Baruch Mathematics Placement Test

Math Placement Part 1 Practice Questions

1. Find the slope of the line that passes through the given points: \( (12,15) \) and \( (10, -3) \).

A) \(-9\)     B) \(\frac{6}{11}\)     C) \(\frac{1}{9}\)     D)  \(9\)    E)  \(\frac{-1}{9} \)

 

2. Find the slope of the line \(11x − 8y =88\).

A) \(\frac{11}{8}\)    B) \(\frac{-11}{8}\)    C) \(\frac{8}{11}\)    D) \(\frac{-8}{11}\)    E) \(11\)

 

3. Find an equation of the line with the slope \(m = \frac{1}{9}\) that passes through the point \( (−6,2) \). Write the equation in the form \( Ax + By = C\).

A) \(9x – 9y = -24\)    B) \(x – 9y = -24\)    C) \(x – 9y = 24\)    D) \(x – 9y = -72\)    E) \(x + 9y = -72\)

 

4. Compute \( f(-3) \) when \( f(x) = 4x^2 – 3x  + 6\).  Your answer should be an integer.

\(f(-3) = \)           

 

5. Solve the system of equations:

\(y = 5x – 5\)

\(2y + 10x = -70\)

A) \( (-3, -20)\)    B) \((-20, -3)\)    C) \( (3, -20)\)    D) No solutions    E) Infinitely many solutions

 

6. Solve the system of equations:

\(x + 5y = -22\)

\(3x + 4y = -33\)

\(x = \)           

\(y = \)           

 

7. Simplify the expression \( \displaystyle\frac{(x^2)^4}{(2x)^3} \)

A) \(\displaystyle\frac{x^5}{8}\)    B) \(\displaystyle\frac{x^5}{2}\)    C) \(\displaystyle\frac{x^{11}}{8}\)    D) \(\displaystyle\frac{x^3}{32}\)    E) \(\displaystyle\frac{x^3}{8}\)

 

8. Perform the indicated operation and simplify: \( (26x + 5)  – (-4x^2 – 13x + 5) \)

A) \(4x^2 – 39x\)    B) \(4x^2 + 39x\)    C) \(4x^2 + 39x – 10\)    D) \(4x^2 + 13x + 10\)    E) \(-4x^2 + 13x + 10\)

 

9. If you multiply \( (6x – 1)(x^2 – 2x + 1)\) and then simplify the result, one of the terms will be:

A) \(-13x^2\)    B) \(-12x^2\)    C) \(13x^2\)    D) \(-11x^2\)    E) \(-9x^2\)

 

10. Multiply \( (8z + 3)^2\).

A) \(64z^2 + 48z + 9\)    B) \(8z^2 + 9\)    C) \(8z^2 + 48z + 9\)    D) \(16z^2 + 16z + 9\)    E) \(64z + 9\)

 

11. Find \(x\) in the given triangle:

A) \(13\)    B) \(\sqrt{104}\)    C) \(7\)    D) \(5\)    E) \(12\)

 

12. If you factor the polynomials \(x^2 – 6x – 27\) and \(x^2 + 15x + 54\), which of the following is NOT a factor of either polynomial?

A) \(x + 1\)    B) \(x + 9\)    C) \(x – 9\)    D) \(x + 3\)    E) \(x + 6\)

 

13. If you factor \(7x^2 + 5x – 2\), one of the factors is:

A) \(7x – 1\)    B) \(7x – 2\)    C) \(x – 9\)    D) \(x + 3\)    E) \(x – 2\)

 

14. A boat can travel 48 miles with the current in 4 hours.  The same boat requires 6 hours to travel the same distance against the current.  How fast is the current?

A) 1 mph    B) 2 mph    C) 3 mph    D) 4 mph    E) 5 mph

 

15. Solve the equation \(x^2 – x = 72\).  Both solutions are integers; one is positive, the other is negative.

Negative solution: \(x = \)           

Positive solution:  \(x = \)            

 

16. Find the domain of the function \(\displaystyle f(x) = \frac{\sqrt{x}}{x – 4} \).

A) \((-\infty, 0) \cup (0,4) \)     B) \([0, \infty)\)     C) \([0,4) \cup (4, \infty) \)     D) \((-\infty,4) \cup (4, \infty)\)     E) \( [0,\infty) \cup (4, \infty)\)

 

17. Simplify the rational expression: \(\displaystyle\frac{2x + 2}{6x^2 + 16x + 10}\)

A) \(\displaystyle\frac{2x + 2}{6x^2 + 16x + 10}\)    B) \(\displaystyle\frac{2x + 3}{3x + 2}\)    C) \(\displaystyle\frac{1}{3x + 5}\)    D) \(\displaystyle\frac{2x}{3x + 5}\)    E) \(\displaystyle\frac{2}{3x + 5}\)

 

18. Find the product and simplify: \(\displaystyle\frac{x^2 + 10x + 24}{x^2 + 13x + 42}\cdot\frac{x^2 + 7x}{x^2 – 5x – 36}\)

A) \(\displaystyle\frac{1}{x – 9}\)    B) \(\displaystyle\frac{x}{x – 9}\)    C) \(\displaystyle\frac{x(x + 7)}{x – 9}\)    D) \(\displaystyle\frac{x}{x^2 + 13x + 42}\)    E) \(\displaystyle\frac{x + 8}{x – 9}\)

 

19. Solve the inequality for \(t\), writing your answer in interval notation: \(t^2 – 2t – 3 \leq 0\)

A) \([-1, 3]\)    B) \( (-3, 1) \)    C) \( [-3, 1]\)    D) \( (-\infty, -1) \cup [3, \infty)\)    E) \( (-\infty, -3] \cup [1, \infty) \)

 

20. Find the quotient and simplify: \(\displaystyle \frac{32x^4}{x^2 – 1} \div \frac{x^8}{(x + 1)^2}\)

A) \(\displaystyle\frac{32(x + 1)}{x^4(x – 1)}\)    B) \(\displaystyle\frac{32x^{12}}{(x – 1)(x + 1)^3}\)    C) \(\displaystyle\frac{4(x + 1)}{x^4(x – 1)}\)    D) \(\displaystyle\frac{32x^4(x + 1)}{(x – 1)}\)    E) \(\displaystyle\frac{32x^{32}}{(x – 1)(x + 1)^3}\)

 

21. Simplify: \(\displaystyle\frac{6x + 2}{x^2 + 13x + 40} – \frac{5x – 6}{x^2 + 13x + 40}\)

A) \(\displaystyle\frac{x – 8}{x^2 + 13x + 40}\)    B) \(\displaystyle\frac{1}{x^2 + 13x + 40}\)    C) \(\displaystyle\frac{1}{x + 8}\)    D) \(\displaystyle\frac{x – 4}{x^2 + 13x + 40}\)    E) \(\displaystyle\frac{1}{x + 5}\)

 

22. Solve:  \(x^2 + 8x + 9 = 0 \)

A) \(x = -4\) or \(x = 4\)     B) \(x = 4\) only    C) \(x = -7\) or \(x = -1\)    D) \(x = -4 + \sqrt{7}\) only    E) \(x = -4 + \sqrt{7}\) or \(x = -4 – \sqrt{7}\)

 

23. Simplify: \(\displaystyle\frac{2}{x} + \frac{9}{x – 6}\)

A) \(\displaystyle\frac{11x – 12}{x(x – 6)}\)    B) \(\displaystyle\frac{12x – 11}{x(x – 6)}\)    C) \(\displaystyle\frac{11}{x(x – 6)}\)    D) \(\displaystyle\frac{2x – 3}{x(x – 6)}\)    E) \(\displaystyle\frac{9x – 12}{x(x – 6)}\)

 

24. Solve the equation:  \(\displaystyle1 + \frac{1}{x} = \frac{72}{x^2} \)

A) \(\displaystyle x = 8, -9\)    B) \(\displaystyle x = -8, 9\)    C) \(\displaystyle x = \frac{-1}{8}, \frac{1}{9}\)    D) \(\displaystyle x = 8, 9\)    E) \(\displaystyle x = \frac{-1}{8}, \frac{-1}{9}\)

 

25. Simplify: \(\displaystyle\frac{4 + \frac{2}{x}}{\frac{x}{3} + \frac{1}{6}}\)

A) \(\displaystyle\frac{x}{6}\)    B) \(\displaystyle\frac{x}{12}\)    C) \(\displaystyle\frac{12}{x}\)    D) \(\displaystyle 12\)    E) \(\displaystyle\frac{4x + 2}{2x + 3}\)

 

26. Solve for \(x\):  \(\sqrt{30x + 15} = x + 8\)

A) \(x = 0\) only    B) \(x = 7\) only   C) \(x = 7\) and \(x = 1\)    D) \(x = 7\) and \(x = -7\)    E) No real solution

 

27. Which of the following is equal to \(243^{4/5}\)?

A) \(81\)    B) \(6561\)    C) \(2187\)    D) \(128\)    E) \(162\)

 

28. If \(x\) is a positive real number, which of the following is equal to \(\sqrt{5x^3}\cdot\sqrt{5x^5}\)?

A) \(\sqrt{25x^{15}}\)    B) \(5x^4\)    C) \(\sqrt{5x^4}\)    D) \(x^4\sqrt{10}\)    E) \(5\sqrt{x^{15}}\)

 

29. Simplify \(2\sqrt{5} – 9\sqrt{45}\).

A) \(25\sqrt{5}\)    B) \(11\sqrt{50}\)    C) \(11\sqrt{5}\)    D) \(-25\sqrt{5}\)    E) \(6\sqrt{5}\)

 

30. Multiply and then simplify if possible: \( (\sqrt{13} + 2)\cdot (\sqrt{13} – 2)\)

A) \(17\)    B) \(9 + 2\sqrt{13}\)    C) \(11\)    D) \(9\)    E) \(9 – 2\sqrt{13}\)

 

 

Math Placement Part 2 Practice Questions

31. Of the following numbers, which is the greatest?

A) \(\cos(0)\)     B) \(\cos(\frac{\pi}{6})\)     C) \(\cos(\frac{\pi}{4})\)     D)  \(\cos(\frac{\pi}{2})\)    E)  \(\cos(\pi)\)

 

32. In the right triangle shown below, what is the value of \(\tan \theta\)?

A) \(x\)     B) \(x\sqrt{x^2 – 1}\)     C) \(x^2 + 1\)     D)  \(\frac{1}{2}(x^2 – 1)\)    E)  \(\sqrt{x^2 – 1}\)

 

33. If \(7^{2x} = 3\), then \(x = \)

A) \(\frac{3}{14}\)     B) \(\frac{7}{6}\)     C) \(\frac{1}{2}\log_3(7)\)     D)  \(\frac{1}{2}\log_7(3)\)    E)  \(3\log_7(\frac{1}{2})\)

 

34. Which of the following is NOT equal to \(\sin(20^{\circ})\)?

A) \(\cos(70^{\circ})\)     B) \(\sin(160^{\circ})\)     C) \(\sin(380^{\circ})\)     D)  \(\sin(-20^{\circ})\)    E)  \(\sin(-340^{\circ})\)

 

35. The expression \(\frac{1}{2}\ln(4x^6) – 2\ln(x^3)\) is equal to

A) \(\ln(2x^6 – 2x^3)\)     B) \( \ln(2x^3 + x^6)\)     C) \(\ln(2x^3)\)     D)  \(\ln (\frac{2}{x^3})\)    E)  \(-\frac{3}{2}\ln(4x^3)\)

 

36. Find the exact value of \(\displaystyle\sin(\frac{23\pi}{6})\).

A) \(\displaystyle\frac{1}{2}\)     B) \(\displaystyle\frac{-1}{2}\)     C) \\displaystyle\frac{-\sqrt{3}}{2}\)     D)  \(\displaystyle\frac{\sqrt{3}}{2}\)    E)  \(\displaystyle 1\)

 

37. Evaluate \(\displaystyle\log_4 \frac{1}{\sqrt{8}}\).  Give your answer as reduced fraction.

\(\displaystyle\log_4 \frac{1}{\sqrt{8}} =\)           

 

38. Which of the following is equal to \(\sin \theta \tan \theta + \cos \theta\)?

A) \(\tan \theta\)     B) \(\sin \theta\)     C) \(\sec \theta\)     D)  \(\csc \theta\)    E)  \(\cot \theta\)

 

39. What is the radian measure of a \(10^{\circ}\) angle?

A) \(\displaystyle\frac{\pi}{8}\)     B) \(\displaystyle\frac{\pi}{10}\)     C) \(\displaystyle\frac{\pi}{18}\)     D)  \(\displaystyle\frac{\pi}{20}\)    E)  \(\displaystyle\frac{\pi}{36}\)

 

40. If \(\displaystyle f(x) = \frac{2}{x}\), simplify \(\displaystyle \frac{f(1 + h) – f(1)}{h}\) for \(h \neq 0\).

A) \(\displaystyle \frac{2}{h + 1}\)     B) \(\displaystyle \frac{-2}{h + 1}\)     C) \(\displaystyle \frac{2}{h^2 + h}\)     D)  \(\displaystyle \frac{1}{h^2 + h}\)    E)  \(\displaystyle \frac{2h}{h + 1}\)

 

41. A town has an initial population of 300. The population is expected to double every 4 years. Which of the following equations gives the expected population, \(p\), in the town after \(t\) years?

A) \(p = 2(300^{t/4})\)     B) \(p = 300(2^{4t})\)     C) \(p = 300(2^{t/4})\)     D)  \(p = 300(4^{2t})\)    E)  \(p = 300e^{4t}\)

 

42. Suppose that the world’s oil reserves at time \(t = 1\) are 1980 billion barrels, and that these reserves are decreasing at a constant rate. If the reserves at time \(t = 3\) are projected to be 1920 billion barrels, at what time should the reserves be 1710 billion barrels?  Your answer should be an integer.

\(t = \)           

 

43. Simplify \(\displaystyle\frac{x – 3}{\sqrt{x} + \sqrt{3}}\) so that no radicals appear in the denominator.  (Assume that \(x \geq 0\).)

A) \(\displaystyle\sqrt{x} + \sqrt{3}\)     B) \(\displaystyle\sqrt{x} – \sqrt{3}\)     C) \(\displaystyle\frac{\sqrt{x} – \sqrt{3}}{x + 3}\)     D)  \(\displaystyle\frac{\sqrt{x} + \sqrt{3}}{x + 3}\)    E)  \(\displaystyle\frac{\sqrt{x} – \sqrt{3}}{x – 3}\)

 

44. Solve for \(x\):  \(\log_2(x) + \log_2(x – 7) = 3\)

A) \(x = 1\) and \(x = 8\)     B) \(x = -1\) only     C) \(x = 8\) only     D)  \(x = 3\) and \(x = 4\)    E)  \(x = -1\) and \(x = 8\)

 

45. If \(\displaystyle \cot \theta = -\sqrt{7}\) and \(\theta\) lies in quadrant IV, compute \(\displaystyle \sin \theta\).

A) \(\displaystyle \frac{1}{\sqrt{8}}\)     B) \(\displaystyle \sqrt{7}\)     C) \(\displaystyle \frac{1}{\sqrt{7}}\)     D)  \(\displaystyle \frac{-1}{\sqrt{8}}\)    E)  \(\displaystyle \frac{-\sqrt{7}}{\sqrt{8}}\)

 

Part 1 Solutions

1. D
2. A
3. B
4. 51
5. A
6. x = -7, y = -3
7. A
8. B
9. A
10. A
11. D
12. A
13. B
14. B
15. Negative solution: x = -8; Positive solution: x = 9
16. C
17. C
18. B
19. A
20. A
21. E
22. E
23. A
24. A
25. C
26. B
27. A
28. B
29. D
30. D

 

Part 2 Solutions

31. A
32. E
33. D
34. D
35. D
36. B
37. \(\displaystyle \frac{-3}{4}\)
38. C
39. C
40. B
41. C
42. t = 10
43. B
44. C
45. D