January 2018 Algebra Two Regents

1. The operator of the local mall wants to find out how many of the mall’s employees make purchases in the food court when they are working. She hopes to use these data to increase the rent and attract new food vendors. In total, there are $1023$ employees who work at the mall. The best method to obtain a random sample of the employees would be to survey

(1) all $170$ employees at each of the larger stores
(2) $50\%$ of the $90$ employees of the food court
(3) every employee
(4) every $30th$ employee entering each mall entrance for one week

2. What is the solution set for $x$ in the equation below?
$\sqrt{x + 1} - 1 = x$

(1) ${1}$
(2) ${0}$
(3) ${-1,0}$
(4) ${0,1}$

3. For the system shown below, what is the value of $z$ ?

$y = 2x + 14$ ,
$3x - 4z = 2$ ,
$3x - y = 16$

(1) $5$
(2) $2$
(3) $6$
(4) $4$

4. The hours of daylight, $y$ , in Utica in days, $x$ , from January 1, 2013 can be modeled by the equation $y = 3.06 \sin (0.017x - 1.40) + 12.23$ . How many hours of daylight, to the nearest tenth, does this model predict for February 14, 2013?

(1) $9.4$
(2) $10.4$
(3) $12.1$
(4) $12.2$

5. A certain pain reliever is taken in $220$ mg dosages and has a half-life of $12$ hours. The function $A = 220 (\dfrac{1}{2})^\frac{t}{12}$ can be used to model this situation, where $A$ is the amount of pain reliever in milligrams remaining in the body after $t$ hours.

According to this function, which statement is true?

(1) Every hour, the amount of pain reliever remaining is cut in half.
(2) In $12$ hours, there is no pain reliever remaining in the body.
(3) In $24$ hours, there is no pain reliever remaining in the body.
(4) In $12$ hours, $110$ mg of pain reliever is remaining.

6. The expression $(x + a)(x + b)$ can not be written as

(1) $a(x + b) + x(x + b)$
(2) $x^{2} + abx + ab$
(3) $x^{2} + (a + b)x + ab$
(4) $x(x + a)+ b(x + a)$

7. There are $440$ students at Thomas Paine High School enrolled in U.S. History. On the April report card, the student’s grades are approximately normally distributed with a mean of $79$ and a standard deviation of $7$ . Students who earn a grade less than or equal to $64.9$ must attend summer school. The number of students who must attend summer school for U.S. History is closest to

(1) $3$
(2) $5$
(3) $10$
(4) $22$

8. For a given time, $x$ , in seconds, an electric current, $y$ , can be represented by $y = 2.5(1 - 2.7^{-.10x})$ . Which equation is not equivalent?

(1) $y = 2.5 - 2.5(2.7^{-.10x})$
(2) $y = 2.5 - 2.5((2.7^{2})^{-.05x})$
(3) $y = 2.5 - 2.5(\dfrac{1}{2.7^{.10x}})$
(4) $y = 2.5 - 2.5(2.7^{-2})(2.7^{.05x})$

9. What is the quotient when $10x^{3} - 3x^{2} - 7x + 3$ is divided by $2x - 1$ ?

(1) $5x^{2} + x + 3$
(2) $5x^{2} - x + 3$
(3) $5x^{2} - x - 3$
(4) $5x^{2} + x - 3$

10. Judith puts $\$5000$ into an investment account with interest compounded continuously. Which approximate annual rate is needed for the account to grow to $\$9110$ after $30$ ?

(1) $2\%$
(2) $2.2\%$
(3) $0.02\%$
(4) $0.022\%$

11. If $n = \sqrt{a^{5}}$ and $m = a$ , where $a\textgreater 0$ , an expression for $\dfrac{n}{m}$ could be

(1) $a\frac{5}{2}$
(2) $a^{4}$
(3) $\sqrt[3]{a^{2}}$
(4) $\sqrt{a^{3}}$

12. The solutions to $x + 3 - \dfrac{4}{x - 1} = 5$ are

(1) $\dfrac{3}{2} \pm \dfrac{\sqrt{17}}{2}$
(2) $\dfrac{3}{2} \pm \dfrac{\sqrt{17}}{2}i$
(3) $\dfrac{3}{2} \pm \dfrac{\sqrt{33}}{2}$
(4) $\dfrac{3}{2} \pm \dfrac{\sqrt{33}}{2}i$

13. If $ae^{bt}=c$ , where $a, b$ , and $c$ are positive, then $t$ equals

(1) $In(\dfrac{c}{ab})$
(2) $In(\dfrac{cb}{a})$
(3) $\dfrac{In(\dfrac{c}{a})}{b}$
(4) $\dfrac{In(\dfrac{c}{a})}{Inb}$

14. For which values of $x$ , rounded to the nearest hundredth, will $|x^{2} - 9| - 3 = log_{3}x$ ?

(1) $2.29$ and $3.63$
(2) $2.37$ and $3.54$
(3) $2.84$ and $3.17$
(4) $2.92$ and $3.06$

15. The terminal side of $0$ , an angle in standard position, intersects the unit circle at $P(-\dfrac{1}{3}, -\dfrac{\sqrt{8}}{3})$ . What is the value of sec $0$ ?

(1) $-3$
(2) $-\dfrac{3\sqrt{8}}{8}$
(3) $-\dfrac{1}{3}$
(4) $-\dfrac{\sqrt{8}}{3}$

16. What is the equation of the directrix for the parabola $-8(y - 3) = (x + 4)^{2}$ ?

(1) $y = 5$
(2) $y = 1$
(3) $y = -2$
(4) $y = -6$

17. The function below models the average price of gas in a small town since January 1st.

$G(t) = -0.0049t^{4} + 0.0923t^{3} - 0.56t^{2} + 1.166t + 3.23$ , where $0\leq t \leq 10$ .

If $G(t)$ is the average price of gas in dollard and $t$ represents the number of months since January 1st, the absolute maximum $G(t)$ reaches over the given domain is about

(1) $\$1.60$
(2) $\$3.92$
(3) $\$4.01$
(4) $\$7.73$

18. Written in simplest form, $\dfrac{c^{2} - d^{2}}{d^{2} + cd - 2c^{2}}$ where $c \neq d$ , is equivalent to

(1) $\dfrac{c + d}{d + 2c}$
(2) $\dfrac{c - d}{d + 2c}$
(3) $\dfrac{-c - d}{d + 2c}$
(4) $\dfrac{-c + d}{d + 2c}$

19. If $p(x)=2x^{3} - 3x + 5$ , what is the remainder of $p(x) \div (x - 5)$ ?

(1) $-230$
(2) $0$
(3) $40$
(4) $240$

20. The results of simulating tossing a coin $10$ times, recording the number of heads, and repeating this $50$ times are shown in the graph below.

Based on the results of the simulation, which statement is false?

(1) Five heads occured most often, which is consistent with the theoretical probability of obtaining a heads.
(2) Eight heads is unusual, as it falls outside the middle $95\%$ of the data.
(3) Obtaining three heads or fewer occured $28\%$ of the time.
(4) Seven heads is not unusual, as it falls within the middle $95\%$ of the data.

21. What is the inverse of $f(x) = -6(x - 2)$ ?

(1) $f^{-1}(x) = -2 - \dfrac{x}{6}$
(2) $f^{-1}(x) = 2 - \dfrac{x}{6}$
(3) $f^{-1}(x) = \dfrac{1}{-6(x - 2)}$
(4) $f^{-1}(x) = 6(x + 2)$

22. Brian deposited $1$ cent into an empty non-interest bearing bank account on the first day of the month. He then additionally deposited $3$ cents on the second day, $9$ cents on the third day, and $27$ cents on the fourth day. What would be the total amount of money in the account at the end of the $20th$ day if the pattern continued?

(1) $\$11,622,614.67$
(2) $\$17,433,922.00$
(3) $\$116,226,146.80$
(4) $\$1,743,392,200.00$

23. If the function $g(x) = ab^{x}$ represents exponential growth, which statement about $g(x)$ is false?

(1) $a \textgreater 0$ and $b\textgreater 1$
(2) The $y$ -intercept is $(0, a)$ .
(3) The asymptote is $y = 0$ .
(4) The $x$ -intercept is $(b, 0)$ .

24. At her job, Pat earns $\$25,000$ the first year and receives a raise of $\$1000$ each year. The explicit formula for the $n$ th term of this sequence is $a_{n} = 25,000 + (n - 1)1000$ . Which rule best represents the equivalent recursive formula?

(1) $a_{n} = 24,000 + 1000n$
(2) $a_{n} = 25,000 + 1000n$
(3) $a_{1} = 25,000, a_{n - 1} + 1000$
(4) $a_{1} = 25,000, a_{n + 1} + 1000$