# No Calculator Section ## Problem 1 If $\frac{2 x-3}{2}=k-1$ and $k=5$, what is the value of $2 x ?$ A) 4 B) $5.5$ C) 8 D) 11 > [!sol]- Click for Solution! > $\frac{2 x-3}{2}=5-1 \rightarrow 2 x-3=8 \rightarrow 2 x=\boxed{11\quad \mathbf{(D)}}$ ## Problem 2 $(5+3 i)-(8-2 i)=a+b i$ In the equation above, $a$ and $b$ are real numbers. If $i=\sqrt{-1}$, what is the value of $b$ ? A) $-1$ B) 1 C) $-5$ D) 5 > [!sol]- Click for Solution! > $5+3 i-8+2 i=a+b i \rightarrow-3+5 i=a+b i$. Therefore, $a=-3$ and $\boxed{b=5\quad \mathbf{(D)}}$. ## Problem 3 If Claire paid $k$ dollars for a computer that was only 20 dollars more than half the original price, what was the original price, in dollars? A) $k+20$ B) $k-40$ C) $2 k-20$ D) $2 k-40$ > [!sol]- Click for Solution! > Original price: $x, k=\frac{1}{2} x+20 \rightarrow k-20=\frac{1}{2} x \rightarrow x=2(k-20) \rightarrow \boxed{x=2 k-40\quad\mathbf{(D)}}$ ## Problem 4 Jenny is on the school swim team and has swim practice $m$ hours in the morning and $p$ hours in the evening each day. The schedule is the same each day. If she swims a total of $k$ hours for five days, which of the following is the expression for $m$ ? A) $\frac{k-p}{5}$ B) $\frac{k-5 p}{5}$ C) $k-5 p$ D) $5(k-p)$ > [!sol]- Click for Solution! > $m+p=\frac{k}{5} \rightarrow m=\frac{k}{5}-p \rightarrow m=\boxed{\frac{k-5 p}{5}\quad\mathbf{(B)}}$ ## Problem 5 A certain business is marketing its product and has determined that, when it raised the selling price of its product, its sales went down. The number of units sold, $P$, is modeled by the equation $P=1200-20$ s, where $s$ is the selling price, in dollars. Based on this model, what is the decrease in selling price from 700 units sold to 900 units sold? A) 5 B) 10 C) 15 D) 20 > [!sol]- Click for Solution! > When $P=700,700=1200-20 s \rightarrow 20 s=500 \rightarrow s=25$. > When $P=900,900=1200-20 s \rightarrow 20 s=300 \rightarrow s=15$. $\quad 25-15=10$. > The decrease in selling price is $\boxed{\$10\quad\mathbf{(B)}}$. ## Problem 6 $\left(x^2+y^2\right)^2-\left(x^2-y^2\right)^2$ Which of the following is equivalent to the expression above? A) $x^4-y^4$ B) $2\left(x^2+y^2\right)$ C) $2 x^2 y^2$ D) $4 x^2 y^2$ > [!sol]- Click for Solution! > $\left(x^2+y^2\right)^2-\left(x^2-y^2\right)^2=x^4+2 x^2 y^2+y^4-\left(x^4-2 x^2 y^2+y^4\right)=\boxed{4 x^2 y^2\quad\mathbf{(D)}}$ ## Problem 7 Kimberly earns $k$ dollars per week. At this rate how many weeks will it take her to earn $p$ dollars? A) $\frac{p}{k}$ B) $\frac{k}{p}$ C) $k p$ D) $\frac{10 p}{k}$ > [!sol]- Click for Solution! > Proportion. $\frac{\$}{\text { week }}=\frac{k}{1}=\frac{p}{x} \rightarrow x=\boxed{\frac{p}{k}\text{ weeks}\quad\mathbf{(A})}$ ## Problem 8 If $\frac{2 a}{b}=5$, what is the value of $\frac{5 b}{a} ?$ A) 2 B) 4 C) 10 D) $12.5$ > [!sol]- Click for Solution! > $\frac{2 a}{b}=5 \rightarrow \frac{a}{b}=\frac{5}{2} \rightarrow \frac{b}{a}=\frac{2}{5}$. Therefore, $\frac{5 b}{a}=5\left(\frac{b}{a}\right)=5\left(\frac{2}{5}\right)=\boxed{2\quad\mathbf{(A)}}$ ## Problem 9 $\begin{aligned}& 2 x+b y=10 \\& a x+4 y=15\end{aligned}$ In the system of equations above, $a$ and $b$ are constants and $a=2 b$. If the system has no solution, which of the following could be a possible value of $a?$ A) $-2$ B) $\frac{1}{2}$ C) 4 D) 8 > [!sol]- Click for Solution! > Make the slopes the same, but keep the lines different: $\frac{2}{a}=\frac{b}{4} \neq \frac{10}{15}$. Therefore, $\frac{2}{a}=\frac{b}{4} \rightarrow a b=8$. Because $a=2 b,(2 b) b=8\rightarrow2 b^2=8 \rightarrow b^2=4 \rightarrow b=\pm 2$. Now $a=2 b=\pm 4$. Possible value of $a$ is 4 or $-4$. > We know that $\frac{2}{4} \neq \frac{10}{15}$ at $a=\boxed{4\quad\mathbf{(D)}}$. ## Problem 10 $f(x)=a x^2-15$ For the function $f$ defined above, $a$ is a constant and $f(3)=10$. Which of the following is equal to the value of $f(5)?$ A) $f(0)$ B) $f(3)$ C) $f(-3)$ D) $f(-5)$ > [!sol]- Click for Solution! > Since axis of symmetry is $x=0$, the graph is symmetry in $y$-axis. Therefore $f(5)=\boxed{f(-5)\quad\mathbf{(D)}}$. ## Problem 11 A certain job can be done in 20 hours by 4 people. How many people are needed to do the same job in 10 hours? A) 2 B) 4 C) 8 D) 10 > [!sol]- Click for Solution! > Inverse variation: ( \# of people ) $\times$ ( \# of hours ) $=$ Constant. Therefore, $20 \times 4=10 \times p \rightarrow p=\boxed{8\text{ people}\quad\mathbf{(C)}}$ ## Problem 12 Which of the following is equivalent to $f(x)=x^2-6 x+7$ ? A) $f(x)=(x+3)^2+5$ B) $f(x)=(x-3)^2+2$ C) $f(x)=(x-3)^2-2$ D) $f(x)=(x-7)(x+1)$ > [!sol]- Click for Solution! > Choice $\mathrm{C}$ is the vertex form of the equation. In order to have vertex form, $f(x)=\left(x^2-6 x+9\right)+(7-9) \rightarrow \boxed{f(x)=(x-3)^2-2\quad\mathbf{(C)}}$. ## Problem 13 If $24 x^2-k x+16=(3 x+4)(a x-b)$ for all values of $x$, where $a, b$, and $k$ are constants, what is the value of $k$ ? A) $-44$ B) $-12$ C) 12 D) 44 > [!sol]- Click for Solution! Identical equation has infinitely many solutions. Because $(3 x+4)(a x-b)=3 a x^2+(4 a-3 b) x-4 b$, $24 x^2-k x+16=3 a x^2+(4 a-3 b) x-4 b$. Therefore, $3 a=24 \rightarrow a=8$ and $-4 b=16 \rightarrow b=-4$. Now $-k=4 a-3 b \rightarrow-k=4(8)-3(-4)=48 \rightarrow k=\boxed{-44\quad\mathbf{(A)}}$. ## Problem 14 In the $x y$-plane, the equation of line $\ell$ is $x+3 y=5$. If line $m$ is perpendicular to line $\ell$, what is a possible equation of line $m$ ? A) $y=-\frac{1}{3} x+2$ B) $y=\frac{1}{3} x-1$ C) $y=-3 x+1$ D) $y=3 x+\frac{2}{3}$ > [!sol]- Click for Solution! > Because $x+3 y=5 \rightarrow y=-\frac{1}{3} x+\frac{5}{3}$, slope is $-\frac{1}{3}$. Slope of the perpendicular line must have slope of 3 , which is negative reciprocal of the other. The line $\boxed{y=3 x+\frac{2}{3}\quad\mathbf{(D)}}$ ## Problem 15 If $a+b=8$ and $\frac{27^a}{3^b}=81$, what is the value of $a$ ? A) 3 B) 4 C) 5 D) 6 > [!sol]- Click for Solution! > Since $\frac{27^a}{3^b}=\frac{3^{3 a}}{3^b}=3^{3 a-b}$ and $81=3^4, 3^{3 a-b}=3^4 \rightarrow 3 a-b=4$. > When you solve system of equation by addition, > $ > \begin{aligned} > a+b &=8 \\ > 3 a-b&=4 \\\hline > 4 a &=12 \quad\rightarrow\quad \boxed{a=3 \quad\mathbf{(A)}} > \end{aligned} > $ ## Problem 16 In a right triangle, one of the angles is $x^{\circ}$. If $\tan x^{\circ}=\frac{5}{12}$, what is the value of $\sin x^{\circ}$? ![[gridin.svg]] > [!sol]- Click for Solution! > $\tan x^{\circ}=\frac{5}{12}\implies \sin x^\circ =5k, \cos x^\circ=12k\implies(5k)^2+(12k)^2=1$ > $25k^2+144k^2=169k^2=1\implies k=\frac1{13}\implies\sin x^\circ=\boxed{\frac{5}{13}\approx.384}$ ## Problem 17 Dawson needs to measure the height of a building near his house. ![[chung-2018-t1-nc-p17.png]]He chooses a point $P$ on the ground where he can visually align the roof of his car with the edge of the building roof. The height of the car is 4 feet and the distance from point $P$ to point $O$ is 10 feet, as shown in the figure above. If the distance from point $O$ to point $R$ is 80 feet, and the height of the building is 80 feet, what is the value of k? ![[gridin.svg]] > [!sol]- Click for Solution! > Proportion: $\frac{4}{10}=\frac{k}{10+80} \rightarrow 10 k=360 \rightarrow k=\boxed{36}$. ## Problem 18 If $a(x+1)+b(x-1)=7 x$ for all real number $x$, where $a$ and $b$ are constants, what is the value of $a?$![[gridin.svg]] > [!sol]- Click for Solution! > $a(x+1)+b(x-1)=(a+b) x+a-6 . \text { Then }(a+b) x+a-6=7 x$ Since the equation is true for all real $x$, both side expressions must be same.Therefore, $a+b=7$ and $a-b=0$. (system of equations)$\begin{aligned}& a+b=7 \\& a-b=0 \\& \hline 2 a=7\\~ \end{aligned} \rightarrow a=\boxed{\frac{7}{2}, \text { or } 3.5}$ ## Problem 19 According to the formula $p=\frac{4}{3} k+81$, if the value of $p$ is increased by 16 , by how much does the value of $k$ increase?![[gridin.svg]] > [!sol]- Click for Solution! > From the equation $p=\frac{4}{3} k+81, \frac{4}{3}$ is slope. > By definition of slope: $\frac{\Delta p}{\Delta k}=\frac{4}{3} \rightarrow \frac{16}{\Delta k}=\frac{4}{3} \rightarrow 4 \Delta k=48 \rightarrow \Delta k=12$ > *OR* > When $k=0, p=81$. After increased by $16, p=97$.When $p=97, \rightarrow 97=\frac{4}{3} k+81$, or $\frac{4}{3} k=16 \rightarrow k=\frac{3}{4}(16)=12$. ## Problem 20 $ \begin{aligned} & x^2+y^2=56 \\ & y=\sqrt{x} \end{aligned} $ According to the system of equations above, what is the value of $x$?![[gridin.svg]] > [!sol]- Click for Solution! > Substitution: > $\left.\begin{array}{l}x^2+y^2=56 \\y=\sqrt{x}\end{array}\right\} \rightarrow x^2+x=56 \rightarrow x^2+x-56=0 \rightarrow(x+8)(x-7)=0$ > Therefore, $x=-8$ or $x=7$. But $\sqrt{-8}$ is undefined. Only $x=7$ is the solution. > You can take a look at the graphs.![[diagram-20230211.svg]] # Calculator Section ## Problem 1 $ \begin{array}{|l|l|} \hline x & f(x) \\ \hline 1 & 6 \\ \hline 2 & 10 \\ \hline 3 & 14 \\ \hline 4 & 18 \\ \hline 5 & 22 \\ \hline \end{array} $ The selected values of a function shown in the table above represent a linear function. Which of the following equals $f(10)?$ A) 36 B) 40 C) 42 D) 44 > [!sol]- Click for Solution! > We define the linear equation $y=m x+b$, with slope $m=4$. Substitute any point in the table. > If you choose $(1,6), y=4 x+b \rightarrow 6=4(1)+b \rightarrow b=2$. Therefore the equation is $y=4 x+2$. > When $x=10, f(10)=4(10)+2=\boxed{42\quad\mathbf{(C)}}$. ## Problem 2 If $3(a+2 b-c)=12$, what is the value of $a+2 b$ in terms of $c$ ? A) $3 c-4$ B) $c-12$ C) $c+4$ D) $c-4$ > [!sol]- Click for Solution! > $3(a+2 b-c)=12 \rightarrow a+2 b-c=4 \rightarrow a+2 b=\boxed{c+4\quad\mathbf{(C)}}$ ## Problem 3 ![[chung-2018-t1-yc-p3.svg]]In the figure above, lines $\ell$ and $m$ are parallel. If the measure of $\angle 1$ is twice the measure of $\angle 2$, what is the measure of $\angle 1$ ? A) $100^{\circ}$ B) $120^{\circ}$ C) $135^{\circ}$ D) $145^{\circ}$ > [!sol]- Click for Solution! > Define: $\angle 2=x$ and $\angle 1=2 x . \angle 1+\angle 2=180 \rightarrow x+2 x=180 \rightarrow x=60$ Therefore $\angle 1=2 x=2(60)=\boxed{120\quad\mathbf{(B)}}$. ## Problem 4 If $8^n \times 4^2=2^{10}$, what is the value of $n$ ? A) 2 B) 3 C) 4 D) 5 > [!sol]- Click for Solution! > $8^n \times 4^2=2^{10} \rightarrow 2^{3 n} \times 2^4=2^{10} \rightarrow 2^{3 n+4}=2^{10}$. From the equation $3 n+4=10 \rightarrow n=\boxed{2\quad\mathbf{(A)}}$ ## Problem 5 For what value of $n$ is $|n+4|+1$ less than 0 ? A) $-5$ B) $-4$ C) 3 D) There is no such value of $n$. > [!sol]- Click for Solution! > Since $|n+4| \geq 0,|n+4|+1 \geq 1$. It cannot be less than 1. $\boxed{\text{There is no such value of }n\quad\mathbf{(D)}}$ ## Problem 6 ![[chung-2018-t1-yc-p6.svg]] The equation of the graph of line $\ell$ in the $x y$-plane above is $y=m x+6$, where $m$ is a constant. If the line passes through a point $(3,2)$, what is the value of $m$ ? A) $-\frac{4}{3}$ B) $-\frac{2}{3}$ C) $-\frac{1}{2}$ D) $-\frac{1}{4}$ > [!sol]- Click for Solution! > From the equation $y=m x+6$, you can see $y$-intercept is $(0,6)$. Therefore, slope $m=\frac{6-2}{0-3}=\boxed{-\frac{4}{3}\quad\mathbf{(A)}}$. ## Problem 7 In Ms. Lee's class, the number of boys is more than twice the number of girls. There are at least 7 girls and there are no more than 15 boys. How many students are in the class? A) 19 B) 20 C) 21 D) 22 > [!sol]- Click for Solution! > \# of boys $=2 \times(\#$ of girls $) \rightarrow b=2 g$. But $g \geq 7$ and $b \leq 15$. > You can use a table as follows. > $ > \begin{array}{rrrrrr} > \text { max \#of girls }&=&7 & 8 & 9 & 10 &\cdots \\ > \text { \# of boys }&=&15 & 16 & 18 & 20 &\cdots > \end{array} > $ > From the table, \# of boys cannot be more than 15. Therefore, \# of students is $7+15=\boxed{22\quad\mathbf{(D)}}$. ## Problem 8 ![[Screenshot 2023-02-10 at 10.24.41 PM.png]]The graph above shows the test scores of 20 students. Based on the histogram above, what is the average (arithmetic mean) score on the test? A) 70 B) 73 C) 75 D) 78 > [!sol]- Click for Solution! Average $=\frac{\text { Total score }}{20}=\frac{30(2)+70(4)+100(6)+80(5)+40(3)}{20}=\frac{1460}{20}=\boxed{73\quad\mathbf{(B)}}$ > [!info] Questions 9 and 10 refer to the following information. $h(t)=-16 t^2+128 t+320$A science class determined that the motion of a ball launched from the top of a 10 -story building could be described by the function above, where $t$ represents the time the ball is in the air in seconds and $h$, the height in feet of the ball above the ground. ## Problem 9 What is the number of seconds it takes for the ball to reach its peak? A) 2 B) 4 C) 8 D) 10 > [!sol]- Click for Solution! >Since the peak is on axis of symmetry, $t=\frac{-b}{2 a}=\frac{-128}{2(-16)}=\boxed{4\quad\mathbf{(B)}}$ ## Problem 10 At what time will the ball hit the ground? A) 5 B) 8 C) 10 D) 12 > [!sol]- Click for Solution! > When the ball hit the ground, the height is $0 .-16 t^2+128 t+320=0 \rightarrow t^2-8 t-20=0$ $(t-10)(t+2)=0 \rightarrow t=-2$ or 10 . Therefore, $t=\boxed{10\quad\mathbf{(C)}}$ ## Problem 11 The perimeter of a rectangle is $54 \mathrm{~cm}$. If the length is $2 \mathrm{~cm}$ more than its width, what is the area of the rectangle? A) $181.25 \mathrm{~cm}^2$ B) $728 \mathrm{~cm}^2$ C) $800 \mathrm{~cm}^2$ D) $820 \mathrm{~cm}^2$ > [!sol]- Click for Solution! > If width $=x$, then length $=x+2 . x+(x+2)=27 \rightarrow 2 x+2=27 \rightarrow x=12.5$ and $x+2=14.5$. Therefore, the area of the rectangle is $12.5 \times 14.5=\boxed{181.25}$. ## Problem 12 $ \begin{aligned} & 3 x-y>0 \\ & 2 x+y>1 \end{aligned} $ Which of the following is NOT a solution of the system of inequalities above? A) $(3,0)$ B) $(2,5)$ C) $(0,-3)$ D) $(5,-8)$ > [!sol]- Click for Solution! > You can check by substituting the coordinates into the inequalities. For $\boxed{(0,-3)\quad\mathbf{(C)}}$, they are not true. ## Problem 13 $\begin{array}{|c|c|} \hline \boldsymbol x & \boldsymbol y \\ \hline 0 & 2 \\ \hline k & 14 \\ \hline k+2 & 17 \\ \hline \end{array}$ The table above shows the point $(x, y)$ represented on a straight line. If the point $(16, m)$ lies on the same line, what is the value of $m$ ? A) 26 B) 24 C) 22 D) 20 > [!sol]- Click for Solution! Slopes between any two points are constant. First you need to find slope or the value of $k$.$\text { slope }=\frac{17-14}{(k+2)-k}=\frac{3}{2} . \text { Therefore, } \frac{m-2}{16-0}=\frac{3}{2} \rightarrow 2 m-4=48 \rightarrow 2 m=52 \rightarrow m=26$The linear equation is $y=\frac{3}{2} x+2$. By substituting $(16, m), m=\frac{3}{2}(16)+2=\boxed{26\quad\mathbf{(A)}}$. ## Problem 14 James spent $\frac{3}{4}$ of his allowance on a music CD. He spent $\frac{2}{3}$ of what was left on a hamburger. If this left him $P$ dollars, which of the following was his allowance in dollars? A) $12 P$ B) $14 P$ C) $16 P$ D) $18 P$ > [!sol]- Click for Solution! If original price $=k$, then $\frac{1}{3}\left(\frac{1}{4} k\right)=P \rightarrow \frac{1}{12} k=P \rightarrow k=\boxed{12 P\quad\mathbf{(A)}}$. > [!info] Questions 15 and 16 refer to the following in formation. > Radioactive decay is an exponential function where the amount , $y$, of radioactive material is reduced by onehalf over a certain period of time $t$. Material $M$ has a half- life of 50 years. ## Problem 15 If there are 800 grams of radioactive material $\mathrm{M}$, which of the following best represents the decay equation? A) $y=800-400 t$ B) $y=800\left(\frac{1}{2}\right)^t$ C) $y=800\left(\frac{1}{2}\right)^{\frac{t}{50}}$ D) $y=800(1-0.5 t)$ > [!sol]- Click for Solution! > The equation of radioactive decay is $p=p_0\left(\frac{1}{2}\right)^{t / n},$ where $n$ is a half-life period. $\boxed{y=800\left(\frac{1}{2}\right)^{\frac{t}{50}}\quad\mathbf{(C)}}$ ## Problem 16 If there are 800 grams of radioactive material $M$, then how much of this material would remain radioactive after 200 years? A) 25 grams B) 50 grams C) 100 grams D) 200 grams > [!sol]- Click for Solution! > $y=800\left(\frac{1}{2}\right)^{200 / 50}=800\left(\frac{1}{2}\right)^4=800\left(\frac{1}{16}\right)=\boxed{50\quad\mathbf{(B)}}$ ## Problem 17 ![[Chung-2018-t1-yc-p17.svg]]Note: Figure not drawn to scale. In the $x y$-plane above, the area of $\triangle O P Q$ is 3 . What is the value of $k$ ? A) 2 B) 4 C) 6 D) 8 > [!sol]- Click for Solution! > Area: $\frac{1}{2}(k+1) k=3 \rightarrow k^2+k=6 \rightarrow k^2+k-6=0 \rightarrow(k+3)(k-2)=0$ $k=-3$ or 2. But $k>0 \rightarrow k=\boxed{2\quad\mathbf{(A)}}$. ## Problem 18 A circle in the $x y$-plane with center $(4,0)$ passes through point $(7,4)$. Which of the following is the equation of the circle? A) $(x-4)^2+y^2=9$ B) $(x-4)^2+y^2=25$ C) $(x-4)^2+y^2=5$ D) $(x+4)^2+y^2=5$ > [!sol]- Click for Solution! > $\text { Since } r^2=(7-4)^2+(4-0)^2=25 \text {, the equation is }\boxed{(x-4)^2+y^2=25\quad\mathbf{(B)}}$ ## Problem 19 ![[Chung-2018-t1-yc-p19.svg]]The graph of the function $f$ is shown in the $x y$-plane above. Which of the following is the average rate of change between $x=-3$ and $x=6?$ A) $\frac{2}{9}$ B) $\frac{8}{9}$ C) 2 D) It cannot be determined from the given information. > [!sol]- Click for Solution! > When $x=-3, y=3$. When $x=6, y=5$. Average rate of change (slope between two points) is $\frac{5-3}{6-(-3)}=\boxed{\frac{2}{9}\quad\mathbf{(A)}}$ ## Problem 20 Emily traveled 60 miles on the highway and 16 miles on the local roads to reach her destination. On the highway, she traveled 30 miles faster than on the local roads. If her speed on local roads is 20 miles per hour, then what was her average speed, in miles per hour, during her entire trip? A) 24 B) 25 C) 35 D) 38 > [!sol]- Click for Solution! > Average speed $=\frac{\text { total distance }}{\text { total time }}, t_1=\frac{60}{50}=1.2$ on the highway, and $t_2=\frac{16}{20}=0.8$ on local roads. > Therefore, Average speed $=\frac{60+16}{1.2+0.8}=\frac{76}{2}=\boxed{38 \mathrm{mph}\quad\mathbf{(D)}}$ ## Problem 21 For O.K theater tickets, a ticket for an adult is 5 dollars more than a ticket for a child. If a group of 6 adults and 10 children pay a total of 142 dollars, what is the cost, in dollars, of a ticket for one adult and one child? A) 19 B) 18 C) 17 D) 16 > [!sol]- Click for Solution! > Child ticket $=\$ k$ and adult ticket $=\$(k+5)$. $6(k+5)+10 k=142 \rightarrow 16 k=112 \rightarrow k=7$ and $k+5=12$ > For one adult and one child, $7+12=\boxed{\$ 19\quad\mathbf{(A)}}$. ## Problem 22 ![[Screenshot 2023-02-11 at 7.34.26 AM.png]]The scatterplot above shows the distance traveled in hours for 10 taxi drivers and the line of best fit for the data. Which of the following is closest to the average speed, in miles per hour, for the drivers? A) 54 B) 59 C) 65 D) 68 > [!sol]- Click for Solution! > ## Problem 23 For a polynomial $p(x)$, the value of $p(-5)=0$. Which of the following must be true about $p(x)$ ? A) $(x-5)$ is a factor of $p(x)$. B) $(x+5)$ is a factor of $p(x)$. C) $x$ is a factor of $p(x)$. D) When $p(x)$ is divided by $(x+5)$, the remainder is $-5$. > [!sol]- Click for Solution! > $p(-5)=0$ means "$p(x)$ has a factor of $(x+5)$." see [[factor theorem]] ## Problem 24 $ \begin{aligned} & y \leq 3 x+\frac{1}{2} \\ & y \geq \frac{1}{2} x+3 \end{aligned} $ If the system of inequalities above is graphed in the $x y$-plane, which quadrant contains solutions to the system? A) Quadrant I B) Quadrant II C) Quadrant III D) Quadrant IV > [!sol]- Click for Solution! > Best solved with a quick sketch. > ![[Chung-2018-t1-yc-s24.svg]] ## Problem 25 ![[Chung-2018-t1-yc-p25.svg]]The figure above shows the graph of the piece-wise function $f$ defined for $-4 \leq x \leq 5.$ For which of the following values of $x$ is $f(x)<|f(x)|$ ? A) $-3$ B) $-1.3$ C) $2.5$ D) $3.7$ > [!sol]- Click for Solution! > For the interval $x\in\left(-2.8,1\right]\ni \boxed{-1.3\quad\mathbf{(B)}}$, $f(x)<|f(x)|$ ## Problem 26 If $a$ and $b$ are positive integers and $a^2-b^2=24$, which of the following could be the smallest value of $a$ ? A) 4 B) 5 C) 7 D) 8 > [!sol]- Click for Solution! > $a^2-b^2=24 \rightarrow(a+b)(a-b)=24$. Since $a$ and $b$ are positive integers, only two arrangements are possible. > $\begin{cases}a+b&=12\\a-b&=2\\\hline2a&=14\quad\implies a=7\end{cases} \text{\quad or\quad} \begin{cases}a+b&=6\\a-b&=4\\\hline2a&=10\quad\implies a=\boxed{5\quad\mathbf{(B)}}\end{cases}$ ## Problem 27 ![[Chung-2018-t1-yc-p7.svg]]The graph of $y=\frac{1}{3} x(x-6)$ is shown in the $x y$-plane above. Which of the following are the coordinates of vertex $P$ ? A) $(3,-2)$ B) $(2,-4)$ C) $(3,-3)$ D) $(3,-4.5)$ > [!sol]- Click for Solution! > Since axis of symmetry is $x=3$, the $y$-coordinate is $y=\frac{1}{3}(3)(3-6)=-3$. Therefore, the coordinates of vertex $P$ is $\boxed{(3,-3)\quad\mathbf{(C)}}$. ## Problem 28 ![[Chung-2018-t1-yc-p28.svg]] In right triangle $A B C$ above, if $B C=10$ and $\angle C=30^{\circ}$, what is the approximate perimeter of the triangle? A) 20 B) $23.7$ C) $25.8$ D) $27.2$ > [!sol]- Click for Solution! > [[Special Right Triangles]] > $30^{\circ}-60^{\circ}-90^{\circ}: A B=5, A C=5 \sqrt{3} \text {, and } B C=10$ > Perimeter $=15+5 \sqrt{3} \simeq 23.7 \text {. }$ ## Problem 29 ![[Chung-2018-t1-yc-p29.svg]] The figure above shows a rectangular solid with width $x$, length $y$, and height $z$. If $x y=20, y z=10$, and $x z=18$, what is the volume of the solid? A) 60 B) 70 C) 80 D) 90 > [!sol]- Click for Solution! > $\left.\begin{array}{l}x y=20 \\ y z=10 \\ x z=18\end{array}\right\}$ multiply together and get $(x y z)^2=3600$ > Squareroot to get $V = x y z=\boxed{60\quad\mathbf{(A)}}$. ## Problem 30 ![[Chung-2018-t1-yc-p30.svg]] The figure above shows a sector $O$ with radius $r$. If the area of the sector is $3 \pi$, what is the approximate perimeter of the sector? A) 10 B) $12.5$ C) $13.4$ D) $15.6$ > [!sol]- Click for Solution! > Since the [[area of a circle]] is $4 \pi$, the [[radius]] is 2. > [[Length of an arc]] is part of [[circumference]] $\frac{3}{4} \times C=\frac{3}{4} \times 4 \pi =3 \pi$. > Therefore, [[perimeter]] is $P=3 \pi+2+2 \simeq \boxed{13.4\quad\mathbf{(C)}}$. ## Problem 31 Kara needs three hours to mow and trim Mrs. Tayler's lawn. One day she asked her friend Peter to work with her. When Peter worked with her, the job took only one hour. How long would it take Peter, in hours, to complete the job himself?![[gridin.svg]] > [!sol] Click for Solution! > If Peter takes $x$ hours to complete the job, $\frac{1}{3}+\frac{1}{x}=\frac{1}{1} \rightarrow \frac{1}{x}=\frac{2}{3} \rightarrow x=\boxed{\frac{3}{2}=1.5}$. > (I guess we assume Peter has his own mower, otherwise, Peter's mowing speed has not been properly assessed.) ## Problem 32 Twenty members of a math club are planning a trip to an amusing park that has an admission price of $\$ 10$ per person. The club members going on the trip must share the $\$ 500$ cost of a bus and the admission price for 2 supervisors who will accompany them on the trip. What is the cost, in dollars, for each member?![[gridin.svg]] > [!sol]- Click for Solution! > $\$ 10+\frac{500+20}{20}=\boxed{\$ 36}$ ## Problem 33 ![[Chung-2018-t1-yc-p33.svg]] The graph of a linear function $f$ is shown in the $x y$-plane above. If $f(k)=1$, what is the value of $f(-2 k) ?$![[gridin.svg]] > [!sol]- Click for Solution! > Since $f(1)=1, k=1 . \quad f(-2 k)=f(-2)=\boxed7$. ## Problem 34 If the average of $2 a$ and $b$ is equal to 50 percent of $4 b$, what is the value of $\frac{a}{b}?$![[gridin.svg]] > [!sol]- Click for Solution! > $\frac{2 a+b}{2}=0.5(4 b) \rightarrow 2 a+b=4 b \rightarrow 2 a=3 b$. If $a=3$, then $b=2$. Therefore, $\frac{a}{b}=\boxed{\frac{3}{2}=1.5}$ ## Problem 35 ![[Chung-2018-t1-yc-p35.svg]] Two cylinders shown above have the same volume. If the radius of cylinder II is twice the radius of cylinder I and the height of cylinder I is 10 , what is the height $h$ of cylinder $\mathrm{X}?$![[gridin.svg]] > [!sol]- Click for Solution! > $\pi r^2(10)=\pi(2 r)^2 h \rightarrow 10=4 h \rightarrow h=\boxed{2.5}$ ## Problem 36 $g(x)=\frac{x^2-3 x+2}{(x+2)^2-8 x}$ For what value of $x$ is the function above undefined?![[gridin.svg]] > [!sol]- Click for Solution! > $(x+2)^2-8 x=0 \rightarrow x^2+4 x+4-8 x=0 \rightarrow x^2-4 x+4=0 \rightarrow(x-2)^2=0$ > $x=2$ is the answer. > [!info] Questions 37 and 38 refer to the following information. > Suppose Claire deposits a principal amount of $P$ dollars in a bank account that pays compound interest. If the annual interest is $r$ (expressed as a decimal) and the bank makes interest payments $n$ times every year, she would have an amount of money equal to $R$ after $t$ years, given by $R(t)=P\left(1+\frac{r}{n}\right)^{n t}$ ## Problem 37 If she deposit $\$ 2,000$ into an account paying $4 \%$ annual interest compounded annually, what is the amount of interest after one year? (Disregard the \$ sign when gridding your answer.)![[gridin.svg]] > [!sol]- Click for Solution! > $ > \text { Interest }=2000 \times 0.4=\boxed{80} > $ ## Problem 38 If she deposits $\$2,000$ into an account paying $4\%$ annual interest compounded quarterly, what is her account balance after one year? (Round your answer to the nearest dollar and disregard the \$ sign when gridding your answer.)![[gridin.svg]] > [!sol]- Click for Solution! > Since $n=4$, balance $=2000\left(1+\frac{0.04}{4}\right)^{4(1)} \simeq \boxed{2081}$