1. Simplify: $4^a \cdot 8^{-a}$
A. $\frac{1}{2^a}$
B. $2^a$
C. $1$
D. $32^{-a^2}$
E. nota
2. If $b=\frac{c d-a}{c-1}$ and $d=a c^{n-1,} c \neq 1$. Express $b$ in terms af a, $c$, and $n$ only.
A. $b=a\left(\frac{c^{n}-1}{c-1}\right)$
B. $b=a\left(\frac{c^{2 n-1}-1}{c-1}\right)$
C. $b=a\left(\frac{c^n}{c-1}\right)$
D. $b=a\left(c^{n-1}-1\right)$
E. nota
3. An open $b a x$ is made by cutting a 4-inch square from each corner of a square piece of tin. If the volume of the box must be 900 cubic inches, what must be the length of one side of this square of tin.
A. 15 inches
B. 19 inches
C. 23 inches
D. 34 inches
E. nota
4. Let $P$ be a polynomial function such that, for all real $x$;
$P\left(x^2-1\right)=x^4+5 x^2+3$
For. al1 real $x, P\left(x^2+1\right)$ is:
A. $x^4+9 x^2+17$
B. $x^4+x^2-3$
C. $x^4+9 x^2+13$
D. $x^4+x^2+3$
E. nota
5. If $f(x)=1 / x$ and $g(x)=\left(x^2+x-6\right)^{-1}$, what is the domain of $f[g(x)]$ if $x \in \mathbb{R} ?$
A. $\mathbb{R}$
B. $\mathbb{R}, x\neq 0$
C. $\mathbb{R}, x\neq 0,3$
D. $\mathbb{R}, x\neq -3,0,2$
E. nota
6. Find the 4th term of the expansion of : $(x-2 y)^{12}$
A. $-5280 x^7 y^3$
B. $-1760 x^9 y^3$
C. $-1320 x^9 y^3$
D. 660 $x^9 y^3$
E. nota
7. Find the a 33 element of the adjoint of $A$ if $A=\left[\begin{array}{ccc}1 & 3 & 1 \\ 2 & 2 & 2 \\ 1 & 3 & 3\end{array}\right]$
A. $-1 / 8$
B. $0$
C. $1 / 2$
D. $-4$
E. nota
8. Evaluate $i^{434}-i^{431}$
A. $0$
B. $1+i$
C. $-1-i$
D. $-1+i$
E. nota
9. Find the slope of the line perpendicular to the line passing though $(1, 3)$ and $(-2,7)$.
A. $4 / 3$
B. $-4 / 3$
C. $3 / 4$
D. $-3 / 4$
E. nota
10. If the partial sums of an infinite series are recursively defined as follows:
$
S_1=1,S_{n+1}=S_n+(1 / 3)^n \text { for } n=1,2,3, \ldots .
$
Find the number to which this series converges.
A. $2 / 3$
B. 1
C. $3 / 2$
D. does not exist
E. nota
11. What is the sum of the first 50 positive odd integers?
A. 1275
B. 2475
C. 2500
D. 5000
E. nota
12. How many distinguishable permutations of 6 objects taken 2 at a time are possible if 3 of the objects are identical?
A. 12
B. 13
C. 20
D. 30
E. nota
13. Which of the following statements is always true?
A. For any real numbers $a, b$, and $x$, except $x \neq-2$, and $x \neq-2(a+b)$ then: $\frac{3(a+b)}{x+2(a+b)}=\frac{3}{x+2}$
B. For any real number $x, \sqrt{x^2}=x$
C. If $a$ and $b$ are real numbers $a^2 \neq b^2, a \neq b$, then: $\frac{a+b}{a-b}=-\frac{(a+b)^2}{b^2-a^2}$
D. If $x$ is a real number, then $-x$ is negative
E. nota
14. If i is a root of the equation $x^4-2 x^3-2 x-1=0$, find the other roots.
A. $\{-i,~ 2,~ -2\}$
B. $\{-i,~ \sqrt{2},~ -\sqrt{2}\}$
C. $\{-i,~ 1+\sqrt{2},~ 1-\sqrt{2}\}$
D. $\{-i,~ 2+i,~ 2-i\}$
E. nota
15. If $n !=n\cdot(n-1)\cdot(n-2)\cdots (1)$, in what digit does $794 !$ end?
A. 1
B. 2
C. 4
D. 6
E. nota
16. If two geometric means were inserted between $128$ and $-2$, their sum could be:
A. $-24$
B. $24$
C. $12$
D. $-12$
E. nota
17. Solve for $x$ if: $x^{\log x} =\frac{x^4}{10000}$
A. 1
B. 10
C. 100
D. 1000
E. nota
16. What is the area bounded by the graph of $9 x^2+16 y^2=144$ ?
A. $12 \pi$
B. $16 \pi$
C. $25 \pi$
D. $144 \pi$
E. nota
19. During half of a trip a car travels at $70~\mathrm{mph}$. After receiving a speeding ticket, it travels the other half at $55~\mathrm{mph}$. What is the average speed of the car for the entire trip?
A. $61.6$
B. $62.5$
C. $63.6$
D. $66$
E. nota
20. A drawer contains 9 red socks and six blue socks. What is the probability that if two socks are picked (without looking) from the drawer, both of the socks will be red?
A. $8 / 25$
B. $3 / 5$
C. $12 / 35$
D. $3 / 7$
E. nota
21. What is the radius of the circle defined by the equation:
$
4 x^2-8 x+4 y^2+4 y-27=0
$
A. 8
B. 4
c. $4 \sqrt{2}$
D. $2 \sqrt{2}$
E. nota
22. A brine contains 54 pounds of water and 6 pounds of salt. Salt is added to obtain a $20 \%$ salt solution. How many pounds of salt were added?
A. $7 \frac{1}{2}$
B. 6
C. 3
D. 2
E. nota
23. If $\log _4|2 x+5|-\log _4|3 x+1|=\frac{1}{2}$, solve for $x$
A. 0
B. $-\frac{1}{3}$
C. $-\frac{5}{2}$
D. $\frac{3}{4}$
E. nota
24. If $f(x)=2 x^4-3 x^2+A x+B$, find values for $A$ and $B$ such that $P(-2)=24$ and $P(2)=20$.
A. $A=O, B=0$
E. $A=O, B=-2$
C. $A=-1, B=2$
D. $A=1, E=-2$ E. not
25. What is the first term of an arithmetic sequence whose seventh term is 3 and whose eleventh term is 1 ?
A. 6 B. $7 \frac{1}{2} \quad$ C. $6 \frac{1}{2} \quad$ D. $6 \frac{1}{4} \quad$ E. nota
26. Find the sum of all the roots of the equation: $x^5-8 x^3-6 x^2+7 x+6=0$
A. $-8$
B. 0
C. 6
D. 8
E. nota
27. If $A=\left[\begin{array}{rr}6 & -4 \\ -2 & 0\end{array}\right]$, and $I=$ identity matrix $x$, find $x$ such that $A X=I$
A. $\left[\begin{array}{cc}-3 / 4 & -1 / 4 \\ 1 / 2 & 0\end{array}\right]$
B. $\left[\begin{array}{cc}0 & -1 / 2 \\ 1 / 4 & 3 / 4\end{array}\right]$
C. $\left[\begin{array}{rr}0 & -4 \\ 2 & 6\end{array}\right]$
D. $\left[\begin{array}{rr}-6 & -2 \\ 4 & 0\end{array}\right] \quad$ E. not
28. In how many ways can the letters in the word EXCELLENT be arranged?
A. 15,120
B. 30,240
C. 60,480
D. 362,880 E. nota
29. If $U=\{1,2,3,4,5,6,7,8,9,10\}, A=\{1,3,5,7,9\}$, $\mathrm{B}=\{2,4,6,8,10\}$, and $\mathrm{C}=\{1,4,7,10\}$, find:
$(A^{\prime} \cap C) \cup(B^{\prime} \cap C)$
A. $A^{\prime}$
B. $B \cap \cap C$
C. $B \cap C^{\prime}$
D. C
E. nota
30. $\frac{5+2 i}{6-i}=?$ where $i=\sqrt{-1}$
A. $\frac{32-7 i}{37}$
B. $\frac{32+17 i}{37}$
c. $\frac{32+17 i}{35}$
D. $\frac{28+17 i}{37}$
E. nota