Precalculus Test

1.
If \(5x - \dfrac{1}{2} (x+4) = 1\), then \(x = \)
(A)
\(\dfrac{5}{9}\)
(B)
\(- \dfrac{2}{9}\)
(C)
\(\dfrac{2}{3}\)
(D)
\(\dfrac{1}{3}\)
2.
\(\left(\dfrac{-16a^2}{39b}\right)\left(\dfrac{13b^4}{8a^3}\right) \div (4b^3) =\)
(A)
\(- \dfrac{1}{6a}\)
(B)
\(- 6a\)
(C)
\(- \dfrac{8b^6}{3a}\)
(D)
\(- \dfrac{8b^{11}}{3a}\)
3.
In the system of equations\( \left\{ \begin{array}{l l} 5x-y=1\\ x+3y=13 \end{array} \right.\ , y =\)
(A)
\(- \dfrac{7}{8}\)
(B)
\(1\)
(C)
\(4\)
(D)
No Solution
4.
If \(f(x) = 2 x^2 - 1\) and \(g(x) = 5x + 2\), then \(f(g(-1))=\)
(A)
\(-19\)
(B)
\(-13\)
(C)
\(17\)
(D)
\(7\)
5.
If \(x = -7\), then \(2 | x + 5 | - | 1 - 2x | = \)
(A)
\(-19\)
(B)
\(-11\)
(C)
\(19\)
(D)
\(-9\)
6.
One of the roots of \(20x^2 + 3x -2 = 0\) is
(A)
\(\dfrac{1}{4}\)
(B)
\(\dfrac{2}{5}\)
(C)
\(-\dfrac{1}{4}\)
(D)
\(4\)
7.
When \(3x^2 - 5x + 8\) is divided by \(x + 2\), the remainder is
(A)
\(3x - 11\)
(B)
\(-14\)
(C)
\(0\)
(D)
\(30\)
8.
\(m^{-2}(m^{-3} + m^{-1})=\)
(A)
\(m^6 + m^2\)
(B)
\(\dfrac{1}{m^5} + \dfrac{1}{m^3}\)
(C)
\(\dfrac{1}{m^5 + m^3}\)
(D)
\(m^8\)
9.
The inequality \(3x - 6 > 5x\) is equivalent to
(A)
\(x < 3\)
(B)
\(x > 3\)
(C)
\(x > -3\)
(D)
\(x < -3\)
10.
The inequality \(|x+1| < 7\) is equivalent to
(A)
\(0 < x < 6\)
(B)
\(-8 < x < 6\)
(C)
\(-6 < x < 6\)
(D)
\(-1 < x < 6\)
11.
For how many values of \(\theta\) between \(0\) and \(2 \pi\) radians is \(sin \theta = - cos \theta\)?
(A)
\(1\)
(B)
\(2\)
(C)
\(3\)
(D)
\(4\)
12.
\(\dfrac{\dfrac{x^2 + x}{5x - 20}}{\dfrac{x + 1}{4 - x}}=\)
(A)
\(\dfrac{x(x+1)^2}{(5x - 20)(4 - x)}\)
(B)
\(- \dfrac{x}{5}\)
(C)
\(\dfrac{x(4 - x)}{5(x - 4)}\)
(D)
\(\dfrac{x}{5}\)
13.
One of the roots of \(x^2 = x + 5\) is
(A)
\(\dfrac{-1 + \sqrt{21}}{2}\)
(B)
\(\dfrac{-1 - \sqrt{6}}{2}\)
(C)
\(\dfrac{1 - \sqrt{21}}{2}\)
(D)
\(\dfrac{1 + \sqrt{6}}{2}\)
14.
\(-2^4+3^{-1}+4^0=\)
(A)
\(\dfrac{61}{48}\)
(B)
\(- \dfrac{44}{3}\)
(C)
\(\dfrac{52}{3}\)
(D)
\(- \dfrac{47}{3}\)
15.
The inequality \(x^2 - x > 6\) is equivalent to
(A)
\(x < -2\) or \(x > 3\)
(B)
\(-2 < x < 3\)
(C)
\(x < -2\)
(D)
\(x > 3\)
16.
Which of the following could be the graph of \(y = (x - 2)^2 - 1\)?
(A)
(B)
(C)
(D)
17.
If \(4^x = 5\), then \(x =\)
(A)
\(\dfrac{4}{\sqrt{5}}\)
(B)
\(\dfrac{5}{\sqrt{4}}\)
(C)
\(\dfrac{log 5}{log 4}\)
(D)
\(\dfrac{log 4}{log 5}\)
18.
\(\dfrac{5}{x^2 - 5x + 6} - \dfrac{5}{x - 3} =\)
(A)
\(\dfrac{5}{x - 2}\)
(B)
\(\dfrac{5}{2 - x}\)
(C)
\(\dfrac{-5 - 5x}{x^2 - 5x + 6}\)
(D)
\(0\)
19.
Which of the following could be a portion of the graph of \(y = \left(\dfrac{1}{2}\right)^x \)?
(A)
(B)
(C)
(D)
20.
The number of moles of an ideal gas \(n\) is directly proportional to its volume \(V\) and inversely proportional to its temperature \(T\). Which of the following could be the variation equation?
(A)
\(n = \dfrac{kT}{V}\)
(B)
\(n = \dfrac{kV^3}{T}\)
(C)
\(n = \dfrac{kV}{T}\)
(D)
\(n = kVT\)
21.
If \(f(x) = \sqrt{4x - 16}\), for what value of \(x\) does \(f(x) = 8\)?
(A)
\(\pm t\)
(B)
\(4\)
(C)
\(20\)
(D)
\(\{-16, 20\}\)
22.
The length of a rectangle is 1 less than twice the width. The perimeter of the rectangle is 200 feet. If \(x\) represents the width of the rectangle, then an equation that can be used to find the length and the width of the rectangle is
(A)
\((2x - 1) x = 200\)
(B)
\((1 - 2x) x = 200\)
(C)
\(2 (1 - 2x) + 2x = 200\)
(D)
\(2 (2x - 1) + 2x = 200\)
23.
If \(y = \dfrac{x + 2}{x - 3}\), then \(x = \)
(A)
\(\dfrac{y + 2}{y - 3}\)
(B)
\(\dfrac{3y + 2}{y - 1}\)
(C)
\(\dfrac{x + 5}{y}\)
(D)
\(xy - 3y + 2\)
24.
An equation of the line in the figure shown is
(A)
\(2x - 3y = 12\)
(B)
\(3x - 2y = 8\)
(C)
\(2x - 3y = 18\)
(D)
\(\dfrac{2}{3} x + y = -4\)
25.
Which of the following could be a portion of the graph of \(y = -cos2x\)?
(A)
(B)
(C)
(D)
26.
If \(f(x) = x^2 - x + 4\), then \(f(a+h) =\)
(A)
\(a^2 - a + 4 + h\)
(B)
\(a^2 + 2ah + h^2 - a - h + 4\)
(C)
\(a^2 + h^2 - a - h + 4\)
(D)
\((a + h)^2 - a + 4 + h\)
27.
\(log_{3} \dfrac{1}{9}=\)
(A)
\(2\)
(B)
\(-2\)
(C)
\(3\)
(D)
\(\sqrt{2}\)
28.
In the figure shown below, \(x =\)
(A)
\(8\)
(B)
\(24\)
(C)
\(\dfrac{27}{8}\)
(D)
\(\dfrac{96}{9}\)
29.
\(tan(\theta + \pi) =\)
(A)
\(- cot \theta\)
(B)
\(- tan \theta\)
(C)
\(tan \theta\)
(D)
\(cot \theta\)
30.
The graph of \(y = f(x)\) is shown in the figure below. Which of the following could be the graph of \(y = f(-x)\)?
(A)
(B)
(C)
(D)
31.
\((3m^{x-1})(2m^{2x})=\)
(A)
\(6m^{2x^2-2x}\)
(B)
\(5m^{2x^2-2x}\)
(C)
\(6m^{3x-1}\)
(D)
\(6(2m)^{3x-1}\)
32.
The number of books in a school library is 180 after a 25% increase. Find the number of books before the increase.
(A)
\(45\)
(B)
\(72\)
(C)
\(144\)
(D)
\(135\)
33.
If \(3 log_{b} x - log_{b} y = log_{b} z\), then \(z = \)
(A)
\(x^3 - y\)
(B)
\(\dfrac{3x}{y}\)
(C)
\(3x - y\)
(D)
\(\dfrac{x^3}{y}\)
34.
\(\sqrt{36x^6 y^2 - 64x^4}\)
(A)
\(6x^3 y - 8x^2\)
(B)
\(2x^2 \sqrt{9x^2y^2 - 16}\)
(C)
\(2x^2y \sqrt{9x^2 - 16}\)
(D)
\(6x^3 y - 8x\)
35.
\(\dfrac{1 + \dfrac{4}{x} - \dfrac{45}{x^2}}{1+ \dfrac{2}{x} - \dfrac{35}{x^2}} =\)
(A)
\(-8\)
(B)
\(\dfrac{x + 9}{x + 7}\)
(C)
\(\dfrac{5}{4}\)
(D)
\(\dfrac{x - 41}{x - 33}\)
36.
\(\dfrac{x}{\sqrt[3]{4x}}=\)
(A)
\(\dfrac{x^2 \sqrt[3]{2}}{2}\)
(B)
\(\dfrac{x \sqrt[3]{2}}{2x}\)
(C)
\(\dfrac{\sqrt[3]{2x}}{8x}\)
(D)
\(\dfrac{\sqrt[3]{2x^2}}{2}\)
37.
In the figure shown below, \(tan \theta =\)

(A)
\(\dfrac{\sqrt{x^2 - a^2}}{a}\)
(B)
\(\dfrac{\sqrt{x^2 + a^2}}{a}\)
(C)
\(\dfrac{\sqrt{x^2 - a^2}}{x}\)
(D)
\(\dfrac{\sqrt{x^2 + a^2}}{x}\)
38.
If \(3^{x-12} = 9^{2x}\), then \(x = \)
(A)
\(-12\)
(B)
\(-4\)
(C)
No Solution
(D)
\(4\)
39.
If \(log_{2} x + log_{2} (x - 2) = 3\),
(A)
\(4\)
(B)
\(2\)
(C)
\(-2\)
(D)
\(\dfrac{5}{2}\)
40.
The domain of the function below is:
(A)
\((-\infty, \infty)\)
(B)
\([-2,2)\)
(C)
\([-2,1]\)
(D)
\([-4,-1]\)
41.
The graph of the piecewise function\( f(x) = \left\{ \begin{array}{l l} -2x + 4 & \quad \textrm{if $x > 0$} \\ 3 & \quad \textrm{if $x \leq 0$} \end{array} \right.\ \) is
(A)
(B)
(C)
(D)
42.
If \(sin \theta = \dfrac{3}{5}\) and \(tan \theta < 0\), then \(cos \theta =\)
(A)
\(-\dfrac{5}{3}\)
(B)
\(-\dfrac{2}{5}\)
(C)
\(-\dfrac{4}{5}\)
(D)
\(-\dfrac{3}{4}\)
43.
An equation of the circle with center \((-3, 0)\) and radius \(5\) is
(A)
\((x+3)^2 + y^2 = \sqrt{5}\)
(B)
\((x-3)^2 + y^2 = 10\)
(C)
\((x+3)^2 + y^2 = 25\)
(D)
\((y+3)^2 + x^2 = 25\)
44.
If \(\left(x + \dfrac{1}{2}\right)^3 = \dfrac{1}{64}\), then \(x = \)
(A)
\(-\dfrac{1}{4}\)
(B)
\(\dfrac{1}{8}\)
(C)
\(\dfrac{1}{2}\)
(D)
\(\dfrac{\sqrt[3]{-7}}{4}\)
45.
\(\sqrt{\sqrt{2}}\)
(A)
\(2\)
(B)
\(16\)
(C)
\(\sqrt[4]{2}\)
(D)
\(\sqrt[4]{4}\)