Mathematics Department Mathematics Placement Examination

(See here for a Postscript version of this document)

Sample Exam Questions: All questions are multiple-choice, one correct answer per question. No credit is lost by guessing. The test is likely to gauge current skills accurately for those students who review the mathematics they have studied. Following are sample questions with labels corresponding to the topics described earlier. To save space, some of the sample questions involve several of the topics. In fact, in Part III of the MPE, unless otherwise noted, each question involves exactly one of the topics. Thus, some of the sample problems are likely to be more difficult than the problems you will encounter.

USE THESE QUESTIONS TO DETERMINE WHICH PART OF THE PLACEMENT EXAM (I-II OR II-III) IS APPROPRIATE FOR YOU! Unless you are completely familiar with the material covered in Part II of the exam, you should NOT take Part III.

Part I

1.

(1)      ${\displaystyle (\frac{1}{2})( \frac{5}{6})( \frac{8}{5}) = }$

$${\rm a)}\; \frac{5}{12} \quad {\rm b)} \; \frac{1}{2} \quad {... ...} \quad {\rm d)} \; \frac{4}{3} \quad {\rm e)} \; \frac{8}{3}$$

2.

(1+7)     x³(2x^-2+4x)=

$$\begin{array}{ll} {\rm a)} \; 2x+4x^4 & \quad {\rm b)}\; 2x^{... ...uad {\rm d)}\; 2x+4x \\ {\rm e)} \; 2x^{-5}+4x^4 & \end{array}$$

3.

(2+4+6)     x²+x-6 has a solution
${\rm a)}\; {\rm between} \, -\! 5\; {\rm and}\, -\!2 \\ {\rm b)} \; {\rm betwe... ...\ {\rm d)} \; {\rm between \; 5\; and\; 7} \\ {\rm e)} \; {\rm more\; than\; 7}$

4.

(3)     The line 2x=3y-5 has slope

$${\rm a)} \; 1.5 \quad {\rm b)} -2.5 \quad {\rm c)} \; 5/3 \quad {\rm d)} \; 2/3 \quad {\rm e)} \; 5/2$$

5.

(5)     Given the equation of two lines: x+2y=5 and 2x-3y=-4, the point of intersection has (x,y) coordinates:

$$\begin{array}{l} {\rm a)} \; (3,1) \qquad {\rm b)}\; (1,2) \q... ...1) \qquad {\rm d)} \; (5,2) \\ {\rm e)} \; (-1,3) \end{array}$$

Answers: c, a, a, d, b

Part II

1.

(1+2)     If F(x)=x^1/2x^-2/3, then F(2⁶)=

$${\rm a)} \; 1 \quad {\rm b)}\; 1/2 \quad {\rm c)} \; 2 \quad {\rm d)}\; 4 \quad {\rm e)} \; 1/8$$

2.

(3+4)     The minimum value of x²-2x+3 is

$${\rm a)}\; 1 \quad {\rm b)} \; 2 \quad {\rm c)} \; 3 \quad {\rm d)} \; 4 \quad {\rm e)} \; 5$$

3.

(5+6+7)      $\ln{(e^3/e^4)}$ is equal to

$$\begin{array}{lll} {\rm a)} \; (3/4) & {\rm b)} \; \ln{3} - \... ...}/\ln{4} \\ {\rm d)}\; e^{3/4} & {\rm e)} \; -1 & \end{array}$$

4.

(8)     The solutions of |x-2| = 5 are

$$\begin{array}{lll} {\rm a)} \; 7 & {\rm b)} \; 3 \; {\rm and}... ... \\ {\rm d)} \; \sqrt{21} & {\rm e)} \; \sqrt{29} \end{array}$$

5.

(9+10)     If $\cos{(\pi/3)}=1/2$, then $\sin{120^{\circ}}$ is

$$\begin{array}{lll} {\rm a)}\; 2\sin{(\pi/3)}\cos{(\pi/3)} & {... ... c)} \; 1/2 \\ {\rm d)} \; -1/2 & {\rm e)} \; -2/3 \end{array}$$

Answers: b, b, e, b, a

Part III

1.

(125.1)
     $\lim_{x\rightarrow 2}(4-x^2)/(x^2-3x+2)=$

$${\rm a)}\; 4 \quad {\rm b)} \; 0 \quad {\rm c)} \; -4 \quad {\rm d)} \; -1 \quad {\rm e)} \; \infty$$

2.

(125.2)     The equation of the tangent line to the graph of y=(x-1)/x at x=1 is

$$\begin{array}{lll} {\rm a)}\; y=-(x-1) & {\rm b)}\; y=(x-1) &... ...y=1/x^2 & {\rm e)} \;{\rm none\; of\; the\; above} \end{array}$$

3.

(125.3+4+9)    x and y are related by the equation $y^2+\sin{(x^2-1)}=4$. The derivative dy/dx at the point x=-1, y=-2 is

$$\begin{array}{l} {\rm a)}\; -2 \qquad {\rm b)} \; 1/2 \qquad ... ...\; \qquad {\rm e)} \; {\rm none\; of\; the\; above}\end{array}$$

4.

(125.5+7+8) Consider the function f(x)= ${\displaystyle \frac{x}{x^2-1}}$. Its graph has the following property:

$$\begin{array}{l} {\rm a)}\; {\rm increasing \; for } \, x> 1 ... ...axis} \\ {\rm e)} \; {\rm none \; of\; the\; above} \end{array}$$

5.

(125.6) The maximum of the function f(x) = x³-6x²+9x+1 on the interval [0,2] is

$${\rm a)}\; 1 \quad {\rm b)} \; 3 \quad {\rm c)} \; 5 \quad {\rm d)} \; 7 \quad {\rm e)} \; 9$$

6.

(125.10+11) The derivative of $e^x \ln(x)$ is

$$\begin{array}{lll} {\rm a)} \; e^x/x & {\rm b)} 1 & {\rm c)} ... ...\ln(x) \\ {\rm d)} e^x \ln(x) + e^x/x & {\rm e)} x \end{array}$$

7.

(131.1)      ${\displaystyle \frac{d}{dx}\int_0 ^x \sin{(t^2)} dt}$ is

$$\begin{array}{lll} {\rm a)}\; \sin{t^2} & {\rm b)} \; \sin{x^... ...\rm d)} \; \cos{x^2} -1 & {\rm e)} \; -\cos{x^2} +1 \end{array}$$

8.

(131.2)      ${\displaystyle \int_0^2 [\frac{d}{dt} \cos{(t^2)}] dt}$ equals

$$\begin{array}{ll} {\rm a)}\; \cos 4 & {\rm b)} \; (\cos 4) -1... ...& {\rm d)} \; -\sin 4 \\ {\rm e)} \; -(\sin 4) + 1 \end{array}$$

9.

(131.3+6) The area of the triangle with vertices (0,0), (1,3), and (2,2) is
${\rm a)} \; 1 \quad {\rm b)}\; 1.5 \quad {\rm c)} \; 2 \quad {\rm d)}\; 2.5 \\ \quad {\rm e)} \; 3$

10.

(131.4) $\int (3x^2 - 2x) dx$ equals

$$\begin{array}{ll} {\rm a)}\; x^3-x^2+C & {\rm b)} \; 6x-2 + C... ...2+C & {\rm d)} \; 3x -2 + C \\ {\rm e)} \; x^6-x^2 \end{array}$$

11.

(131.5+7)     The area under the graph of y=1/x from x=1 to x=2 is

${\rm a)} \; \ln{2} \quad {\rm b)}\; \ln{1} \quad {\rm c)} \; -\frac{1}{4}+1 \quad {\rm d)}\; \frac{1}{4}-1 \\ \quad {\rm e)} \; \frac{1}{2}$

12.

(132.1+6)      ${\displaystyle \int x \cos x dx =}$

$$\begin{array}{ll} {\rm a)} \; \sin x +C & {\rm b)}\; x\sin x ... ... +C \\ {\rm e)} \; {\rm none\; of \; the\; above} & \end{array}$$

13.

(132.3)     What is the volume of the solid generated by revolving about the x-axis the region under the graph of 3x-x² from x=0 to x=3? ${\rm a)} \; 2\pi {\displaystyle \int_0 ^3 (3x-x^2)dx} \\ \quad {\rm b)} \; {\d... ...tyle \int_0 ^3 (3x-x^2)^2 dx} \\ {\rm e)} \; {\rm none \; of \; the \; above}$

14.

(132.2+5) A force F(x) = e^x moves an object from x = 1to x = 2. The work done by this force is

$$\begin{array}{lll} {\rm a)}\; e^2 & {\rm b)} \; e^2 - e & {\r... ... e - e^2 \\ \\ {\rm d)} \; kx^2/2 & {\rm e)} \; 0 \end{array}$$

15.

(132.4+6)    A particle moves so that its position at t=0 is 1 and its velocity at time t is $t\cos{t}$. Then its position at time $t=\pi/2$ is

$${\rm a)} \; \frac{\pi}{2} \quad {\rm b)} \; \frac{\pi}{2} +1 ... ...frac{\pi ^2}{2} \quad {\rm e)}\; {\rm none\; of\; the\; above}$$

16.

(132.7)      ${\displaystyle \int \frac{xdx}{1+4x^2}=}$

$$\begin{array}{ll} {\rm a)} \; \frac{1}{8}\ln{(1+4x^2)} +C & {... ...+C\\ {\rm e)} \; {\rm none \; of \; the \; above} & \end{array}$$

17.

(127.1) For which real values of p does $\int_0^1 dx/x^p$converge?
${\rm a)} \; \{p: p > 0\} \\ {\rm b)} \; \{p: p > 1\} \\ {\rm c)} \; \{p: p < 0\} \\ {\rm d)} \; \{p: p < 1\} \\ {\rm e)} \; {\rm all} \; p$

18.

(127.2)      $\lim_{x\rightarrow 0}{\rm sin}^{2}x/(1-\cos{x})=$

$${\rm a)}\; 1 \quad {\rm b)} \; 1/2 \quad {\rm c)} \; -1/2 \quad {\rm d)} \; 2 \quad {\rm e)} \; 0$$

19.

(127.3+5)      ${\displaystyle \sum _{n=0}^{\infty} \frac{x^n}{3^{2n}}} =$

$$\begin{array}{lll} {\rm a)} \; \frac{1}{1-x/9} & \quad {\rm b... ...} \\ {\rm d)}\; \infty & \quad {\rm e)} \; 1-x/9 & \end{array}$$

20.

(127.4)     The coefficient of x⁴ in the Maclaurin series for f(x)=e^-x/2 is

$$\begin{array}{lll} {\rm a)} \; -1/24 & \quad {\rm b)} \; 1/24... ... \\ {\rm d)}\; -1/384 & \quad {\rm e)} \; 1/384 & \end{array}$$

21.

(127.6)     The complex number i has one of its square roots equal to

$$\begin{array}{lll} {\rm a)}\; 1+i & {\rm b)} \; 1-i & {\rm c... ...\ {\rm d)} \; 2(1+i) & {\rm e)} \; (1+i)/\sqrt{2} \end{array}$$

22.

(127.7)     The volume V(t) of some gas expands at a rate that is twice V(t), in cubic meters per second. At time t=10 seconds, its volume is known to be 1,000 cubic meters. What is the volume at t=5 seconds in cubic meter ($e\sim 2.7$, $e^2\sim 7.4$)?

$$\begin{array}{ll} {\rm a)}\; {\rm less\; than} \, 1 & {\rm b)... ...30} \\ {\rm e)} \; {\rm between \; 30\; and\; 40} \end{array}$$

23.

(127,8+9) A function y satisfies ${{dy}\over{dx}} = 2y/x$ and y(1) = 3. What is y(2)? ${\rm a)} \; 3 \quad {\rm b)}\; 6 \quad {\rm c)} \; 12 \quad {\rm d)}\; e^2 \quad {\rm e)} \; e^3$

Answers: c, b, d, b, c, d, b, b, c, a, a, d, c, b, a, a, d, d, a, e, e, a, c