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%TCIDATA{<META NAME="Title" CONTENT="AMATYC - February 1998">}
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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
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\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
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\lhead{\large Test 2}
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\rhead{\large February 1998 -  page   \ \thepage}
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\begin{document}


\begin{enumerate}
\item Which of the following is NOT a solution for $\dfrac{2}{x+4}\geq -1$?

A) \ $-10$ \ \ \ \ \ \ \ \ B) \ $-6$ \ \ \ \ \ \ \ \ C) \ $-5%
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$ \ \ \ \ \ \ \ \ \ D) \ $-2$ \ \ \ \ \ \ \ \ E) \ $0$

\item The graph of \ $3y^{2}+12y+x+16=0$ \ is

A) \ a parabola, opening right, with vertex at $\left( 4,-2\right) $

B) \ a parabola, opening right, with vertex at $\left( -2,4\right) $

C) \ a parabola, opening left, with vertex at $\left( -4,-2\right) 
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$

D) \ a parabola, opening left, with vertex at $\left( -2,-4\right) $

E) \ none of these

\item Given that $\left[ 
\begin{array}{cc}
2 & 1 \\ 
3 & -1%
\end{array}%
\right] \cdot \left[ 
\begin{array}{cc}
j & -2 \\ 
1 & 4%
\end{array}%
\right] =\left[ 
\begin{array}{cc}
7 & 0 \\ 
k & -10%
\end{array}%
\right] $ \ , find \ $3j+5k$ .

A) \ $11$\ \ \ \ \ \ \ \ B) \ $\dfrac{51}{2}$ \ \ \ \ \ \ \ \ \ C) \ $29$ \
\ \ \ \ \ \ \ \ \ D) \ $49%
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$\ \ \ \ \ \ \ \ \ \ E) \ none of these

\item What would be the approximate dollar value of a straight trail of
pennies from Borough of Manhattan CC in New York to Pasadena City College in
California, where each penny is horizontal and tangent to the previous penny
in the trail?

A) \ $\$$ $200\,000$\ \ \ \ B) \ $\$$ $2\,000\,000%
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$ \ \ \ \ C) \ $\$$ $20\,000\,000$\ \ \ \ \ \ \ D) \ $\$$ $200\,000\,000$\ \
\ \ E) \ $\$$ $2\,000\,000\,000$

\item The mean of four numbers is $107$, the median is $83,$ and the mode is 
$51$. \ Find the range.

A) \ $160%
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$\ \ \ \ \ \ \ \ B) \ $184$ \ \ \ \ \ \ C) \ $187$\ \ \ \ \ \ \ \ \ \ \ D) \ 
$211$ \ \ \ \ \ E) \ $262$

\item Which of the following is NOT equivalent to $\cos \left( \dfrac{3\pi }{%
2}+x\right) $?

A) \ $\sin x$\ \ \ \ \ \ \ \ B) \ $\cos \left( x-\dfrac{\pi }{2}\right) $ \
\ \ \ \ \ \ \ C) \ $\cos \left( \dfrac{\pi }{2}-x\right) $\ \ \ \ \ \ \ \ \
D) \ $\sin \left( 2\pi -x\right) 
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$ \ \ \ \ \ \ \ \ E) \ $\sin \left( \pi -x\right) $

\item How many different scores are possible for an individual taking this
exam? (It's OK te refer back to the instructions!)

A) \ $94$\ \ \ \ \ \ \ \ B) \ $95%
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$ \ \ \ \ \ \ C) \ $96$\ \ \ \ \ \ \ \ \ \ \ D) \ $100$ \ \ \ \ \ E) \ $101$

\item If $f$ is a linear function such that $f\left( 3\right) =0$ \ and \ $%
f\left( 10\right) \not=0,$ \ find \ $\dfrac{f\left( -2\right) +f\left(
0\right) +f\left( 8\right) }{f\left( 1\right) +f\left( 5\right) +f\left(
6\right) }$

A) \ $-1%
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$ \ \ \ \ \ \ \ \ B) \ $0$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{1}{2}$\ \ \ \ \
\ \ \ \ \ \ \ D) \ $1$ \ \ \ \ \ \ \ \ \ \ \ E) \ $3$

\item Six cards each have one letter printed on each side of the card. \ The
table below shows the letters that are on each card. \ (For example, the
first card has an A on one side and an M on the other.)

\begin{tabular}{cccccc}
A & A & A & A & Y & B \\ 
M & T & Y & C & C & T%
\end{tabular}

The six cards are tossed into the air and and fall randomly to the ground. \
What is the probability that the resullting letters can be arranged to spell
AMATYC?

A) \ $\dfrac{1}{64}$ \ \ \ \ \ \ \ \ \ B) \ $\dfrac{1}{32}%
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$ \ \ \ \ \ \ \ \ \ \ C) \ $\dfrac{1}{16}$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $%
\dfrac{1}{8}$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these

\item The distance from the vertex of \ $x+2y^{2}+8y+3=0$ \ to the center of
\ $x^{2}+2y^{2}-4y=3$ \ is

A) \ $\sqrt{26}$ \ \ \ \ \ \ \ \ \ B) \ $\sqrt{34}%
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$ \ \ \ \ \ \ \ \ \ \ C) \ $6$ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $\sqrt{58}$ \ \
\ \ \ \ \ \ \ \ \ \ E) \ $8$

\item Suppose that $P$ and $Q$ are both fourth degree polynomial functions,
each having four distinct real zeros but with three zeros in common. \ Then
the graph of \ $y=\dfrac{P\left( x\right) }{Q\left( x\right) }$ \ lies on

A) \ a line\ \ \ \ \ B) \ a parabola\ \ \ \ \ C) \ a hyperbola$%
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$\ \ \ \ \ \ D) \ a semi-circle\ \ \ \ \ \ E) none of these

\item Given that $\cos B=-\dfrac{2}{7},$ \ $\sin B<0$, \ and \ $0<B<2\pi $,
\ find \ $\cos ^{-1}\left( -\dfrac{2}{7}\right) $ \ in terms of $B$.

A) \ $B$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $2\pi -B%
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$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $-B$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $%
B-\pi $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) \ none of these

\item A \ $3\times 3$ \ magic square uses the integers $1,2,...,9$ \ once
each in such a way that each column, each row, and each diagonal sums to $15 
$. \ Find the value of $n$ for the magic square, a portion of which is shown
below.

\begin{tabular}[t]{|c|c|c|}
\hline
& $1$ &  \\ \hline
$n$ &  &  \\ \hline
&  & $4$ \\ \hline
\end{tabular}

A) \ $2$ \ \ \ \ \ \ \ \ \ \ \ \ B) \ $5$ \ \ \ \ \ \ \ \ \ \ \ \ \ C) \ $6$
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D) \ $7%
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$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E) $8$

\item On a trip to the post office, Jorge spent $\$$ $6.91$ \ on \ $32$-cent
and $23$-cent stamps. \ How many stamps did he buy?

A) \ $22$ \ \ \ \ \ \ \ \ \ B) \ $23%
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$ \ \ \ \ \ \ \ \ \ \ C) \ $24$\ \ \ \ \ \ \ \ \ \ D) \ $25$ \ \ \ \ \ \ \ \
\ \ \ \ E) \ $26$

\item Let \ $\ f\left( x\right) =\left\{ 
\begin{array}{cc}
22-3^{x} & \text{ \ if \ }x\geq 3\text{ \ \ \ \ \ \ \ \ \ } \\ 
-\dfrac{5}{3}x & \text{ \ if \ }-3<x<3 \\ 
2^{-x}-3 & \text{if \ }x\leq -3%
\end{array}%
\right. \allowbreak $. \ Find $f^{-1}\left( 10\right) $.

A) \ $-6.5$ \ \ \ \ \ B) \ $-\ln \left( 6.5\right) $ \ \ \ \ \ \ C) \ $\log
_{3}12$\ \ \ \ \ D) \ $-6$\ \ \ \ \ \ \ \ \ \ E) \ $-\log _{2}13%
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$

\item How many times does the graph of $x^{2}y+xy^{2}+3xy+5x-7y=5$ \
intersect the graph of \ $y=2x$?

A) \ $0$ \ \ \ \ \ \ \ \ \ B) \ $1$ \ \ \ \ \ \ \ \ \ \ C) \ $2$ \ \ \ \ \ \
\ \ \ \ D) \ $3%
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$ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $4$

\item $ABCD$ quadrilateral with $AD=10.0$, \ $CD=10.0,$ \ $\angle
A=118.00^{\circ },$ \ $\angle B=22.00^{\circ }$, \ and \ $\angle
C=130.00^{\circ }$. \ Find $BC$ accurate to three significant figures.

A) \ $18.8$ \ \ \ \ \ \ \ \ \ B) \ $21.8$ \ \ \ \ \ \ \ \ \ \ C) \ $23.4$ \
\ \ \ \ \ \ \ \ D) \ $25.9$\ \ \ \ \ \ \ \ \ \ \ \ E) \ $36.1%
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$

\item For what values of $k$ will the line $x+2y=k$ \ be tangent to the
circle \ $x^{2}+y^{2}=9$?

A) \ $\pm 5\sqrt{2}$ \ \ \ \ \ \ \ \ \ \ B) \ $\pm 4\sqrt{3}$ \ \ \ \ \ \ \
\ \ C) \ $\pm 3\sqrt{5}%
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$ \ \ \ \ \ \ \ \ \ \ \ D) \ $\pm 2\sqrt{6}$\ \ \ \ \ \ \ \ \ \ \ \ \ E) \ $%
\pm 2\sqrt{7}$

\item A circle with area one is surrounded by four squares, each with area
one and each externally tangent to the circle at equally spaced points
around the circle. \ Find the area, to the nearest hundredth of a square
unit, of the circle circumscribing the four squares. \ 

A) \ $8.47%
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$\ \ \ \ \ B) \ $8.51$ \ \ \ \ \ C) \ $8.59$ \ \ \ \ \ D) \ $8.67$\ \ \ \ \
\ E) \ $8.72$

\item What is the minimum value for $n$ such that the probability that there
is a birthday today among a group of $n$ people is greater than or equal to $%
0.5$?

A) \ $23$\ \ \ \ \ B) \ $128$ \ \ \ \ \ C) \ $183$ \ \ \ \ \ D) \ $253%
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$\ \ \ \ \ \ E) \ $282$\pagebreak
\end{enumerate}

Answers

\bigskip

\begin{tabular}{lllll}
1. & C & ~~~~~~~~~ & 11. & C \\ 
2. & C &  & 12. & B \\ 
3. & D &  & 13. & D \\ 
4. & B &  & 14. & B \\ 
5. & A &  & 15. & E \\ 
6. & D &  & 16. & D \\ 
7. & B &  & 17. & E \\ 
8. & A &  & 18. & C \\ 
9. & B &  & 19. & A \\ 
10. & B &  & 20. & D%
\end{tabular}

\bigskip

\end{document}