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 two lines $l_{1}$: $(3+m)x-4y=5-3m$ and $l_{2}$: $2x-y=8$ are parallel, find the value of the real number $m$. Express your answer as a single real number. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer $n$ such that $\sqrt{18n}$ is an integer. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the vertices of triangle ABC as A(2,0), B(0,4), and C(4,1), where AE is the angle bisector of angle BAC, calculate the coordinates of a directional vector of line AE. Express your answer as a vector in the form (x, y). | (1,3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $\cos 45^{\circ}\cos 15^{\circ}+\sin 45^{\circ}\sin 15^{\circ}$ using the cosine formula for the difference of two angles. Express your answer as a fraction or decimal value. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the modulus of the complex number $z= \frac{|1-i|}{3+i}$. Express your answer as a simplified fraction with a square root. | \frac{\sqrt{5}}{5} | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<.01$? Express your answer as a single integer. | 2501 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
There are 1000 numbers arranged in a row, where for any three consecutive numbers, the middle number is equal to the sum of the two surrounding numbers. If the first and second numbers are both 1, calculate the sum of these 1000 numbers. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
How many lines pass through point P(1, 2) and have equal intercepts on the x-axis and y-axis? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is the largest negative integer, $b$ is the integer with the smallest absolute value, and $c$ is the smallest positive integer, find the value of $|a+b-c|$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(3,1)$, $\overrightarrow{b}=(1,3)$, $\overrightarrow{c}=(k,7)$, if $(\overrightarrow{a}+2\overrightarrow{c})\parallel \overrightarrow{b}$, find the value of $k$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A sample of a certain size is drawn from a group of students to analyze their academic performance. The sum of the frequencies of the first three groups is 20, and the sum of their frequencies is 0.4. Calculate the size of the sample drawn. Express your answer as a whole number. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Find the square root of 4, including both the positive and negative roots. Express your answer in the form \\boxed{\\pm x}, where x is the numerical value of the square root. | \\pm 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = a*sin(x) + b*3x + 4, where a and b are real numbers, and f(lg log3 10) = 5, find the value of f(lg lg 3). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi}{3}$, and $|\overrightarrow{a}| = \frac{1}{2}, |\overrightarrow{b}| = 4$, calculate the value of $\overrightarrow{a} \cdot \overrightarrow{b}$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (-1, 2) and b = (λ, 1), if a + b is parallel to a, find the value of λ. Express your answer as a single value. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
What is the fixed point that the line $kx-y-3k+3=0$ passes through, regardless of the value of $k$? Express your answer as an ordered pair $(x, y)$. | (3,3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, $F_{1}$, $F_{2}$ are its left and right foci, point $P$ is a point on the right branch of the hyperbola, point $M$ is the center of the circle, circle $M$ is the inscribed circle of triangle $PF_{1}F_{2}$, $PM$ line intersects the ... | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n} that satisfies a_1 = 1/4 and a_3a_5 = 4(a_4-1), find the value of a_2. Express your answer as a fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Fill the numbers 1, 2, 3, 4 into the squares labeled 1, 2, 3, 4, with one number per square, such that the label of each square is different from the number filled in it. Calculate the total number of ways to fill the squares under this condition. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Four families, A, B, C, and D, each have two children, making a total of 8 individuals. They plan to use two cars, labeled as Alpha and Beta, for an outing. Each car can seat 4 individuals, and the arrangement of the 4 children in the same car does not consider the seating order. However, the twin sisters from family A... | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Carve a cube with an edge length of 1 into a sphere of the largest possible volume. Calculate the volume of this sphere. Express your answer as a fraction with pi. | \dfrac{\pi}{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $b$ and $c$ are in the interval $[-1, 1]$, calculate the probability that the equation $x^{2} + 2bx + c^{2} = 0$ has real roots. Express your answer as a simplified fraction. | \dfrac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Express the decimal number 0.0000005 in scientific notation. Use the format $a \times 10^{b}$. | $5 \times 10^{-7}$ | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Express the decimal degree value 21.24° in degrees, minutes, and seconds. Provide your answer in the format °'\", with whole numbers for minutes and seconds. | 21°14'24" | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\overrightarrow{a} = (2, 1)$$, $$\overrightarrow{a} \cdot \overrightarrow{b} = 10$$, and $$|\overrightarrow{a} + \overrightarrow{b}| = 5\sqrt{2}$$, calculate the magnitude of $$\overrightarrow{b}$$, denoted as $$|\overrightarrow{b}|$$. Express your answer as a numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
When two fair dice are rolled at the same time, what is the probability that the sum of the points obtained is 6? Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$. | \frac{5}{36} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=2^{x}$, the common difference of the arithmetic sequence $\{a_n\}$ is $2$. If $f(a_2+a_4+a_6+a_8+a_{10})=4$, find the value of $\log_2[f(a_1)⋅f(a_2)⋅f(a_3)⋅…⋅f(a_{10})]$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The school wants to select 6 students from 10 to participate in a study seminar. However, two students, A and B, cannot attend at the same time. Calculate the total number of different selection methods. Express your answer as a numerical value. | 140 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{(x,y)|x^{2}+y^{2}=1\}$ and $B=\{(x,y)|y=x\}$, find the number of elements in $A\cap B$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
In the aftermath of an earthquake, 3 doctors and 4 nurses are arranged to assist injured teachers and students at 3 schools that were severely affected. Each school is required to have at least 1 doctor and 1 nurse. Calculate the total number of different arrangements. Express your answer as a single integer. | 216 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $\alpha \in \left(0,\pi \right)$ and $\cos\alpha=-\frac{1}{2}$, find the value of $\alpha$. Express your answer in radians as a fraction of $\pi$. | \frac{2\pi}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
5 people stand in a row, and there are exactly 1 person between person A and person B. What is the total number of arrangements? Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
When a number is divided by 7, the quotient is 12 and the remainder is 5. What is the number? Express your answer as a whole number. | 89 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the monotonically decreasing interval of the function y = 2^(x^2 - 5x - 6). Express your answer as an interval. | (-\infty, \frac {5}{2}) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the definition of an inflection point, find the equation of the line on which the inflection point M(x0, f(x0)) of the function f(x) = 3x + 4sin(x) - cos(x) lies. Express your answer in the form y = mx, where m is the slope of the line. | y = 3x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the point (n,a_n) is on the graph of the function y=2x-13, find the minimum value of the sum of the first n terms of the sequence {a_n}, S_n. Express your answer as a single integer. | -36 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
From the set {1,2,3,…,10}, select three different numbers such that at least two of them are adjacent. Find the number of different ways to do this. Express your answer as a whole number. | 64 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{1+ \sqrt{3}i}{ \sqrt{3}+i}$, find the complex conjugate of $z$. Express your answer as a complex number in the form $\boxed{a+bi}$. | \frac{\sqrt{3}}{2}-\frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Wenlu, Xander, Yasser, and Zoe make the following statements:
Wenlu: "Xander is lying."
Xander: "Yasser is lying."
Yasser: "Zoe is telling the truth."
Zoe: "Wenlu is telling the truth."
Determine the possible numbers of people telling the truth. Express your answer as a list or range of possible numbers. | 1, 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Michael has $280 in $20 bills. How many $20 bills does he have? Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Real numbers n and m are consecutive integers such that n < √26 < m. Find the value of m + n. Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
All edges of a tetrahedron have a length of $\sqrt{2}$, and the four vertices are on the same sphere. Calculate the surface area of this sphere. Express your answer in terms of $\pi$. | 3 \pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $a=(1,0)$ and $b=(-\frac{1}{2},\frac{\sqrt{3}}{2})$, calculate the angle between $a$ and $a+b$. Express your answer in radians using pi notation (e.g. pi/3, 2*pi/3, etc.). | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the coefficient of $x^{6}y^{2}$ in the expansion of $(x^{2}+xy+2y)^{5}$. | 40 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a \in \mathbb{R}$, the complex number $z = \frac{a - i}{1 - i}$ is a purely imaginary number ($i$ is the imaginary unit). Find the value of $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(A C = C B = 10 \, \text{m}\), where \(A C\) and \(C B\) are each the diameter of the small equal semi-circles. The diameter of the larger semi-circle is \(A B\). Calculate the difference in lengths between the semi-circular arc from \(A\) to \(B\) and the combined semi-circular arcs from \(A\) to \(C\... | 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through the point (1,0) and is perpendicular to the line x - 2y - 2 = 0. Express your answer in standard form (Ax + By + C = 0). | 2x + y - 2 = 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Three sets of parallel lines and another set of five parallel lines intersect on a plane. How many parallelograms can be formed by selecting two lines from one set and two lines from the other set? Express your answer as a single numerical value. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
From 10 events, one event is chosen at random. If the probability that this event is a certain event is 0.2, and the probability that it is an impossible event is 0.3, then find the number of random events among these 10 events. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
For the function $y= \begin{cases} x+2, & x\leqslant -1 \\ x^{2}, & -1 < x < 2 \\ 2x, & x\geqslant 2\end{cases}$, find the exact value(s) of $x$ that satisfy $f(x)=1$. Express your answer as exact numerical values, separating multiple values with commas if applicable. | -1, 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
The number of integers A that satisfy the inequality 10^4 ≤ A ≤ 10^5 is x * 10^4 + 1. Find the value of x. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
A student sets up a game on a computer where a ball is dropped from a height of 100 meters, and each time it falls to the ground, it bounces back to half of its previous height. Calculate the total distance the ball has traveled by the time it hits the ground for the 8th time. Express your answer in meters (m), rounded... | 298.4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Each of the following 15 cards has a letter on one side and a positive integer on the other side. What is the minimum number of cards that need to be turned over to check if the following statement is true: "If a card has a lower case letter on one side, then it has an odd integer on the other side." Express your answe... | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Suppose $S$ cans of soda can be purchased from a vending machine for $Q$ quarters. Express the number of cans of soda that can be purchased for $D$ dollars in terms of $S$, $Q$, and $D$, given that $1$ dollar is equivalent to $4$ quarters. Provide your answer as an algebraic expression. | \frac{4DS}{Q} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given a positive real number $m$, if $x^{10}=a_{0}+a_{1}(m-x)+a_{2}(m-x)^{2}+…+a_{10}(m-x)^{10}$, where $a_{8}=180$, calculate the value of $m$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The sum of three numbers is 98. The ratio of the first to the second is 2/3, and the ratio of the second to the third is 5/8. Find the value of the second number. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression \(\frac{4 \times 3}{2+1}\) and provide your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
At Mathville Junior High School, 30 boys and 20 girls wrote the Pascal Contest. Certificates were awarded to 30% of the boys and 40% of the girls. What percentage of all participating students received certificates? Express your answer as a percentage. | 34 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A scalene triangle ABC has two heights with lengths 4 and 12. If the length of the third height is also an integer, what is the maximum possible length of it? Provide your answer as a single integer. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (1,2), b = (1,0), and c = (3,4), find the real number λ such that (a + λb) || c. Express your answer as a single real number. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Let $ABC$ be a triangle with circumradius $1$. If the center of the circle passing through $A$, $C$, and the orthocenter of $\triangle ABC$ lies on the circumcircle of $\triangle ABC$, what is the length of $AC$? Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2a-3b=2$, find the value of the expression $8-6a+9b$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A person has 4 keys, of which 2 can open a door. If a key is randomly selected to try to open the door, and the key that cannot open the door is discarded, calculate the probability of opening the door on the second try. Express your answer as a fraction in simplest form. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the opposite of the number $a$ is $-2022$, find the value of $a$. Express your answer as a single integer. | 2022 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
A railway freight station is organizing and dispatching six coal freight trains into two groups, each consisting of three trains, with trains 'A' and 'B' in the same group. If the group containing trains 'A' and 'B' departs first, calculate the total number of different possible departure sequences for the six trains. | 144 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $y=f\left(x\right)$ is $\left[-4,5\right]$, determine the domain of the function $y=\frac{f(x-1)}{\sqrt{x+2}}$. Express your answer in interval notation. | (-2, 6] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $a_n$, $a_5 + a_6 = 4$ and $a_{15} + a_{16} = 16$. Find the value of $a_{25} + a_{26}$. Express your answer as a single numerical value. | 64 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, and $(\overrightarrow{a}-2\overrightarrow{b})\perp\overrightarrow{a}$, $(\overrightarrow{b}-2\overrightarrow{a})\perp\overrightarrow{b}$, calculate the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$. Express your answer... | \dfrac{\pi}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given $|\overrightarrow{AB}|=3$, points $A$ and $B$ move respectively on the $x$-axis and $y$-axis, with $O$ being the origin, and $\overrightarrow{OP}=\frac{1}{3}\overrightarrow{OA}+\frac{2}{3}\overrightarrow{OB}$. Find the equation of the trajectory of the moving point $P$ in standard form. | x^2+ \frac{y^2}{4}=1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a proportional function y=kx (k!=0), where the function values of y decrease as x increases, analyze the graph of the linear function y=2x+k and identify the quadrants through which it passes. Express your answer as a list of quadrants (e.g. Quadrants I, III, IV). | Quadrants I, III, IV | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Let \( a_1, a_2, \cdots, a_k \) be a finite arithmetic sequence. Given \( a_4 + a_7 + a_{10} = 17 \), \( a_4 + a_5 + a_6 + \cdots + a_{14} = 77 \), and \( a_k = 13 \), determine the value of \( k \). Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In a circle with a radius of 3cm, there is an arc AB with a length of $\frac{\pi}{2}cm$. Calculate the central angle corresponding to this arc AB. Express your answer in degrees (°) as a whole number. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an even function on the real numbers $\mathbb{R}$, and for $x \geq 0$, $f(x+2) = f(x)$. If $f(x) = \log_2(x + 1)$ when $x \in [0, 2)$, then calculate the value of $f(-2013) + f(2014)$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular box has a total length of its 12 edges equal to 140 and a diagonal length of 21. Find the surface area of the box. Express your answer as a whole number. | 784 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Find the product of two different prime numbers whose sum is 10. Express your answer as a single integer. | 21 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the locus of the midpoint of the line segment connecting point P(4, -2) and any point on the circle x^2 + y^2 = 4. Express your answer as a standard equation of a circle. | (x - 2)^2 + (y + 1)^2 = 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( P Q = 19 \), \( Q R = 18 \), and \( P R = 17 \). Point \( S \) is on \( P Q \), point \( T \) is on \( P R \), and point \( U \) is on \( S T \) such that \( Q S = S U \) and \( U T = T R \). Calculate the perimeter of \(\triangle P S T\). Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
If \(a\) and \(b\) are natural numbers not exceeding 10, how many pairs \((a, b)\) satisfy the equation \(a x = b\) such that the solution \(x\) is less than \(\frac{1}{2}\) but greater than \(\frac{1}{3}\)? Provide your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the exact value of sin(405°) + cos(-270°). Express your answer as a single value, possibly involving square roots. | \dfrac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In the right triangular prism $ABC-A_1B_1C_1$, if $\angle BAC = 90^\circ$ and $AB = AC = AA_1$, determine the angle formed by the skew lines $BA_1$ and $AC_1$. Express your answer in degrees. | 60^\circ | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the vertices of rectangle PQRS as P(15,55), Q(26,55), and R(26,35), calculate the area of this rectangle. Express your answer as a whole number. | 220 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $(2-i)^2-(1+3i)$. Express your answer as a complex number in the form $a+bi$. | 2-7i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the circle x^2 + y^2 + 2x - 4y = 0 is symmetric about the line 3x + y + m = 0, find the value of the real number m. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the lengths of the sides of quadrilateral ABCD are given as follows: AD = 4, AB = 5, BC = 10, CD = 7. Calculate the perimeter of ABCD. Express your answer as a whole number. | 26 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (x, \sqrt{3})$ and $\overrightarrow{b} = (x, -\sqrt{3})$, if $(2\overrightarrow{a} + \overrightarrow{b}) \perp \overrightarrow{b}$, calculate the magnitude of vector $\overrightarrow{a}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, a_1 = 2. When n is an odd positive integer, a_{n+1} = a_n + 2; when n is an even positive integer, a_{n+1} = 2a_n. Calculate the value of a_6. | 22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
When tossing two fair coins simultaneously four times, let X be the number of times exactly one coin lands heads up and the other lands tails up. Calculate the expected value of X. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $-3\left(x-xy\right)$. Express your answer as a simplified algebraic expression. | -3x + 3xy | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of real roots for the cubic equation \(x^{3} - \sqrt{3} x^{2} + x - \left(1 + \frac{\sqrt{3}}{9}\right) = 0\). Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Equations"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
The equation of an ellipse is given by $\frac {x^{2}}{5}$+ $\frac {y^{2}}{4}$$=1$. Find the eccentricity of this ellipse. Express your answer as a fraction or decimal. | \frac{\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In a class, a speech competition is organized with 5 students, including students A, B, and C. The order of the speeches is determined by drawing lots. Given the condition that "Student A is not the first to speak, and student B is not the last to speak," what is the probability that student C will speak first? Express... | \frac{3}{13} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\{a\_n\}$ where all terms are positive, $a\_2$, $\frac{a\_3}{2}$, $a\_1$ form an arithmetic sequence. Find the value of $\frac{a\_4 + a\_5}{a\_3 + a\_4}$. Express your answer as a fraction or decimal value. | \frac{1 + \sqrt{5}}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If the random variable ξ follows the normal distribution N(0, σ^2) and the probability P(-2 < ξ ≤ 0) = 0.4, then find the probability P(ξ > 2). Express your answer as a decimal value. | 0.1 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given that the lateral surface of a cone, when unfolded, is a semicircle with an area of $2\pi$, find the area of the base of the cone. Express your answer in terms of $\pi$. | \pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the graph of the function f(x) = x^2 + 1/x at the point (1, f(1)). Express your answer in standard form (Ax + By + C = 0). | x - y + 1 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the function y = x^3 - 3x^2 + 3 at the point (1, 1). Express your answer in the slope-intercept form (y = mx + b), where m and b are the slope and y-intercept of the tangent line, respectively. | y = -3x + 4 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
There are two red balls numbered 1 and 2 and three black balls numbered 1, 2, and 3 in a bag (all these 5 balls have no other difference except for color and number). Each time you draw a ball from the bag without replacement, what is the probability that one black ball and one red ball are drawn in the first two draws... | \frac{3}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In a plane with a Cartesian coordinate system, consider two sets: A, which consists of all points on a circle, and B, which consists of all points on a certain line. Determine the maximum number of elements in the intersection of sets A and B. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given two 7-digit numbers, $\underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1}$ and $\underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C}$, each of which is a multiple of 3, find a possible value of C that satisfies the divisibi... | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = 2(x + 1)^2 - 3, translate it one unit to the right and three units up. Express the equation of the new function in standard function notation (e.g., y = f(x)). | y = 2x^2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
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