $2.00$ moles of $Y$ produces $1.75$ moles of $Z$. Calculate the percent yield of this reaction. Express your answer as a percentage (%). | 87.5\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{(1-i)^{2}}{1+i}$, find the value of $|z|$. Express your answer as a numerical value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\sqrt{2}$, $\sqrt{5}$, $2\sqrt{2}$, $\sqrt{11}$, $\ldots $, what is the term number of $4\sqrt{2}$? Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Xiaoming has participated in 20 chess matches, winning 95% of them. If Xiaoming wins all his future matches, the winning percentage will exactly be 96%. Calculate the number of additional matches Xiaoming needs to play to achieve this winning percentage. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the opposite of the real number 2023. The opposite of a real number x is defined as the number that, when added to x, results in zero. Express your answer as a single integer. | -2023 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{a_{n}\}$ defined by $a_{1} = 3$ and $a_{n} = 3^{a_{n-1}}$ for $n \geq 2$, calculate the last digit of $a_{1990}$. Express your answer as a single digit. | 7 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=i(1+2i)$, find the imaginary part of $z$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(0,1) and B(3,2), and vector AC = (-4,-3), find vector BC. Express your answer as a vector in the form (x, y). | (-7,-4) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(\frac{a+b}{a-b} = 3\), calculate the ratio \(\frac{a}{b}\). Express your answer as a simple numerical ratio. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $\sqrt{8}+|-2| imes \cos 45^{\circ}$. Express your answer in simplest radical form if necessary. | 3\sqrt{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex numbers z1 = m^2 - 3m + m^2i and z2 = 4 + (5m + 6)i, where m is a real number, and i is the imaginary unit, if z1 - z2 = 0, calculate the value of m. | -1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $$C: \frac{x^{2}}{m+1} - \frac{y^{2}}{m} = 1 (m > 0)$$ whose left focus $F$ lies on the circle $x^2 + y^2 - 2x - 6y - 15 = 0$, calculate the eccentricity of the hyperbola $C$. Provide your answer as a fraction or decimal, and include necessary calculations and formulas to support your solution. | \frac{3\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
What is the decimal equivalent of thirty-six hundredths? Express your answer as a decimal. | 0.36 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A line passing through the focus F of the parabola y^2 = 4x at an angle of π/3 intersects the parabola at points A and B (with point A above the x-axis). Find the ratio |AF|/|BF|. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
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