Question 1

Which of the following is the correct set of real solutions to the equation $$\sqrt{x^2$ - 5} = 6 - x$$?

Accepted Answer

$$x = 7, x = -1$$

Question 2

Given the quadratic equation $$3x^2 - 7x - 8 = 0$, which of the following is the correct solution using the quadratic formula?

Accepted Answer

$$x = 4, x = -\frac{2}{3}$$

Question 3

Which of the following lists all possible rational roots for $$f(x) = 6x^4$ + 12x + 9$$ according to the Rational Root Theorem?

Accepted Answer

$$\pm1, \pm3, \pm9, \pm\frac{1}{2}, \pm\frac{3}{2}, \pm\frac{9}{2}, \pm\frac{1}{3}, \pm\frac{3}{3}, \pm\frac{9}{3}, \pm\frac{1}{6}, \pm\frac{3}{6}, \pm\frac{9}{6}$$

Question 4

For the rational function $$f(x) = \frac{2x^2$ + 10}{x^2$ - 4}$$, which of the following correctly describes its horizontal and vertical asymptotes?

Accepted Answer

Horizontal: $$y = 2$$, Vertical: $$x = 2$$, $$x = -2$$

Question 5

Given $$f(x) = 5x^3$ - 35x^2$ + 31x + 1$$ and $$x = 3/5 is a $$zero, which method would be most efficient to verify this and find the other zeros?

Accepted Answer

Synthetic division

Question 6

Which of the following is the correct factorization of $$f(x) = x^3$ + 4x^2$ + 6x + 4$$?

Accepted Answer

$$(x + 2)^2$(x + 1)$$

Question 7

Given $$f(x) = (x-1)^2$(x-2)^3$$, what is the multiplicity of the zero at $$x = 2$?

Accepted Answer

Multiplicity 3

Question 8

A ramp is built at a $$35^{\circ}$$ angle of elevation and has a horizontal length of 6 meters. What is the vertical height of the ramp?

Accepted Answer

$$6 \sin 35$$^{\circ}$$

Question 9

Given $$\sin 	heta = \frac{3}{5}$$ and $$	heta is in $$Quadrant II, what is $$\cos 	heta$$?

Accepted Answer

$$-\frac{4}{5}$$

Question 10

Which of the following is the correct amplitude, period, and phase shift for $$f(x) = 4 \sin(2x - \frac{\pi}{4})$$?

Accepted Answer

Amplitude: $$4$$, Period: $$\pi$$, Phase shift: $$\frac{\pi}{8}$$ right

Question 11

Evaluate $$\cos(\sin$$^{-1}$$(-\frac{2}{3})).$$

Accepted Answer

$$\frac{\sqrt{5}}{3}$$

Question 12

Which of the following is the correct polar form of $$z = -4 + 4i$$?

Accepted Answer

$$4\sqrt{2}(\cos 135$$^{\circ}$$ + i \sin 135$$^{\circ}$$)$$

Question 13

Given the sequence $$a_n = 4a_{n-1} - a_{n-2}$$ with $$a_0 = 2$, a_1$ = 3$$, what is $$a_3$$?

Accepted Answer

$$35$$

Question 14

Determine whether the infinite geometric series $$\sum_{k=1}^{\infty} \left(\frac{1}{3}\right)^k$$ converges, and if so, find its sum.

Accepted Answer

Converges, sum is $$\frac{1}{2}$$

Question 15

Given the binomial $$(3x + 2y)^6$, what is the coefficient of the x^3$y^3$ term?

Accepted Answer

$$1080$$

Question 16

Find all solutions in the interval $$[0, 2\pi)$$ for $$4\sin^2$ x = 1 + \sin x.$$

Accepted Answer

$$x = \frac{\pi}{6}, x = \frac{5\pi}{6}, x = \frac{7\pi}{6}, x = \frac{11\pi}{6}$$

Question 17

A vector $$\vec{v} = 6\hat{i} + 7\hat{j}. $$What is the magnitude of $$\vec{v}$$?

Accepted Answer

$$\sqrt{85}$$

Question 18

Given the arithmetic sequence with $$a_6 = 58$ and a_0$ = 6$$, what is the common difference $$d$$?

Accepted Answer

$$d = 8.67$$

Question 19

Expand $$(2x - 3)^4$ and select the correct coefficient for the x^2$ term.

Accepted Answer

$$-72$$

Precalculus Final Exam Review: Comprehensive Study Notes

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