August 2018 Algebra Two Regents

1. The solution of $87e^{0.3x} = 5918$ , to the nearest thousandth, is

(1) $0.583$
(2) $1.945$
(3) $4.220$
(4) $14.066$

2. A researcher randomly divides $50$ bean plants into two groups. He puts one group by a window to receive natural light and the second group under artificial light. He records the growth of the plants weekly. Which data collection method is described in this situation?

(1) observational study
(2) controlled experiment
(3) survey
(4) systematic sample

3. If $f(x) = x^{2} + 9$ and $g(x) = x + 3$ , which operation would not result in a polynomial expression?

(1) $f(x) + g(x)$
(2) $f(x) - g(x)$
(3) $f(x) \bullet g(x)$
(4) $f(x) \div g(x)$

4. Consider the function $p(x) = 3x^{3} + {x^2} - 5x$ and the graph of $y = m(x)$ below.

Which statement is true?

(1) $p(x)$ has three real roots and $m(x)$ has two real roots.
(2) $p(x)$ has one real root and $m(x)$ has two real roots.
(3) $p(x)$ has two real roots and $m(x)$ has three real roots.
(4) $p(x)$ has three real roots and $m(x)$ has four real roots.

5. Which expression is equivalent to $\dfrac {2x^{4} + 8x^3 - 25x^{2} + 6x + 14}{x + 6}$ ?

(1) $2x^{3} + 4x^{2} + x - 12 + \dfrac{86}{x + 6}$
(2) $2x^{3} - 4x^{2} - x + 14$
(3) $2x^{3} - 4x^{2} - {x} + \dfrac{14}{x + 6}$
(4) $2x^{3} - 4x^{2} - x$

6. Given $f(x) = \dfrac{1}{2}x + 8$ , which equation represents the inverse, $g(x)$ ?

(1) $g(x) = 2x - 8$
(2) $g(x) = 2x - 16$
(3) $g(x) = -\dfrac{1}{2}x + 8$
(4) $g(x) = -\dfrac{1}{2}x - 16$

7. The value(s) of $x$ that satisfy $\sqrt{x^{2} - 4x- 5} = 2x - 10$ are

(1) ${5}$
(2) ${7}$
(3) ${5, 7}$
(4) ${3, 5, 7}$

8. Stephanie found that the number of white-winged crossbills in an area can be represented by the formula $C = 550(1.08)^{t}$ , where $t$ represents the number of years since 2010. Which equation correctly represents the number of white-winged crossbills in terms of the monthly rate of population growth?

(1) $C = 550(1.00643)^{t}$
(2) $C = 550(1.00643)^{12t}$
(3) $C = 550(1.00643)^{\frac{t}{12}}$
(4) $C = 550(1.00643)^{t + 12}$

9. The roots of the equation $3x^{2} + 2x = -7$ are

(1) $-2,-\dfrac{1}{3}$
(2) $-\dfrac{7}{3},1$
(3) $-\dfrac{1}{3} \pm \dfrac{2i}{\sqrt{5}}{3}$
(4) $-\dfrac{1}{3} \pm \dfrac{\sqrt{11}}{3}$

10. The average depreciation rate of a new boat is approximately $8\%$ per year. If a new boat is purchased at a price of $\$75,000$ , which model is a recursive formula representing the value of the boat $n$ years after it was purchased?

(1) $a_{n} = 75,000(0.08)^{n}$
(2) $a_{0} = 75,000$ ,
$a_{n} = (0.92)^{n}$
(3) $a_{n} = 75,000(1.08)^{n}$
(4) $a_{0} = 75,000$ ,
$a_{n} = 0.92(a_{n} - 1)$

11. Given $\cos \theta = \dfrac{7}{25}$ , where $\theta$ is an angle in standard position terminating in quadrant IV, and $\sin^{2}\theta + \cos^{2}\theta = 1$ , what is the value of $\tan\theta$ ?

(1) $-\dfrac{24}{25}$
(2) $-\dfrac{24}{7}$
(3) $\dfrac{24}{25}$
(4) $\dfrac{24}{7}$

12. For $x\textgreater 0$ , which expression is equivalent to $\dfrac{\sqrt[3]{x^{2}}\bullet \sqrt{x^{5}}}{\sqrt[6]{x}}$ ?

(1) $x$
(2) $x^{\frac{3}{2}}$
(3) $x^{3}$
(4) $x^{10}$

13. Jake wants to buy a car and hopes to save at least $\$5000$ for a down payment. The table below summarizes the amount of money he plans to save each week.

Based on this plan, which expression should he use to determine how much he has saved in $n$ weeks?

(1) $\dfrac{2 - 2(2.5^{n})}{1 - 2.5}$
(2) $\dfrac{2 - 2(2.5^{n-1})}{1 - 2.5}$
(3) $\dfrac{1 - 2.5^{n}}{1 - 2.5}$
(4) $\dfrac{1 - 2.5^{n-1}}{1 - 2.5}$

14. Which expression is equivalent to $x^{6} y^{4} (x^{4} - 16) - 9(x^{4} - 16)$ ?

(1) $x^{10} y^{4} - 16x^{6} y^{4} - 9x^{4} - 144$
(2) $(x^{6}y^{4} - 9)(x + 2)^{3}(x - 2)$
(3) $(x^{3} y^{2} + 3)(x^{3} y^{2} - 3)(x + 2)(x - 2)^{2}$
(4) $(x^{3} y^{3} + 3)(x^{3} y^{2} - 3)(x^{2} + 4)(x^{2} + 4)$

15. If $A = -3 + 5i, B = 4 - 2i$ , and $C = 1 + 6i$ , where $i$ is the imaginary unit, then $A - BC$ equals

(1) $5 - 17i$
(2) $5 + 27i$
(3) $-19 - 17i$
(4) $-19+ 27i$

16. Which sketch best represents the graph of $x = 3^{y}$ ?

(1)
(2)
(3)
(4)

17. The graph below represents national and New York State average gas prices.

If New York State’s gas prices are modeled by $G(x)$ and $C \textgreater 0$ , which expression best approximates the national average $x$ months from August 2014?

(1) $G(x + C)$
(2) $G(x) + C$
(3) $G(x - C)$
(4) $G(x) - C$

18. Data for the students enrolled in a local high school are shown in the Venn diagram below.

If a student from high school is selected at random, what is the probability that the student is a sophomore given that the student is enrolled in Algebra II?

(1) $\dfrac{85}{210}$
(2) $\dfrac{85}{295}$
(3) $\dfrac{85}{405}$
(4) $\dfrac{85}{1600}$

19. If $p(x) = 2ln(x) - 1$ and $m(x) = ln(x + 6)$ , then what is the solution for $p(x) = m(x)$ ?

(1) $1.65$
(2) $3.14$
(3) $5.62$
(4) no solution

20. Which function’s graph has a period of $8$ and reaches a maximum height of $1$ if at least one full period is graphed?

(1) $y = -4 \cos(\dfrac{\pi}{4}x) - 3$
(2) $y = 4 \cos(\dfrac{\pi}{4}x) + 5$
(3) $y = -4 \cos(8x) - 3$
(4) $y = -4 \cos(8x) + 5$

21. Given $c(m) = m^{3} - 2m^{2} + 4m - 8$ , the solution of $c(m) = 0$ is

(1) $\pm2$
(2) $2$ , only
(3) $2i, 2$
(4) $\pm2i, 2$

22. The height above ground for a person riding a Ferris wheel after $t$ seconds is modeled by $h(t) = 150 \sin(\dfrac{\pi}{45}t + 67.5) + 160$ feet.

How many seconds does it take to go from the bottom of the wheel to the top of the wheel?

(1) $10$
(2) $45$
(3) $90$
(4) $150$

23. The parabola described by the equation $y = \dfrac{1}{12}(x - 2)^{2} + 2$ has the directrix at $y=-1$ . The focus of the parabola is

(1) $(2, -1)$
(2) $(2, 2)$
(3) $(2, 3)$
(4) $(2, 5)$

24. A fast-food restaurant analyzes data to better serve its customers. After its analysis, it discovers that the events $D$ , that a customer uses the drive-thru, and $F$ , that a customer orders French fries, are independent. The following data are given in a report:

$P(f) = 0.8$
$P(F \cap D) = 0.456$

Given this information, $P(F|D)$ is

(1) $0.344$
(2) $0.3648$
(3) $0.57$
(4) $0.8$