$$\theta = \arcsin\left(\frac{3}{4}\right)$$ and $$\theta = \pi - \arcsin\left(\frac{3}{4}\right)$$

$$\begin{pmatrix} 5 & -1 \\ -3 & -7 \end{pmatrix}$$

Question 1

Given the rational function $$f(x) = \frac{x^2$}{2x - 48}$$, which of the following describes all of its vertical asymptotes?

Accepted Answer

$$x = 24$$

Question 2

For the function $$f(x) = \frac{2x^2$ + 8x}{x^2$ - 2x + 1}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 3

Which of the following is the oblique (slant) asymptote for $$f(x) = \frac{x^2$ - 20x - 21}{x - 3}$$?

Accepted Answer

$$y = x - 17$$

Question 4

Solve for $$x$$: $$2^x + 4 = 13$$

Accepted Answer

$$x = \log_2$ 9$$

Question 5

If the population of a colony of bunnies doubles in size every 4 years and the current population is 32, what will the population be in 20 years?

Accepted Answer

$$1024$$

Question 6

Solve for $$	heta in $$the interval $$0 \leq 	heta \u003c 2\pi$$: $$4 \sin 	heta = 3$$

Accepted Answer

$$	heta = \arcsin\left(\frac{3}{4}ight)$$ and $$	heta = \pi - \arcsin\left(\frac{3}{4}ight)$$

Question 7

Given the polar coordinates $$(2, \frac{\pi}{4})$$, what are the rectangular coordinates?

Accepted Answer

$$(\sqrt{2}, \sqrt{2})$$

Question 8

Find the magnitude of the vector $$\vec{v} = -16\hat{i} + 5\hat{j}.$$

Accepted Answer

$$\sqrt{281}$$

Question 9

Given $$\vec{v} = -4\hat{i} + 3\hat{j}$$ and $$\vec{w} = -2\hat{i} - 2\hat{j}$$, what is the dot product $$\vec{v} \cdot \vec{w}$$?

Accepted Answer

$$2$$

Question 10

What is the type of conic section represented by the equation $$16x^2$$ + $$36y^2 - 224x + 288 = 0$?

Accepted Answer

Ellipse

Question 11

Find the value of the determinant of the matrix $$\begin{pmatrix} 5 & 4 & -3 \ 1 & 2 & 1 \ 2 & 1 & 2 \end{pmatrix}.$$

Accepted Answer

$$-29$$

Question 12

Given the sequence $$\left\{ \frac{1}{1-2n} \right\}$$, what is the fourth term?

Accepted Answer

$$-\frac{1}{7}$$

Question 13

Find the 6th term of the geometric sequence $$1, 2, 4, 8, 16, \ldots.$$

Accepted Answer

$$32$$

Question 14

Given the parametric equations $$x = 2t$$, $$y = t^2$ - 1$$, which of the following is the corresponding rectangular equation?

Accepted Answer

$$y = \frac{x^2$}{4} - 1$$

Question 15

Consider the system of equations:
$$\begin{cases} x - 5y + 11z = 29 \ 7x - 6y - 10z = 0 \end{cases}$$
Which statement is true about the system?

Accepted Answer

The system has a unique solution.

Question 16

Given the matrices $$A = \begin{pmatrix} 0 & 2 \ 1 & 5 \end{pmatrix}$$ and $$C = \begin{pmatrix} 5 & 1 \ -2 & -2 \end{pmatrix}$$, what is $$C - A$$?

Accepted Answer

$$\begin{pmatrix} 5 & -1 \ -3 & -7 \end{pmatrix}$$

Question 17

Which of the following is the correct rectangular equation for the parametric equations $$x = \cosh t$$, $$y = t \sinh t$$?

Accepted Answer

$$y = \arccosh(x) \cdot \sqrt{x^2$ - 1}$$

Question 18

Given the equation $$9x^2$$ - $$4y^2 + 12y + 16 = 0$, what type of conic section does it represent?

Accepted Answer

Hyperbola

Question 19

Given the vector $$\vec{a} = \langle -3, 4 \rangle$$, what is the direction angle (in degrees) of $$\vec{a}$$?

Accepted Answer

$$126.87^\circ$$

Question 20

Which of the following is the correct solution for $$\sqrt{2} \cos 	heta + 3 = 6 in $$the interval $$0 \leq 	heta \u003c 2\pi$$?

Accepted Answer

No solution

Precalculus Final Exam Study Guide: Key Topics and Practice Problems

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