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Section 1: No-Calculator Section

Part A: Multiple Choice (Approx. 8-10 Questions)

Instructions: Answer the following multiple-choice questions without using a calculator. Select the best answer.

What is the degree and leading coefficient of the polynomial function $f(x) = 2x^3 - 3x^4 + 5x - 1$ ?

Select one option

Degree 3, Leading Coefficient 2

Degree 4, Leading Coefficient -3

Degree 4, Leading Coefficient 2

Degree 3, Leading Coefficient -3

Describe the end behavior of the function $g(x) = -2x^5 + 7x^3 - x + 10$ .

Select one option

As $x \to \infty$ , $g(x) \to \infty$ and as $x \to -\infty$ , $g(x) \to -\infty$

As $x \to \infty$ , $g(x) \to -\infty$ and as $x \to -\infty$ , $g(x) \to \infty$

As $x \to \infty$ , $g(x) \to -\infty$ and as $x \to -\infty$ , $g(x) \to -\infty$

As $x \to \infty$ , $g(x) \to \infty$ and as $x \to -\infty$ , $g(x) \to \infty$

Which of the following is the logarithmic form of the exponential equation $b^C = A$ ?

Select one option

$\\log_A (b) = C$

$\\log_C (b) = A$

$\\log_b (A) = C$

$\\log_b (C) = A$

Evaluate $\\log_2 (8)$ .

Select one option

What are the vertical asymptotes of the rational function $h(x) = \frac{x-1}{x^2 - 9}$ ?

Select one option

$x = 1$

$x = 3$

$x = -3$ and $x = 3$

$x = -1$

What is the horizontal asymptote of the rational function $k(x) = \frac{2x+5}{x^2 - 4x + 3}$ ?

Select one option

$y = 2$

$y = \frac{5}{3}$

$y = 0$

No horizontal asymptote

What is the exact value of $\\cos(\frac{\pi}{6})$ ?

Select one option

$\frac{1}{2}$

$\frac{\sqrt{2}}{2}$

$\frac{\sqrt{3}}{2}$

Simplify the trigonometric expression $\\sin x \cot x$ .

Select one option

$\sec x$

$\tan x$

$\sin x$

$\cos x$

10.

If $x=2$ is a root of a polynomial $P(x)$ , which of the following is a factor of $P(x)$ ?

Select one option

$x+2$

$x-2$

$2x$

$x$

Part B: Free Response (Approx. 5-7 Questions)

11.

Instructions: For the following free-response questions, show all necessary work to receive full credit. Do not use a calculator.

12.

Divide $P(x) = x^3 - 6x^2 + 11x - 6$ by $(x-2)$ using synthetic division. State the quotient and remainder.

13.

Solve for $x$ : $\\log_5 (x) = 3$ .

14.

Solve for $x$ : $4^{x-1} = 16$ .

15.

Sketch the graph of the rational function $f(x) = \frac{x+2}{x-1}$ . Clearly indicate all asymptotes and intercepts.

16.

Prove the trigonometric identity: $\\frac{\\sin x}{1 + \\cos x} = \\frac{1 - \\cos x}{\\sin x}$ .

17.

Factor the polynomial $P(x) = x^3 + 2x^2 - 5x - 6$ completely, given that $(x+1)$ is a factor.

Section 2: Calculator Permitted Section

Part A: Multiple Choice (Approx. 8-10 Questions)

18.

Instructions: Answer the following multiple-choice questions. A calculator is permitted for this section. Select the best answer.

19.

Identify the type of conic section represented by the equation $y = 3x^2 - 4x + 1$ .

Select one option

Circle

Ellipse

Parabola

Hyperbola

20.

Using the change of base formula, approximate $\\log_7 (23)$ to two decimal places.

Select one option

0.70

1.36

1.77

3.26

21.

Solve for $x$ to two decimal places: $e^{2x} = 400$ .

Select one option

3.00

6.00

20.00

20.09

22.

What is the principal value of $\\arcsin(0.85)$ in radians, rounded to two decimal places?

Select one option

0.85 radians

0.98 radians

1.02 radians

1.11 radians

23.

A manufacturing company's average cost per unit, $C(x)$ , for producing $x$ units of a product is given by $C(x) = \frac{1000 + 0.5x}{x}$ . As the number of units produced increases indefinitely, what does the average cost per unit approach?

Select one option

$y = 1000$

$y = 0$

$y = 0.5$

$y = \infty$

24.

What are the center and radius of the circle given by the equation $(x-2)^2 + (y+3)^2 = 25$ ?

Select one option

Center: $(-2, 3)$ , Radius: $5$

Center: $(2, -3)$ , Radius: $25$

Center: $(2, -3)$ , Radius: $5$

Center: $(-2, 3)$ , Radius: $25$

25.

The population of a certain type of bacteria triples every 2 hours. If a culture starts with 100 bacteria, how long will it take for the population to reach 5000 bacteria? Round to two decimal places.

Select one option

3.91 hours

4.50 hours

5.65 hours

6.21 hours

26.

A ladder 20 meters long leans against a wall. If the base of the ladder is 7 meters from the wall, what is the angle of elevation of the ladder (angle with the ground)? How high up the wall does the ladder reach? Round to two decimal places.

Select one option

20.49 degrees, 6.57 meters

69.51 degrees, 18.39 meters

70.49 degrees, 19.39 meters

19.51 degrees, 18.39 meters

27.

A polynomial has roots at $x=1, x=-2,$ and $x=3$ . Which of the following could be the polynomial?

Select one option

$(x+1)(x-2)(x+3)$

$(x-1)(x+2)(x-3)$

$(x-1)(x-2)(x-3)$

$(x+1)(x+2)(x+3)$

Part B: Free Response (Approx. 5-7 Questions)

28.

Instructions: For the following free-response questions, show all necessary work to receive full credit. A calculator is permitted.

29.

Graph the parabola $(y-1)^2 = 8(x+2)$ . Clearly label the vertex, focus, and directrix.

30.

Solve the rational equation: $\\frac{3}{x-2} + \\frac{x}{x+1} = \\frac{x^2+1}{x^2-x-2}$ . Be sure to check for extraneous solutions.

31.

A population of rabbits grows exponentially. If an initial population of 5000 rabbits grows to 15000 rabbits in 5 years, how long will it take for the population to reach 50,000 rabbits? Round your answer to two decimal places.

32.

Solve the trigonometric equation $2\\sin^2 \\theta - \\sin \\theta - 1 = 0$ for $0 \le \\theta < 2\\pi$ .

33.

Write the equation of an ellipse with foci at $(0, \pm 4)$ and vertices at $(0, \pm 6)$ .

34.

Solve for $x$ : $\\log_2 (x) + \\log_2 (x-4) = 5$ . Be sure to check for extraneous solutions.