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 vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-2,1)$, calculate the value of $\overrightarrow{a}+2\overrightarrow{b}$. Express your answer as an ordered pair (x, y). | (-3, 4) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {i^{2018}}{(1-i)^{2}}$ ($i$ being the imaginary unit), calculate the conjugate of $z$, $\overline {z}$. Express your answer as a complex number in the form $a + bi$, including the imaginary unit $i$ if necessary. | \frac {1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation y^2 = 2px, where p > 0, with vertex at the origin O, passing through the point A(x_0, 2), and focus F such that |AF| = 3|OF|, calculate the value of p. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be a linear function, and f(f(x)) = 4x - 1. If f(3) = -5, find the analytical expression of f(x) in the form f(x) = ax + b. | f(x) = -2x + 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
To arrange the schedule for a class in one day, including Chinese, Mathematics, Politics, English, Physical Education, and Art, a total of 6 lessons, with the requirement that the Mathematics lesson is scheduled in the morning (the first 4 periods) and the Physical Education lesson is scheduled in the afternoon (the la... | 192 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (1+i)z = 2, calculate the value of z. Express your answer as a complex number in the form a + bi. | 1-i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A math interest group consists of 2 female students and 3 male students. Now, 2 students are randomly selected from this group to participate in a math competition. Calculate the probability that the number of female students among the 2 selected is not less than the number of male students. Express your answer as a si... | \frac{7}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function f(x) = $$\frac{\ln(x+2)}{\sqrt{1-3^x}}$$. Express your answer in interval notation. | (-2, 0) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that function f(x) is differentiable, evaluate the limit: $\lim_{\Delta x \to 0} \frac{f(1+\Delta x)-f(1)}{3\Delta x}$. Express your answer as a fraction of f'(1). | \frac{1}{3}f'(1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A circle with the equation $(x-6)^2 + y^2 = 4$ is given. Find the length of the tangent line drawn from the origin to this circle. Express your answer as a simplified radical. | 4\sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $1-2i$, find the imaginary part of its conjugate. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^5y^2$ in the expansion of $(x^2+x+y)^5$. Express your answer as a numerical value. | 30 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, 2)$ and $\overrightarrow{b} = (2, 1)$, if the vector $\overrightarrow{a} - \lambda \overrightarrow{b}$ is collinear with vector $\overrightarrow{c} = (5, -2)$, calculate the value of λ. Express your answer as a numerical value. | \frac{4}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the common ratio of the geometric sequence {an} is (-1/2), find the value of (a1+a3+a5)/(a2+a4+a6). Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac {\overline {z}}{1+i}=2+i$, solve for the complex number $z$. Express your answer in the form $a+bi$. | 1 - 3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance between the two parallel lines $l_1: 4x-3y+2=0$ and $l_2: 4x-3y-1=0$. Express your answer as a fraction or decimal. | \frac{3}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A woman skilled in weaving weaves faster day by day. On the first day, she weaves 5 meters of fabric, and from the second day onwards, she weaves 16/29 meters more fabric each day than the day before. Calculate the total amount of fabric she will have woven in a month (considered as 30 days). Express your answer in met... | 390 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial function y = a x^5 + b x^3 + c x - 5, when x = -3, y = 7. Calculate the value of y when x = 3. Express your answer as a single integer. | -17 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
There is one odd integer \( N \) between 400 and 600 that is divisible by both 5 and 11. Find the sum of the digits of \( N \). Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Number Theory -> Divisibility"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( \theta \) is an angle between 0 and \( \frac{\pi}{2} \), and \( \sin 2 \theta = a \), derive an expression for \( \sin \theta + \cos \theta \) in terms of \( a \). Express your answer as an algebraic expression. | \sqrt{1 + a} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given a binomial distributed random variable X with X ~ B(10, 0.8), find the value of k that maximizes P(x=k). Express your answer as an integer. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Find the fraction that is exactly halfway between 1/5 and 1/3 on the number line. Express your answer as a simplified fraction. | \frac{4}{15} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations:
\[p + 2q + 3r + 4s = k\]
\[4p = 3q = 2r = s,\]
find the smallest value of \(k\) for which \(p, q, r,\) and \(s\) are all positive integers. | 77 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\int_{0}^{\frac{\pi}{2}} (\sin x - a \cos x) dx = 2$, find the value of the real number $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse $\dfrac{x^2}{m^2} + y^2 = 1$ ($m > 1$) and the hyperbola $\dfrac{x^2}{n^2} - y^2 = 1$ ($n > 0$), both sharing a common focus $F_1$. Let $F_2 P$ be the intersection point of the two curves. Calculate the area of the triangle $\triangle F_1 P F_2$. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the number x such that 3 + x = -1. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the identity $(1+x+x^2)^n=a_1x+a_2x^2+ \cdots + a_{2n}x^{2n}$, define $s=a_0+a_2+a_4+\cdots +a_{2n}$. Calculate the value of $s$ in terms of $n$. Express your answer as a mathematical expression. | \frac{3^n+1}{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Two applicants, A and B, apply to a university simultaneously. The probability of applicant A being admitted is 0.6, while the probability for applicant B is 0.7. The admissions of the two are independent of each other. Find the probability that at least one of them gets admitted. Express your answer as a decimal value... | 0.88 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the total number of ways to select 3 volunteers from a group of 5 boys and 2 girls, ensuring that at least 1 girl is selected. Express your answer as a single integer. | 25 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the point P(3-m, m-1), find the range of m such that P is in the second quadrant. Express your answer as an inequality in the format m > value or m < value, or a combination of both. | m > 3 | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overrightarrow{a}=(1,3)$ and $\overrightarrow{b}=(x,-3)$ in the plane, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, calculate the magnitude of the vector $\overrightarrow{a}+2\overrightarrow{b}$. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{10} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $x$ in the equation $x^{x^{x^{.^{.^.}}}} = 2$. Express your answer as a single value. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the probability of a shuttlecock's weight being less than 4.8g is 0.3, and the probability of its weight being less than 4.85g is 0.32, find the probability that the weight is in the range of [4.8, 4.85). Express your answer as a decimal value. | 0.02 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
The Changsha Metro had a passenger volume of 3.021 million on March 10, 2023. Represent this number in scientific notation, expressing your answer in the form $a \times 10^{b}$, where $a$ is a number between 1 and 10, and $b$ is an integer. | 3.021 \times 10^{6} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex numbers $z_{1}=1+i$ and $z_{2}=2+xi$ ($x\in\mathbb{R}$). If $z_{1}\cdot z_{2}$ is a real number, find the value of $x$. Express your answer as a single real number. | -2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
A car uses 8.4 liters of gas for every 100 km it is driven. A mechanic can modify the car's engine for a cost of $400 so that it will only use 6.3 liters of gas per 100 km. If gas costs $0.80 per liter, calculate the minimum distance the owner would have to drive to recover the cost of the modifications. Express your a... | 23809.52 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain region along the east coast of our country, people aged 60 and above account for 20%. Given that a person is over 60 years old, the probability of them having hypertension is 45%. Calculate the probability that a person in this region is both over 60 years old and has hypertension. Express your answer as a... | 9% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
The digits 1, 2, 3, 4 can be arranged to form twenty-four different four-digit numbers. If these twenty-four numbers are then listed from smallest to largest, what is the position of 3142? Express your answer as an ordinal position (e.g., 1st, 2nd, 3rd, etc.). | 14th | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Permutations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Mary's top book shelf holds five books with the following widths, in centimeters: 6, 1/2, 1, 2.5, and 10. Calculate the average book width. Express your answer in centimeters as a decimal number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the average of two numbers is 12 and the average of three other numbers is 7, find the average of all five numbers. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Using the pattern $(x-1)(x^{n}+x^{n-1}+\cdots+x+1)=x^{n+1}-1$, find the units digit of $2^{2023}+2^{2022}+2^{2021}+\cdots+2^{2}+2+1$. Express your answer as a single digit (0-9). | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
In an opaque bag, there are 50 balls of two colors, yellow and white. After multiple attempts of picking balls, Xiaohong found that the frequency of picking yellow balls stabilized around 0.3. Find the possible number of yellow balls in the bag. Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC with AB = AC = 3.6, a point D is taken on AB at a distance 1.2 from A. Point D is joined to E in the prolongation of AC so that triangle AED is equal in area to ABC. Calculate the length of AE. | 10.8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A survey of 150 students asked them to choose between pizza, Thai food, or Greek food. Given that 20% of the students chose pizza and 38% chose Thai food, calculate the number of students who chose Greek food. Express your answer as a whole number. | 63 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the probability distribution of a random variable $\xi$ as shown below, and $E(\xi)=6.3$, calculate the value of $a$.
| $\xi$ | $a$ | $7$ | $9$ |
|-------|-----|-----|-----|
| $P$ | $b$ | $0.1$ | $0.4$ |
Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions for the equation \(\frac{1}{x}+\frac{1}{y}=\frac{1}{7}\). Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of solutions to the equation $$C_{16}^{x^{2}-x} = C_{16}^{5x-5},$$ considering the constraints on the values of x that satisfy the conditions of the binomial coefficient. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations of two circles: $C_1: x^2+y^2+2x-6y=0$ and $C_2: x^2+y^2-4x+2y+4=0$, determine the number of common tangents to both circles. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
A certain ninth-grade school held a student basketball game, using a single-cycle system by class. It was required that each pair of classes play one game. If a total of 45 games were played in the entire competition, calculate the number of participating classes. Express your answer as a whole number. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the rational numbers $-\frac{1}{2}$, $5$, $0$, $-(-3)$, $-2$, $-|-25|$, how many numbers are negative? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Cheenu could run 15 miles in 3 hours and 30 minutes when he was a boy. Now, as an old man, he can walk 10 miles in 4 hours. Calculate the difference in time it takes for him to walk a mile now compared to when he was a boy. Express your answer in minutes. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 2^a * 3^b = 8 * (6^10), where a and b are integers, find the value of b - a. Express your answer as an integer. | -3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Exponential Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations $mn + p^2 + 4 = 0$ and $m - n = 4$, calculate the value of $m + n$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Let $M=\{x|x<9\}$ and $N=\{x|x^2<9\}$. Determine the correct subset relationship between sets M and N. Express your answer as a subset relationship (e.g., M ⊆ N, N ⊆ M, etc.). | N ⊆ M | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $P=\{0,1\}$, and $M=\{x|x\subseteq P\}$, calculate the number of subsets of set $M$. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of positive integer points (points with both coordinates being positive integers) through which the graph of the quadratic function y = -x^2 + 20x - 64 passes. Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Algebra -> Algebra -> Quadratic Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an odd function, and f(x) = e^x - 1 when x >= 0, find the value of f(ln(1/2)). | -1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be a differentiable function, satisfying $\lim_{x \rightarrow 0} \frac{f(1) - f(1 - x)}{2x} = -1$. Find the slope of the tangent line to the curve $y = f(x)$ at the point $(1,f(1))$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, it is known that (a_2+a_5+a_9+a_12=60). Calculate the sum of the first 13 terms (S_13). Express your answer as a single integer. | 195 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given the point P(3-m, m-1), find the range of m such that P is in the second quadrant. Express your answer as an inequality in the format m > a or m < a, where a is a number. | m > 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
When factoring the expression $4m^{3}n-9mn^{3}$ using the method of factoring out the common factor, what is the common factor to be extracted? Express your answer as a mathematical expression. | mn | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=2$, $A=45^{\circ}$, and $B=60^{\circ}$, calculate the length of side $a$. Express your answer as a simplified fraction or exact decimal, including any necessary square roots or trigonometric fun... | \frac{2\sqrt{6}}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In a bag, there are 5 red balls and 3 white balls. Two balls are drawn successively without replacement, one ball each time. If it is known that the first ball drawn is a red ball, calculate the probability of drawing a white ball on the second draw. Express your answer as a simplified fraction. | \frac{3}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex numbers $z_1$ and $z_2$ are symmetric with respect to the imaginary axis in the complex plane, and $z_1 = 1 + i$, find the complex number $z_2$. Express your answer as a complex number in the form $a + bi$. | -1 + i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfying $$\frac {z}{2+ai} = \frac {2}{1+i}$$ ($a\in\mathbb{R}$), if the imaginary part of $z$ is -3, calculate the real part of $z$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 3x^2 + 2(a-1)x + b, find the range of values for the parameter a such that the function is decreasing on the interval (-∞, 1). Express your answer as an inequality, including all necessary mathematical symbols and notations. | a \leq -2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) = x^3 - 3x + a has a minimum value of 1 on the interval x ∈ [-2,0]. Calculate the value of the real number a. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
If the line $ax+y-5=0$ is perpendicular to the line $y=7x-2$, what is the value of $a$? Express your answer as a numerical value. | \frac{1}{7} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
A positive integer whose digits are the same when read forwards or backwards is called a palindrome. An example of a palindrome is 13931. Find the sum of the digits of the next palindrome greater than 13931. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the real numbers $2.1010010001\ldots$, $\frac{{22}}{7}$, $\sqrt{4}$, $\sqrt[3]{9}$, $100\pi$, how many numbers are non-irrational (rational)? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of $\cos {555}^{0}$. Express your answer as a simplified fraction or radical expression. | -\dfrac{\sqrt{6}+\sqrt{2}}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Let \( f(x) = |x-2| + |x-4| - |2x-6| \) for \( 2 \leqslant x \leqslant 8 \). Calculate the sum of the maximum and minimum values of \( f(x) \) within the given interval. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors in the plane, $\overrightarrow{a}$ and $\overrightarrow{b}$, with an angle of $\dfrac{\pi}{3}$ between them, and $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|= \dfrac{1}{2}$, calculate the angle between $\overrightarrow{a}+2\overrightarrow{b}$ and $\overrightarrow{b}$. Express your answer in radians... | \dfrac{\pi}{6} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
What is the arithmetic square root of 64? Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation
$$
(2 \sqrt{3}+\sqrt{2}) x^{2}+2(\sqrt{3}+\sqrt{2}) x+(\sqrt{2}-2 \sqrt{3})=0,
$$
calculate the sum and product of its roots. Express your answer for both values as simplified fractions or exact decimals if necessary. | \frac{2 \sqrt{6}-7}{5} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the solution $\left\{\begin{array}{l}x=-1\\ y=2\end{array}\right.$ to the equation $2x+ky=6$, calculate the value of $k$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $0 < \alpha < \frac{\pi}{2}$ and $-\frac{\pi}{2} < \beta < 0$, and that $\cos (\frac{\pi }{4}+\alpha )=\frac{1}{3}$ and $\cos (\frac{\pi }{4}-\frac{\beta }{2})=\frac{\sqrt{3}}{3}$, calculate the value of $\cos (\alpha +\frac{\beta }{2})$. Express your answer as a fraction or decimal value. | \frac{5\sqrt{3}}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Four consecutive integers have a sum of 2. Find the least of these integers. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin \alpha \cos \alpha = \frac{1}{8}$ and $\frac{5\pi}{4} < \alpha < \frac{3\pi}{2}$, calculate the value of $\cos \alpha - \sin \alpha$. Express your answer as a single value, including any necessary radicals. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coordinates of the point in the complex plane corresponding to the complex number $z= \frac {1-i}{2+i}$. Express your answer as an ordered pair (x, y), where x and y are the real and imaginary parts of the complex number, respectively. | \left(\frac {1}{5}, -\frac {3}{5}\right) | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given **_a_** = (2,1) and **_b_** = (_x_, -2), and **_a_** + **_b_** is parallel to **_2a_** - **_b_**, find the value of _x_. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (6, x)$ and $\overrightarrow{b} = (2, -2)$, and that $(\overrightarrow{a} - \overrightarrow{b}) \perp \overrightarrow{b}$, find the value of $x$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 300 billion in scientific notation. Provide your answer in the format $a \times 10^b$, where $a$ is the coefficient and $b$ is the exponent. | 3 \times 10^{11} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the x-intercept of the line passing through the points (3, 9) and (-1, 1). Express your answer as a fraction or decimal, including the negative sign if applicable. | - \frac {3}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {0,1,2,3,4,5}, M = {0,3,5}, and N = {1,4,5}, find the intersection of M and the complement of N in U, denoted as M ∩ (∁U N). Express your answer as a set, with elements separated by commas and enclosed in curly brackets. | {0,3} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system xOy, find the area of the region enclosed by the curve 2|x| + 3|y| = 5. Express your answer as a fraction or decimal. | \frac{25}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 - bx^2 + 1/2, which has exactly two distinct zeros, find the value of b. | \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number \(z=\dfrac{4-i}{1+i}\), find the imaginary part of its conjugate. Express your answer as a fraction or decimal. | \dfrac{5}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | 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"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies the equation $z(i+1)=\frac{2}{i-1}$, determine the imaginary part of the complex number $z$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
The equations $2x + 7 = 3$ and $bx - 10 = -2$ have the same solution for $x$. Find the value of $b$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of the function f(x) passes through the point (0, 1), find the point on the graph of the function f(4-x) that corresponds to this original point. Express your answer as an ordered pair (x, y). | (4, 1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the random variable ξ follows a normal distribution N(3, σ^2), if P(ξ > m) = a, then find the probability P(ξ > 6 - m). Express your answer as a probability expression in the format P(ξ > 6 - m) =.... | P(ξ > 6 - m) = 1 - a | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, if $\sqrt{3}\sin B+\cos B=2$, then calculate the value of $\tan \frac{A}{2}+\tan \frac{C}{2}+\sqrt{3}\tan \frac{A}{2}\tan \frac{C}{2}$. Express your answer in the form $\boxed{[answer]}$. | \sqrt{3} | big_math | [
"Mathematics -> Trigonometry -> Functions -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle α passes through point P(4, -3), calculate the value of 2sin(α) + cos(α). Express your answer as a simplified fraction. | -\frac{2}{5} | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the imaginary unit $i$ is a root of the equation $x^2+ax+b=0$, where $a, b \in \mathbb{R}$, calculate the value of $a+b$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given circle C: x² + y² - 4x + 3 = 0, find the equation of the circle symmetric to circle C about the line y = -x - 4. Express your answer as a circle equation in standard form. | (x + 4)² + (y + 6)² = 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x|(x-3)(x+1)\geqslant 0\}$ and $B=\left\{ x \mid x < - \frac{4}{5}\right\}$, calculate the intersection $A\cap B$. Express your answer as a set definition. | \{x|x\leqslant -1\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Ronald buys a pizza cut into 12 equal parts. He then cuts each part into 2 equal pieces. If he eats 3 of these pieces, what fraction of the pizza does he eat? Express your answer as a simplified fraction. | \frac{1}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the expression $x = \sqrt{18} - 2$, find the integer part of $x$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
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