instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Given that the domain of the function $y = f(x)$ is $[2, 4]$, determine the domain of $y = f(\log_{\frac{1}{2}}x)$. Express your answer in interval notation, using square brackets for closed intervals. | [\frac{1}{16}, \frac{1}{4}] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 keys strung together, among which only one is useful. If you try to unlock in sequence, discarding each key that fails to unlock until success, what is the maximum value of the number of attempts? Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $x(x-3) = x-3$ to find the solution other than $x=1$. Express your answer as a single value of $x$. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions for the inequality \( |x| < 3 \pi \). Express your answer as a whole number. | 19 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $$a=( \sqrt {3},1),b=(0,1),c=(k, \sqrt {3})$$, if $$\overrightarrow {a}$$+2 $$\overrightarrow {b}$$ is perpendicular to $$\overrightarrow {c}$$, find the value of k. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(0,4)$ and $\overrightarrow{b}=(-3,-3)$, calculate the coordinates of the projection vector of $\overrightarrow{a}$ onto $\overrightarrow{b}$. Express your answer as an ordered pair (x, y). | (2,2) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given sin(α) = 3/5, calculate cos(2α) using the double angle formula for cosine. Express your answer as a fraction. | \frac{7}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The lateral surface of a cylinder unfolds into a square. Find the ratio of its lateral surface area to the base area. Express your answer as a numerical value, including any necessary mathematical constants. | 4\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The perimeter of a right-angled triangle, with one of its legs being 11 and the other two sides also being positive integers, is to be calculated. Find the perimeter of the triangle. Express your answer as a positive integer. | 132 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( \angle A = 3 \angle B \) and \( \angle B = 2 \angle C \). Find the measure of \( \angle B \) in degrees. | 40^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with a common difference of 2, and the sum of the first 100 terms (S100) is 10000, calculate the sum of the odd-numbered terms (a1 + a3 + a5 +... + a99). Express your answer as a single integer. | 4950 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $2^a+2^b=3^c+3^d$, find the maximum number of integers $a,b,c,d$ that can be negative. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Put 4 different colored balls into 3 differently labeled boxes. Calculate the total number of different ways to do this. Express your answer as a single integer. | 81 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles trapezoid where the lower base is twice the length of the upper base, a point inside this trapezoid is called a "hope point" if the distance from this point to either the upper base or lower base is the smallest among the distances to the four sides of the trapezoid. Calculate the probability that a ran... | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the opposite number of the real number a is -1, find the value of a + 1. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
For the integer numbers $i,j,k$ satisfying the condition $i^2+j^2+k^2=2011$, what is the largest possible value of $i+j+k$? Express your answer as a single integer. | 77 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Using the Monte Carlo method to calculate the area enclosed by $y=x^2+1$ and $y=5$, first generate two groups of uniform random numbers between 0 and 1, denoted $a_1=\text{RAND}$ and $b_1=\text{RAND}$, then perform translation and scaling transformations as $a=4a_1-2$ and $b=4b_1+1$. After 1000 experiments, the number ... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given $$\overrightarrow{a} = 5\overrightarrow{e}, \overrightarrow{b} = -3\overrightarrow{e}, \overrightarrow{c} = 4\overrightarrow{e},$$ calculate the result of the expression $2\overrightarrow{a} - 3\overrightarrow{b} + \overrightarrow{c}$. Express your answer as a vector in terms of $\overrightarrow{e}$. | 23\overrightarrow{e} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In a set of data $1$, $2$, $3$, $4$, $5$, $x$, there exists a unique mode, and the average of this set of data is equal to the mode. Find the value of $x$. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $\mathrm{ABCD}$ is a right trapezoid with $\angle \mathrm{DAB} = \angle \mathrm{ABC} = 90^\circ$. A rectangle $\mathrm{ADEF}$ is constructed externally along $\mathrm{AD}$, with an area of 6.36 square centimeters. Line $\mathrm{BE}$ intersects $\mathrm{AD}$ at point $\mathrm{P}$, and line $\mathrm{PC}$ ... | 3.18 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the equation a_3 + a_9 = 15 - a_6 holds. Find the value of a_6. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Among the numbers from 1 to 10, when selecting any 4 numbers, find the total number of ways such that the second largest number is 7. Express your answer as a single integer. | 45 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the cube root of -8. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions (x, y) to the equation 1/x + 1/y = 1/7. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the fraction $\frac{x^2-x}{x-1}$ equals $0$, find the value of $x$ that satisfies this condition. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
A bank plans to introduce a new fixed deposit service. It is predicted that the deposit amount is directly proportional to the deposit interest rate, with a proportionality constant $k$ ($k>0$). The loan interest rate is $4.8\%$. Assuming that all the deposits the bank attracts can be lent out, calculate the deposit in... | 0.024 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise... | 6 | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
During the search for the underwater black box of Malaysia Airlines Flight MH370, 3 underwater robots (A, B, C) and 2 divers (a, b) are assigned to one search mission each. Only one underwater robot or one diver can be deployed at a time. Robot C cannot be deployed first, and A and a must be arranged adjacently. Calcul... | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
How many integers between 10000 and 100000 include the block of digits 178? Express your answer as a single integer. | 280 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable follows the binomial distribution $B(n, p)$, if $E(x) = 30$ and $D(x) = 20$, find the value of $p$. Express your answer as a fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \u221a[3]{5+2\u221a{13}} + \u221a[3]{5-2\u221a{13}} and find its exact value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A box contains 14 disks, each colored red, blue, or green. There are twice as many red disks as green disks, and half as many blue disks as green disks. Find the number of green disks. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, each scale shows the total mass (weight) of the shapes on that scale. Using the information provided by the scales, calculate the mass (weight) of a $\triangle$. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that points P and Q are inside triangle ABC, and the vector equations:
\[
\overrightarrow{PA} + 2\overrightarrow{PB} + 3\overrightarrow{PC} = 2\overrightarrow{QA} + 3\overrightarrow{QB} + 5\overrightarrow{QC} = \overrightarrow{0}
\]
find the value of the ratio |PQ|/|AB|. Express your answer as a simplified frac... | \frac{1}{30} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In right triangle ABC, ∠C=90° and AB=2BC. Find the value of cos A. Express your answer as a fraction or decimal. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The Sunshine Company sells an electronic product with a cost price of 21 yuan. If the product is sold at 90% of its marked price and still yields a 20% profit, then what is the marked price of this electronic product? Express your answer in yuan as a whole number. | 28 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum period of the function $y = \cos^4(x) - \sin^4(x) + 2$. Express your answer as a single value or expression. | \pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the remainder of $1949^{1995}$ when divided by 7. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Peter, Emma, and Kevin are playing chess. Peter won 4 games and lost 2 games; Emma won 3 games and lost 3 games. If Kevin lost 3 games, calculate the number of games Kevin won. | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of functions from the list below that are power functions, where a power function is defined as $y = x^n$ with $n$ being a constant real number:
1. $y = x^2$
2. $y = \left(\frac {1}{2}\right)^x$
3. $y = 4x^2$
4. $y = x^5 + 1$
5. $y = (x - 1)^2$
6. $y = x$
7. $y = a^x$ ($a > 1$)
Express your answe... | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $m$ in the equation $$\frac {2}{1+i}=1+mi \quad (m \in \mathbb{R}, i \text{ denotes the imaginary unit}).$$ Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_n}=C_{200}^n⋅{({\sqrt[3]{6}})^{200-n}}⋅{({\frac{1}{{\sqrt{2}}}})^n}$, where n ranges from 1 to 95, calculate the number of integer terms in the sequence. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} defined by a_1 = 2 and a_{n+1} = a_n + 2n for n >= 1, find the value of a_{100}. Express your answer as a single integer. | 9902 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given set $A=\{x|x \lt -3\}$ and set $B=\{-5,-4,-3,1\}$, find the intersection of sets $A$ and $B$, denoted as $A\cap B$. Express your answer in standard set notation. | \{-5,-4\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression \(8 imes 10^{5} + 4 imes 10^{3} + 9 imes 10 + 5\). Express your answer as a single integer. | 804095 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of all positive divisors of 1184. | 2394 | big_math | [
"Mathematics -> Number Theory -> Divisor Functions -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Half the people in a room left. One third of those remaining started to dance. There were then 12 people who were not dancing. Find the original number of people in the room. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $y=f(x)$ is $[-1,3]$, determine the domain of the function $y=f(3x-2)$. Express your answer in interval notation. | [\frac{1}{3},\frac{5}{3}] | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
The line $2x-my+1-3m=0$ represents a family of lines when $m$ changes. Find the fixed point that all lines pass through. Express your answer as a coordinate point $(x, y)$. | \left(-\frac{1}{2}, -3\right) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given the ordinary generating function $A(t)$ of the sequence $\{a_{0}, a_{1}, a_{2}, \cdots \}$, find the ordinary generating function of the sequence $\{a_{0}, a_{0} + a_{1}, a_{0} + a_{1} + a_{2}, \cdots \}$. Express your answer in terms of $A(t)$. | \frac{1}{1-t} A(t) | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
If $S = 6 \times 10,000 + 5 \times 1,000 + 4 \times 10 + 3 \times 1$, what is the value of $S$? Express your answer as a single integer. | 65043 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that y=f(x) is an odd function defined on R, and it is monotonically increasing on (0, +∞), and f(log2(8)) = 0, find the solution set for the inequality xf(x) > 0. Express your answer in interval notation. | (-∞, -3) ∪ (3, +∞) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In a triangle, its area and its perimeter are numerically equal. Find the radius of its inscribed circle. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given 6 white balls and 4 black balls, calculate the number of ways to draw 4 balls such that at least 2 of them are black. Express your answer as a single numerical value. | 115 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that the magnitudes of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are 2 and 3 respectively, and the angle between them is $60^{\circ}$, find the value of $| \overrightarrow{a}+ \overrightarrow{b}|$. Express your answer as a numerical value. | \sqrt{19} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=-\frac{1}{2}+\frac{{\sqrt{3}}}{2}i$, calculate the value of $\sum_{i=1}^{2023}{{z^i}}$. Express your answer as a complex number in the form $a + bi$. | -\frac{1}{2}+\frac{\sqrt{3}}{2}i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of N such that the number of odd integers between 4 and N is equal to the number of even integers between 1 and 103. Express your answer as a single integer. | 106 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a$ and $b$ that satisfy $| \overrightarrow{a}|=1$, $| \overrightarrow{b}|=2$, and $| \overrightarrow{a}- \overrightarrow{b}|=2$, find the value of $| \overrightarrow{a}+ \overrightarrow{b}|$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{6} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Find the arithmetic square root of 9. Remember that the arithmetic square root only gives the positive root of a number. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the expression $(x+1)(x+2)(x+3)(x+4) + 2019$. Express your answer as a single integer. | 2018 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Three rugs have a combined area of \( 90 \, \text{m}^2 \). When they are laid down to cover completely a floor of area \( 60 \, \text{m}^2 \), the area which is covered by exactly two layers of rug is \( 12 \, \text{m}^2 \). Find the area of the floor covered by exactly three layers of rug. Express your answer in squar... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation y = x^2 + 2x + 1, find all possible values of x when y = 4. Express your answer as a list of values. | -3, 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Two lines with slopes \(\frac{1}{4}\) and \(\frac{5}{4}\) intersect at \((1,1)\). Find the area of the triangle formed by these two lines and the vertical line \(x=5\). Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function $y=\cos \left( \frac {\pi}{3}- \frac {2}{5}x\right)$. Express your answer in terms of π. | 5\pi | big_math | [
"Mathematics -> Precalculus -> Functions -> Trigonometric Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function $f(x)=\frac{1}{x}$. Express your answer as a mathematical expression in the form $\boxed{f'(x)}$. | -\frac{1}{x^2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
A school invites 6 out of 10 students to participate in a learning seminar. However, students A and B cannot attend at the same time. Calculate the number of different invitation methods. Express your answer as a whole number. | 140 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of the interior angles of a dodecagon. Express your answer in degrees (°). | 1800 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Find the absolute value of 2022. Express your answer as a single number. | 2022 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic inequality in x, ax^2 + bx + 1 > 0, with the solution set {x | -1 < x < 1/3}, find the product ab. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 people lining up for a photo, with the requirement that three specific individuals, named A, B, and C, must stand together. Find the number of different arrangements. Express your answer as a whole number. | 144 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, it is known that $A=30^{\circ}$, $B=45^{\circ}$, and $a=1$. Using the Law of Sines, calculate the value of side $b$. Express your answer as a simple radical. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If $(1-i)z=2i$, find the modulus of the complex number $z$. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = a{x}^{3} + \\frac{b}{x} + 3, and it's known that f(4) = 5, calculate the value of f(-4). Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
If $f''(x_{0})=2$, find the value of $\lim\limits_{k\to 0} \frac{f(x_{0}-k)-f(x_{0})}{2k}$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular to each other, $\overrightarrow{a} = (6, -8)$, $|\overrightarrow{b}| = 5$, and the angle between $\overrightarrow{b}$ and the vector $(1, 0)$ is obtuse, find the vector $\overrightarrow{b}$. Express your answer in the format $\... | (-4, -3) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the definite integral of the function f(x) = sin^5(x) + 1 over the interval [-π/2, π/2]. Express your answer as a numerical value. | π | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors and their angle is 60°, calculate the magnitude of the vector sum $|\overrightarrow{a}+2\overrightarrow{b}|$. Express your answer as a numerical value. | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Sammy is allowed to eat five sweets from his very large jar which contains many sweets of each of three flavors - Lemon, Orange, and Strawberry. He wants to eat his five sweets in such a way that no two consecutive sweets have the same flavor. Calculate the number of ways he can do this. Express your answer as a single... | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A regular octahedron is formed by joining the centers of adjoining faces of a cube. Find the ratio of the volume of the octahedron to the volume of the cube. Express your answer as a numerical value. | \frac{1}{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given a=($\frac{1}{2}$)<sup>-0.8</sup>, b=${log_{\frac{1}{2}}}\frac{2}{3}$, c=4<sup>0.3</sup>, what is the relationship between the values of a, b, and c? Express your answer in the format a [relation] b [relation] c (e.g., a < b < c). | b < c < a | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then calculate the distance from \( P \) to \( T \). Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Ben thinks of a number, adds 1 to it, and doubles the result. He gives this answer to Sue. Sue then subtracts 1 from the number she is given and doubles the result to get her answer. If Ben thinks of 6, what should Sue's answer be? Express your answer as a single integer. | 26 | big_math | [
"Mathematics -> Algebra -> Simple Equations"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, line $l$ passes through the points $(-3, 0)$ and $(0, -5)$, and line $l'$ passes through the point $(2, 4)$ and is parallel to the y-axis. Find the coordinates of the intersection point of these two lines. Express your answer as an ordered pair $(x, y)$ and specify the quadrant or ax... | (2, - \frac{25}{3}) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Let the random variable X follow a binomial distribution B(6, 1/2). Find the value of P(X=3). Express your answer as a simplified fraction. | \frac{5}{16} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the real number m for which the function y = e^x + mx has an extremum. Express your answer as an inequality. | m < 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A shooter has probabilities of hitting the 10-ring, 9-ring, and 8-ring of 0.24, 0.28, and 0.19, respectively. Find the probability that this shooter scores at most 8 points in a single shot. Express your answer as a decimal value between 0 and 1. | 0.48 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Solve for $z$ in the equation $\frac {1+i}{z-i}=i$. Express your answer as a complex number in the form $a+bi$. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
What number should be placed in the box to make \( 10^{4} imes 100^{\square}=1000^{6} \)? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the imaginary number z = 1 + bi is a root of the equation x^2 + ax + 3 = 0, where a and b are real numbers, calculate the magnitude |z| of the complex number z. Express your answer as a numerical value. | \sqrt{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of pairs of integer solutions (x, y) that satisfy the equation (x-8)(x-10) = 2^y. | 0 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. Calculate the fraction of the grid that is covered by the triangle. Express your answer as a simplified fraction in the form $\boxed{\frac{a}{b}}$. | \frac{1}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, 2)$, $\overrightarrow{b} = (2, -3)$. If vector $\overrightarrow{c}$ satisfies $(\overrightarrow{c} + \overrightarrow{a}) \parallel \overrightarrow{b}$ and $\overrightarrow{c} \perp (\overrightarrow{a} + \overrightarrow{b})$, then find vector $\overrightarrow{c}$. Express your ans... | \left( -\frac{7}{9}, -\frac{7}{3} \right) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Qaddama is 6 years older than Jack. Jack is 3 years younger than Doug. If Qaddama is 19 years old, how old is Doug? Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^2 + 1 (x ≥ 1), find the value of f^(-1)(10). Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Functions -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 2sin(x) + cos(x), find the value of sin(α) when α takes the maximum value such that f(x) is an increasing function on the interval [0, α]. Express your answer as a fraction, including the square root if necessary. | \frac{2\sqrt{5}}{5} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (4,3) and b = (-1,2), if the vector a + k*b is perpendicular to a - b, calculate the value of k. | \frac{23}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the solution $\left\{\begin{array}{l}x=-2\\ y=1\end{array}\right.$ to the equation $mx+3y=7$, find the value of $m$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
What is the imaginary part of the conjugate of the complex number $(2+i)i$? Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial $P = a^2 + 4a + 2014$, find the minimum value of $P$. Express your answer as a single integer. | 2010 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
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