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Simplify the following expression: $k = \dfrac{p - 9}{5p - 9} - \dfrac{-4p - 20}{5p - 9}$
Hints:
Since the expressions have the same denominator we simply subtract the numerators: $k = \dfrac{p - 9 - (-4p - 20)}{5p - 9}$ Distribute the negative sign: $k = \dfrac{p - 9 + 4p + 20}{5p - 9}$ Combine like terms: $k = \dfra... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
${45} \div {9} = {?}$
Hints:
If we split ${45}$ circles into $9$ equal rows, how many circles are in each row? ${9}$ ${\color{#29ABCA}{1}}$ ${\color{#29ABCA}{2}}$ ${\color{#29ABCA}{3}}$ ${\color{#29ABCA}{4}}$ ${\color{#29ABCA}{5}}$ ${8}$ ${\color{#29ABCA}{6}}$ ${\color{#29ABCA}{7}}$ ${\color{#29ABCA}{8}}$ ${\color{#29... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Simplify the following expression: ${5y-1-y+8}$
Hints:
Rewrite the expression to group the ${y}$ terms and numeric terms: $ {5y - y} {-1 + 8}$ Combine the ${y}$ terms: $ {4y} {-1 + 8}$ Combine the numeric terms: $ {4y} + {7}$ The simplified expression is $4y+7$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Solve for $k$. $ \dfrac{11}{8} = \dfrac{k}{4} $ $k =$
Hints:
Multiply both sides by ${4}$. $ {4} \times \dfrac{11}{8} = \dfrac{k}{4} \times {4} $ $ \dfrac{{4} \times 11}{8} = k $ $k = \dfrac{44}{8}$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Rewrite ${((7^{-5})(6^{7}))^{4}}$ in the form ${7^n \times 6^m}$.
Hints:
${ ((7^{-5})(6^{7}))^{4} = (7^{(-5)(4)})(6^{(7)(4)})} $ ${\hphantom{ ((7^{-5})(6^{7}))^{4}} = 7^{-20} \times 6^{28}} $ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$\Huge{{?} = {90} \times {3}}$
Hints:
$ {{?} = {10} \times {9} \times {3}} $ $ {{?} = {10} \times 27} $ $ {{?} = 270} $ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$ C = \left[\begin{array}{rr}0 & 3 \\ 5 & 0 \\ -2 & 0\end{array}\right]$ $ F = \left[\begin{array}{rr}4 & 0 \\ 3 & 2\end{array}\right]$ What is $ C F$ ?
Hints:
Because $ C$ has dimensions $(3\times2)$ and $ F$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ C F = \left[\begin{array}... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
What is the greatest common factor of $10$ and $6$ ? Another way to say this is: $\operatorname{gcf}(10, 6) = {?}$
Hints:
The greatest common factor is the largest number that is a factor of both $10$ and $6$ The factors of $10$ are $1$ $2$ $5$ , and $10$ The factors of $6$ are $1$ $2$ $3$ , and $6$ Thus, the greatest... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
What is the extraneous solution to these equations? $\dfrac{x^2 + 6}{x - 5} = \dfrac{-5x}{x - 5}$
Hints:
Multiply both sides by $x - 5$ $ \dfrac{x^2 + 6}{x - 5} (x - 5) = \dfrac{-5x}{x - 5} (x - 5)$ $ x^2 + 6 = -5x$ Subtract $-5x$ from both sides: $ x^2 + 6 - (-5x) = -5x - (-5x)$ $ x^2 + 6 + 5x = 0$ Factor the express... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$ (-7)^{2}$
Hints:
$= (-7)\cdot(-7)$ $= 49$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Solve for $r$. $4r-3=3(3r+4)$ $r =$
Hints:
We need to manipulate the equation to get $ r $ by itself. $\begin{aligned} 4r-3 &= 3(3r+4) \\\\ 4r-3 &= 9r+12~~~~~~~~~~\gray{\text{Distribute}}\\\\ 4r-3{-9r} &= 9r+12{-9r} ~~~~~~~~~~\gray{\text{Subtract 9r from each side}}\\\\ -5r-3&=12 ~~~~~~~~~~\gray{\text{Combine like ter... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
As a particle moves along the number line, its position at time $t$ is $s(t)$, its velocity is $v(t)$, and its acceleration is $a(t)= 3t^2$. If $v(0) = 3$ and $s(0) = 1$, what is $s(2)$ ? $s(2)=~$
Hints:
The antiderivative of $~a(t)~$ is $~v(t)=t^3+C\,$. We know that $~v(0) = 3\,$, so $~C=3\,$. Therefore, $~v(t) = t^3... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
What is the inverse of the function $h(x)=\dfrac{3-x}{x+1}$ ? $ h^{-1}(x) =$
Hints:
Let's start by replacing $h(x)$ with $y$. $y=\dfrac{3-x}{x+1}$ Now let's swap $x$ and $y$ and solve for $y$. $\dfrac{3-y}{y+1}=x$ [Why do we swap x and y?] $\begin{aligned} \dfrac{3-y}{y+1}&=x \\\\ 3-y&=x(y+1) \\\\ 3-y&=xy+x \\\\ 3-x&=... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Solve for $v$ : $-\dfrac{9}{8}=v-\dfrac{1}{2}$ $v =$
Hints:
To isolate $v$, we add $\dfrac{1}{2}$ to both sides. $\begin{aligned} -\dfrac{9}{8}&=v-\dfrac{1}{2} \\\\ -\dfrac{9}{8}{+\dfrac{1}{2}}&=v-\dfrac{1}{2}{+\dfrac{1}{2}} \\\\ -\dfrac{9}{8}{+\dfrac{1}{2}}&=v \end{aligned}$ Simplifying, we get: $v = -\dfrac{5}{8}$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Simplify to lowest terms. $\dfrac{84}{48}$
Hints:
There are several ways to tackle this problem. What is the greatest common factor (GCD) of 84 and 48? $84 = 2\cdot2\cdot3\cdot7$ $48 = 2\cdot2\cdot2\cdot2\cdot3$ $\mbox{GCD}(84, 48) = 2\cdot2\cdot3 = 12$ $\dfrac{84}{48} = \dfrac{7 \cdot 12}{ 4\cdot 12}$ $\hphantom{\dfr... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
A certain circle can be represented by the following equation. $x^2+2x+y^2=0$ What is the center of this circle ? $($
Hints:
The strategy We can find the center and radius of a circle by rewriting the given equation in the form of the standard equation of a circle. [What is the standard equation of the circle?] In ord... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Factor the following expression: $8x^2 - 2$
Hints:
We can start by factoring a ${2}$ out of each term: $ {2}({4x^2} - {1})$ The second term is of the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as ${2}({a} + {b}) ({a} - {b})$ What are the values of $a$ and $b$ $ a = \sqrt{4x^2} = 2x... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
You just got a free ticket for a boat ride, and you can bring along $3$ friends! Unfortunately, you have $6$ friends who want to come along. How many different groups of friends could you take with you?
Hints:
There are $3$ places for your friends on the boat, so let's fill those slots one by one. For the first slot, ... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
What is the missing constant term in the perfect square that starts with $x^2-4x$ ?
Hints:
Let $b$ be the missing constant term. Let's assume $x^2-4x+b$ is factored as the perfect square $(x+a)^2$. $\begin{aligned} (x+a)^2&=x^2+{2a}x+{a^2} \\\\ &=x^2{-4}x+ b \end{aligned}$ For the expressions to be the same, ${2a}$ mu... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Teams $ A$, $ B$, and $ C$ are playing a game of strength. Each team has attached a rope to a metal ring and is trying to pull the ring into their own area (team areas shown below). Team $ A$ pulls with force vector ${\vec{a}} = 4\hat{i} + 0\hat{j}$ Team $ B$ pulls with force vector ${\vec{b}} = -2\hat{i} + 4\hat{j}$ T... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$z=-19+3.14i$ What are the real and imaginary parts of $z$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\text{Re}(z)=3.14$ and $\text{Im}(z)=-19$ (Choice B) B $\text{Re}(z)=-19$ and $\text{Im}(z)=3.14i$ (Choice C) C $\text{Re}(z)=-19$ and $\text{Im}(z)=3.14$ (Choice D) D $\text{Re}(z)=3.14i$ and $\text{Im}(z)=-19$... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$\vec u = (-8,2)$ $\vec w = (6, 1)$ $2\vec u + 3\vec w= (~ $
Hints:
Strategy overview This question asks us to perform the following: Scalar multiplication Vector addition Solution steps $\begin{aligned} {2}\vec u + {3}\vec w &= {2}(-8,2) + {3}(6,1) \\\\\\\\ &= (-16, 4) + (18, 3) \\\\\\\\ &= (-16+ 18, 4+ 3) \\\\ &= (2... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
${28} \div {7} = {?}$
Hints:
If we split ${28}$ circles into $7$ equal rows, how many circles are in each row? ${7}$ ${\color{#29ABCA}{1}}$ ${\color{#29ABCA}{2}}$ ${\color{#29ABCA}{3}}$ ${\color{#29ABCA}{4}}$ ${6}$ ${\color{#29ABCA}{5}}$ ${\color{#29ABCA}{6}}$ ${\color{#29ABCA}{7}}$ ${\color{#29ABCA}{8}}$ ${5}$ ${\col... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$- \dfrac{7}{9} \times \dfrac{8}{9}$
Hints:
$= \dfrac{-7 \times 8}{9 \times 9}$ $= \dfrac{-56}{81}$ $= -\dfrac{56}{81}$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
${\dfrac{4}{5} \div 8 =} $
Hints:
Draw ${\dfrac{4}{5}}$. Divide ${\dfrac{4}{5}}$ into $8$ rows. Each of the ${8}$ rows is $\dfrac{4}{40}$ of the whole. We can also use the image to see that ${\dfrac{4}{5}} \div 8$ is the same as $\dfrac{1}{8} \text{ of }{ \dfrac{4}{5}}$. $\dfrac{1}{8} \text{ of } {\dfrac{4}{5}} = \dfr... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$83.6 \times 85.5 = $
Hints:
${8}$ ${3}$ ${6}$ ${8}$ ${5}$ ${5}$ ${\times\vphantom{0}}$ ${.}$ ${.}$ ${0.6}\times {0.5}= {0.30}$ ${0}$ ${3}\times {0.5}+C{0.3}= {1.8}$ ${8}$ ${80}\times {0.5}+C{1}= {41}$ ${1}$ ${4}$ ${0.6}\times {5}= {3.0}$ ${0}$ ${0}$ ${3}\times {5}+C{3}= {18}$ ${8}$ ${80}\times {5}+C{10}= {410}$ ${1}$... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
The following line passes through point $(-8, -6)$ : $y = \dfrac{3}{4} x + b$ What is the value of the $y$ -intercept $b$ ?
Hints:
Substituting $(-8, -6)$ into the equation gives: $-6 = \dfrac{3}{4} \cdot -8 + b$ $-6 = -6 + b$ $b = -6 + 6$ $b = 0$ Plugging in $0$ for $b$, we get $y = \dfrac{3}{4} x + 0$. ${1}$ ${2}$ $... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Marcus is picking songs to play during a slideshow. The songs are each $3\dfrac12$ minutes long. The slideshow is $31\dfrac12$ minutes long. How many songs will play in the slideshow?
Hints:
We can think about this problem like this: $ {\text{number of songs}} = {\text{total time}} \div {\text{length of each song}}$ $... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$f(x)=3x^4-16x^3+24x^2+48$. On which intervals is the graph of $f$ concave down? Choose 1 answer: Choose 1 answer: (Choice A) A $x<-2$ and $x>-\dfrac{2}{3}$ (Choice B) B $\dfrac{2}{3}<x<2$ only (Choice C) C $x<\dfrac{2}{3}$ and $x>2$ (Choice D) D $x>0$ only
Hints:
We can analyze the intervals where $f$ is concave up/d... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$b(n) = 1 \left(-2\right)^{n - 1}$ What is the $4^\text{th}$ term in the sequence?
Hints:
This is an explicit formula. All we have to do is plug $n=4$ in the formula to find the $4^\text{th}$ term. $\begin{aligned} b({4}) &=1(-2)^{{4} - 1} \\\\ &= -8 \end{aligned}$ The $4^\text{th}$ term is $-8$. | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$f(x, y) = \left( -\sin(x)\sin(y), \cos(x)\cos(y) \right)$ Find $F$ such that $f = \nabla F$. $F(x, y) =$ $ + \, C$
Hints:
We know that $\nabla F = f$. Therefore: $\begin{aligned} F_x &= -\sin(x)\sin(y) \\ \\ F_y &= \cos(x)\cos(y) \end{aligned}$ Let's integrate these two equations. Instead of getting a constant at the... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$\sqrt{\left(\dfrac{9}{8}\right)^2}$ =
Hints:
What number times itself equals $\left(\dfrac{9}{8}\right)^2$ ? Remember that $\left(\dfrac{9}{8}\right)^2$ is the same as $\dfrac{9}{8} \cdot \dfrac{9}{8}$. So, ${\dfrac{9}{8}}\cdot{\dfrac{9}{8}}$ = $\left(\dfrac{9}{8}\right)^2$. $\sqrt{\left(\dfrac{9}{8}\right)^2} = {\df... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Ashley ate 4 slices of cake. Stephanie ate 3 slices. If there were 1 slice remaining, what fraction of the cake was eaten?
Hints:
$\text{fraction of cake eaten} = \dfrac{\text{number of slices eaten}}{\text{number of slices total}}$ Since they ate 7 slices of cake with 1 slice remaining, they must have begun with 8 sl... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
There are $9$ students in a class: $5$ boys and $4$ girls. If the teacher picks a group of $3$ at random, what is the probability that everyone in the group is a boy?
Hints:
One way to solve this problem is to figure out how many different groups there are of only boys, then divide this by the total number of groups y... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Expand.
Hints:
We expand the parentheses using the distributive property : $ A(B+C)= A\cdot B+ A\cdot C$ We can also think about the problem using an area model: $r^2$ $-2$ $8r^2$ Here's how the solution goes, algebraically: $\begin{aligned} &\phantom{=}{8r^2}(r^2-2) \\\\ &={8r^2}(r^2)+{8r^2}(-2) \\\\ &=8r^4-16r^2 \en... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Combine the like terms to create an equivalent expression. ${4q+3+2q-1}$
Hints:
Combine the ${q}$ terms: 4 q + 3 + 2 q − 1 = = ( 4 + 2 ) q + 3 − 1 6 q + 3 − 1 { \begin{eqnarray} 4{q} +3+ 2{q} - 1 &=& (4 + 2){q} +3 - 1 \\ &=& 6{q} +3 -1 \end{eqnarray}} Combine the numeric terms: $ { 6{q} + {3} - {1} = 6{q} + {2}} $ The... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$\int x^{^{\frac13}}\,dx=$ $+C$
Hints:
The integrand is of the form $x^n$ where $n\neq-1$, so we can use the reverse power rule: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ $\begin{aligned} \int x^{^{{\frac13}}}\,dx&=\dfrac{x^{^{{\frac13}+1}}}{{\dfrac13}+1}+C \\\\ &=\dfrac34x^{^{\frac43}}+C \end{aligned}$ In conclusion, $\i... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Evaluate the double integral. $ \int_0^2 \int_1^{\sqrt{y}} x - 2y^2 \, dx \, dy =$ Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{8}{3} - \dfrac{16\sqrt{2}}{7}$ (Choice B) B $\dfrac{8}{3} - \dfrac{24\sqrt{2}}{7}$ (Choice C) C $\dfrac{16}{3} - \dfrac{24\sqrt{2}}{7}$ (Choice D) D $\dfrac{16}{3} - \dfrac{32\sqrt{2... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
The power generated by an electrical circuit (in watts) as a function of its current $x$ (in amperes) is modeled by: $P(x)=-12x^2+120x$ What is the maximum power generated by the circuit?
Hints:
The circuit's power is modeled by a quadratic function, whose graph is a parabola. The maximum power is reached at the verte... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Which integral gives the length of the graph of $f(t)=\dfrac{1-2t}{t^2}$ between $t=a$ and $t=b$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $ \int_a^b\sqrt{1+\bigg(\dfrac{-2-2t}{t^3}\bigg)^2}~dt$ (Choice B) B $ \int_a^b\sqrt{1+\dfrac{2-2t}{t^3}}~dt$ (Choice C) C $ \int_a^b\sqrt{1+\bigg(\dfrac{2t-2}{t^3}\bigg)^2}~... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$ C = \left[\begin{array}{rrr}1 & 1 & 2 \\ 0 & 0 & 1 \\ 1 & 2 & 1\end{array}\right]$ What is $ C^{-1}$ ?
Hints:
$ = \left[\begin{array}{rrr}2 & -3 & -1 \\ -1 & 1 & 1 \\ 0 & 1 & 0\end{array}\right]$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
A gardener is planting flowers. He plants $30$ flowers in each garden bed. He plants $9$ garden beds. How many flowers does the gardener plant in all?
Hints:
${30}$ flowers is the same as ${3\text{ groups of ten}}$ flowers. There are $9$ garden beds. Total number of flowers: $\begin{aligned}&9\times{3\text{ groups of ... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Subtract. $\dfrac{5}{2} - \dfrac{6}{8} = $
Hints:
Before we can subtract our fractions, they need to have the same denominator. $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{1}{8}$ $\frac{1}{8}$ $\frac{1}{8}$ $\frac{1}{8}$ $\frac{1}{8}$ $\frac{1}{8}$ $\frac{1}{8}$ $\frac{1}... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$-6 \times (-8) = ?$
Hints:
A negative times a negative is a positive. $-6 \times (-8) = 48$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Expand and combine like terms. $(7b^5-b^2)^2=$
Hints:
We can expand this expression using the "perfect square" pattern (where $P$ and $Q$ can be any monomial): $(P+Q)^2=P^2+2PQ+Q^2$ Since we have a minus sign, let's rewrite the binomial as a sum where the second term is negative, then use the pattern. $\begin{aligned}... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Multiply the following complex numbers: $({4-2i}) \cdot ({-5+5i})$
Hints:
Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({4-2i}) \cdot ({-5+5i}) = $ $ ({4} \cdot {-5}) + ({4} \cdot {5}i) + ({-2}i \cdot {-5}) + ({-2}i \cdot {5}i) $ Then simplify the terms: $ (-20) + (20i)... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
A set of middle school student heights are normally distributed with a mean of $150$ centimeters and a standard deviation of $20$ centimeters. Let $X$ represent the height of a randomly selected student from this set. Find $P(X<115)$. You may round your answer to two decimal places.
Hints:
Representing probability wit... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$ {1.36 \div 1.7 = ?} $
Hints:
${1}$ ${7}$ ${1}$ ${3}$ ${6}$ ${.}$ ${.}$ $\text{Bring the decimal up into the}$ $\text{answer (the quotient).}$ ${.}$ ${0}$ $\text{How many times does }17\text{ go into }{136}\text{?}$ ${8}$ ${1}$ ${3}$ ${6}$ $-$ ${0}$ ${136}\div17={8}\text{ with a remainder of }{0}$ $\text{The remainde... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Integrate. $ \int \sec(x)\tan(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $\sec (x) + C$ (Choice B) B $\cot (x) + C$ (Choice C) C $\tan (x) + C$ (Choice D) D $\csc (x) + C$
Hints:
We need a function whose derivative is $\sec(x)\tan(x)$. We know that the derivative of $\sec(x)$ is $\sec(x)\tan(x)$, so let's... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$4$ roly poly bugs $+$ $5$ ladybugs $=$
Hints:
$4+5=9$ There are $9$ bugs in all. | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
You have found the following ages (in years) of all 6 sloths at your local zoo: $ 14,\enspace 21,\enspace 11,\enspace 10,\enspace 2,\enspace 18$ What is the average age of the sloths at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Hints:
Because we have data for all 6 slot... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$\dfrac{d}{dx}\left(\dfrac{2x^2+x-3}{2x+7}\right)=$
Hints:
$\dfrac{2x^2+x-3}{2x+7}$ is a rational expression. To find the derivative of rational expressions, we use the quotient rule : $\begin{aligned} \dfrac{d}{dx}\left[\dfrac{u(x)}{v(x)}\right]&=\dfrac{\dfrac{d}{dx}[u(x)]v(x)-u(x)\dfrac{d}{dx}[v(x)]}{[v(x)]^2} \\\\ ... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
${10} \div {1} = {?}$
Hints:
We can think of ${10} \div {1}$ as putting ${10}$ circles into $1$ group: How many circles are in the group? ${10} \div {1} = {10}$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Given $ m \angle RPS = 9x + 71$, $ m \angle QPR = 2x + 41$, and $ m \angle QPS = 167$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Hints:
From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were gi... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Solve for $x$. Enter the solutions from least to greatest. $2x^2 - 16x + 14 = 0$ $\text{lesser }x = $
Hints:
$\begin{aligned} 2x^2 - 16x + 14&= 0 \\\\ 2(x^2-8x+7)&=0 \end{aligned}$ Now let's factor the expression in the parentheses. $x^2-8x+7$ can be factored as $(x-1)(x-7)$. $\begin{aligned} 2(x-1)(x-7)&=0 \\\\ x-1=0... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Solve for $x$ : $4x + 5 = 7$
Hints:
Subtract $5$ from both sides: $(4x + 5) - 5 = 7 - 5$ $4x = 2$ Divide both sides by $4$ $\dfrac{4x}{4} = \dfrac{2}{4}$ Simplify. $x = \dfrac{1}{2}$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$\vec w = (-3,5)$ $3\vec w= ($
Hints:
In general, the scalar multiple of $k$ times $\vec u$ is this: $k\vec u = k(u_x, u_y) = (ku_x, ku_y)$. So, here's how we find $3 \vec{w}$ : $\begin{aligned} {3}\vec w = {3} \cdot (-3,5) &= \left({3} \cdot (-3), {3} \cdot 5\right) \\\\ &= (-9,15) \end{aligned}$ The answer is $ (-9,... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Multiply and simplify the following complex numbers: $({-1+4i}) \cdot ({4-3i})$
Hints:
Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({-1+4i}) \cdot ({4-3i}) = $ $ ({-1} \cdot {4}) + ({-1} \cdot {-3i}) + ({4i} \cdot {4}) + ({4i} \cdot {-3i}) $ Then simplify the terms: $ ... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
${12} \div {6} = {?}$
Hints:
If we split ${12}$ circles into $6$ equal rows, how many circles are in each row? ${6}$ ${\color{#29ABCA}{1}}$ ${\color{#29ABCA}{2}}$ ${5}$ ${\color{#29ABCA}{3}}$ ${\color{#29ABCA}{4}}$ ${4}$ ${\color{#29ABCA}{5}}$ ${\color{#29ABCA}{6}}$ ${3}$ ${\color{#29ABCA}{7}}$ ${\color{#29ABCA}{8}}$ ... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Factor the quadratic expression completely. $2x^2-13x+20=$
Hints:
Since the terms in the expression do not share a common monomial factor and the coefficient on the leading $x^2$ term is not $1$, let's factor by grouping. The expression ${2}x^2{-13}x{+20}$ is in the form ${A}x^2+{B}x+{C}$. First, we need to find two i... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Evaluate the following expression. $3^3-10+7 =$
Hints:
$\begin{aligned} &\phantom{=}3^3-10+7 \\\\ &=(3\cdot 3 \cdot 3)-10+7 \end{aligned}$ $=27-10+7$ $=17+7$ $=24$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Let $V$ be a simple solid region oriented with outward normals that has a piecewise-smooth boundary surface $S$. $ \iiint_V y\left( x + e^x \right) \, dV$ Use the divergence theorem to rewrite the triple integral as a surface integral. Leave out constant coefficients and extraneous functions of $x$ and $y$. $ \oiint_S ... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
3 people can paint 4 walls in 42 minutes. How many minutes will it take for 6 people to paint 8 walls? Round to the nearest minute.
Hints:
We know the following about the number of walls $w$ painted by $p$ people in $t$ minutes at a constant rate $r$ $w = r \cdot t \cdot p$ $\begin{align*}w &= 4\text{ walls}\\ p &= 3\... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Express this number in scientific notation. $0.3276$
Hints:
Count the zeroes to the right of the decimal point before the leading $\leadingColor{3}$ : there are $0$ zeroes If you count the leading digit $\leadingColor{3}$ , there is $\exponentColor{1}$ digit to the right of the decimal point. So: $0.3276 = \leadingCol... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Express your answer as a mixed number simplified to lowest terms. $19\dfrac{4}{15}+1\dfrac{10}{15} = {?}$
Hints:
Separate the whole numbers from the fractional parts: $= {19} + {\dfrac{4}{15}} + {1} + {\dfrac{10}{15}}$ Bring the whole numbers together and the fractions together: $= {19} + {1} + {\dfrac{4}{15}} + {\dfr... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
A curve in the plane is defined parametrically by the equations $x=6\tan(t)$ and $y=2t+1$. Find $\dfrac{dy}{dx}$. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{\cos^2(t)}{3}$ (Choice B) B $\dfrac{1}{3\cos^2(x)}$ (Choice C) C $3\sec^2(x)$ (Choice D) D $\dfrac{\cot(t)}{3}$
Hints:
In general, to find the derivat... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
The graph of a sinusoidal function has a maximum point at $(0,10)$ and then intersects its midline at $\left(\dfrac{\pi}{4},4\right)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
Hints:
The strategy First, let's use the given information to determine the function's amplitude, midline, a... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Christopher decided to paint some of the rooms at his 12-room inn, Christopher's Place. He discovered he needed $\frac{4}{5}$ of a can of paint per room. If Christopher had 8 cans of paint, how many rooms could he paint?
Hints:
We can divide the cans of paint (8) by the paint needed per room ( $\frac{4}{5}$ of a can) ... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
$\dfrac{17}{10} + \dfrac{8}{100} = {?}$
Hints:
The fractions must have the same denominator before you can add them. Rewrite $\dfrac{17}{10}$ as $\dfrac{170}{100}$ $\dfrac{170}{100} + \dfrac{8}{100} = {?}$ $ = \dfrac{178}{100}$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Solve for $x$ : $8x = 3$
Hints:
Divide both sides by $8$ $ {\dfrac{\color{black}{8x}}{8}} = {\dfrac{\color{black}{3}}{8}} $ Simplify: $\dfrac{\cancel{8}x}{\cancel{8}} = \dfrac{3}{8}$ $x = \dfrac{3}{8}$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Simplify the following expression: $y = \dfrac{8k}{8} \times \dfrac{7}{7k}$
Hints:
When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 8k \times 7 } { 8 \times 7k}$ $y = \dfrac{56k}{56k}$ Simplify: $y = \dfrac{1}{1}$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Let $f(x) = -3x^{2}+8x+9$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
Hints:
The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-3x^{2}+8x+9 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 ... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
${56} \div {8} = {?}$
Hints:
If we split ${56}$ circles into $8$ equal rows, how many circles are in each row? ${8}$ ${\color{#29ABCA}{1}}$ ${\color{#29ABCA}{2}}$ ${\color{#29ABCA}{3}}$ ${\color{#29ABCA}{4}}$ ${\color{#29ABCA}{5}}$ ${\color{#29ABCA}{6}}$ ${\color{#29ABCA}{7}}$ ${7}$ ${\color{#29ABCA}{8}}$ ${\color{#29... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Express your answer in scientific notation. $4.9 \cdot 10^{-5} + 0.0005 = $
Hints:
$\phantom{=}4.9 \cdot 10^{-5} + {0.0005}$ $=4.9 \cdot 10^{-5} +{5\cdot10^{-4}} $ $=4.9 \cdot 10^{-5} + {50\cdot10^{-5}} $ $= (4.9+50)\cdot10^{-5}$ $=54.9\cdot10^{-5}$ $=5.49\cdot10^{-4}$ | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Let $f$ be a transformation from $\mathbb{R}^3$ to $\mathbb{R}^3$. Its Jacobian matrix is given below. $J(f) = \begin{bmatrix} 2xy & x^2 & 0 \\ \\ 0 & -1 & 1 \\ \\ 1 & 0 & 1 \end{bmatrix}$ Find the Jacobian determinant of $f$. $|J(f)| = $ How will $f$ expand or contract space around the point $\left( 1, \dfrac{1}{4}, 2... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Rewrite the equation by completing the square. $x^{2}+10x+16 = 0$ $(x + $
Hints:
Begin by moving the constant term to the right side of the equation. $x^2 + 10x = -16$ We complete the square by taking half of the coefficient of our $x$ term, squaring it, and adding it to both sides of the equation. Since the coefficie... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
Find the unit vector in the direction of $\vec{v}=\left( 12, -5 \right)$. $($ $~,$ $)$
Hints:
Getting started A unit vector has a magnitude (or length) of $1$. Dividing $\vec v$ by its magnitude will find a vector in the same direction as $\vec v$ but with a magnitude of $1$ : $\text{Unit vector in the direction of } ... | @article{hendrycks2021measuring,
title={Measuring Mathematical Problem Solving With the MATH Dataset},
author={Hendrycks, Dan and Burns, Collin and Kadavath, Saurav and Arora, Akul and Basart, Steven and Tang, Eric and Song, Dawn and Steinhardt, Jacob},
journal={arXiv preprint arXiv:2103.03874},
year={2021}
} |
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