\section*{1. For ALL APPLICANTS.}
For each part of the question on pages $3 - 7$ you will be given five possible answers, just one of which is correct. Indicate for each part $\mathbf { A } - \mathbf { J }$ which answer (a), (b), (c), (d), or (e) you think is correct with a tick $( \sqrt { } )$ in the corresponding column in the table below. Please show any rough working in the space provided between the parts.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
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 & (a) & (b) & (c) & (d) & (e) \\
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A &  &  &  &  &  \\
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B &  &  &  &  &  \\
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C &  &  &  &  &  \\
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D &  &  &  &  &  \\
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E &  &  &  &  &  \\
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F &  &  &  &  &  \\
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G &  &  &  &  &  \\
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H &  &  &  &  &  \\
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I &  &  &  &  &  \\
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J &  &  &  &  &  \\
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\end{tabular}
\end{center}

A. A square has centre ( 3,4 ) and one corner at ( 1,5 ). Another corner is at\\
(a) $( 1,3 )$,\\
(b) $( 5,5 )$,\\
(c) $( 4,2 )$,\\
(d) $( 2,2 )$,\\
(e) $( 5,2 )$.\\
B. What is the value of $\int _ { 0 } ^ { 1 } \left( e ^ { x } - x \right) \left( e ^ { x } + x \right) \mathrm { d } x$ ?\\
(a) $\frac { 3 e ^ { 2 } - 2 } { 6 }$,\\
(b) $\frac { 3 e ^ { 2 } + 2 } { 6 }$,\\
(c) $\frac { 2 e ^ { 2 } - 3 } { 6 }$,\\
(d) $\frac { 3 e ^ { 2 } - 5 } { 6 }$,\\
(e) $\frac { e ^ { 2 } + 3 } { 6 }$.\\
C. The sum

$$1 - 4 + 9 - 16 + \cdots + 99 ^ { 2 } - 100 ^ { 2 }$$

equals\\
(a) - 101\\
(b) - 1000\\
(c) -1111\\
(d) - 4545\\
(e) $\quad - 5050$.\\
D. The largest value achieved by $3 \cos ^ { 2 } x + 2 \sin x + 1$ equals\\
(a) $\frac { 11 } { 5 }$,\\
(b) $\frac { 13 } { 3 }$,\\
(c) $\frac { 12 } { 5 }$,\\
(d) $\frac { 14 } { 9 }$,\\
(e) $\frac { 12 } { 7 }$.\\
E. A line is tangent to the parabola $y = x ^ { 2 }$ at the point $\left( a , a ^ { 2 } \right)$ where $a > 0$. The area of the region bounded by the parabola, the tangent line, and the $x$-axis equals\\
(a) $\frac { a ^ { 2 } } { 3 }$,\\
(b) $\frac { 2 a ^ { 2 } } { 3 }$,\\
(c) $\frac { a ^ { 3 } } { 12 }$,\\
(d) $\frac { 5 a ^ { 3 } } { 6 }$,\\
(e) $\frac { a ^ { 4 } } { 10 }$.\\
F. Which of the following expressions is equal to $\log _ { 10 } ( 10 \times 9 \times 8 \times \cdots \times 2 \times 1 )$ ?\\
(a) $1 + 5 \log _ { 10 } 2 + 4 \log _ { 10 } 6$,\\
(b) $1 + 4 \log _ { 10 } 2 + 2 \log _ { 10 } 6 + \log _ { 10 } 7$,\\
(c) $2 + 2 \log _ { 10 } 2 + 4 \log _ { 10 } 6 + \log _ { 10 } 7$,\\
(d) $2 + 6 \log _ { 10 } 2 + 4 \log _ { 10 } 6 + \log _ { 10 } 7$,\\
(e) $2 + 6 \log _ { 10 } 2 + 4 \log _ { 10 } 6$.\\
G. A cubic has equation $y = x ^ { 3 } + a x ^ { 2 } + b x + c$ and has turning points at $( 1,2 )$ and $( 3 , d )$ for some $d$. What is the value of $d$ ?\\
(a) - 4 ,\\
(b) - 2 ,\\
(c) 0 ,\\
(d) 2 ,\\
(e) 4 .\\
H. The following five graphs are, in some order, plots of $y = f ( x ) , y = g ( x ) , y = h ( x )$, $y = \frac { \mathrm { d } f } { \mathrm {~d} x }$ and $y = \frac { \mathrm { d } g } { \mathrm {~d} x }$; that is, three unknown functions and the derivatives of the first two of those functions. Which graph is a plot of $h ( x )$ ?\\
\includegraphics[max width=\textwidth, alt={}, center]{197f6ad6-3a31-43db-af23-48285e63cd42-06_465_412_1612_230}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(b)}
  \includegraphics[alt={},max width=\textwidth]{197f6ad6-3a31-43db-af23-48285e63cd42-06_410_417_1667_797}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(c)}
  \includegraphics[alt={},max width=\textwidth]{197f6ad6-3a31-43db-af23-48285e63cd42-06_408_415_1667_1363}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(d)}
  \includegraphics[alt={},max width=\textwidth]{197f6ad6-3a31-43db-af23-48285e63cd42-06_417_419_2163_502}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(e)}
  \includegraphics[alt={},max width=\textwidth]{197f6ad6-3a31-43db-af23-48285e63cd42-06_410_417_2174_1071}
\end{center}
\end{figure}

I. In the range $- 90 ^ { \circ } < x < 90 ^ { \circ }$, how many values of $x$ are there for which the sum to infinity

$$\frac { 1 } { \tan x } + \frac { 1 } { \tan ^ { 2 } x } + \frac { 1 } { \tan ^ { 3 } x } + \ldots$$

equals $\tan x$ ?\\
(a) 0\\
(b) 1\\
(c) 2\\
(d) 3\\
(e) 4 .\\
J. Consider a square with side length 2 and centre ( 0,0 ), and a circle with radius $r$ and centre $( 0,0 )$. Let $A ( r )$ be the area of the region that is inside the circle but outside the square, and let $B ( r )$ be the area of the region that is inside the square but outside the circle. Which of the following is a sketch of $A ( r ) + B ( r )$ ?\\
\includegraphics[max width=\textwidth, alt={}, center]{197f6ad6-3a31-43db-af23-48285e63cd42-07_478_472_1535_230}\\
\includegraphics[max width=\textwidth, alt={}, center]{197f6ad6-3a31-43db-af23-48285e63cd42-07_462_499_1537_826}\\
\includegraphics[max width=\textwidth, alt={}, center]{197f6ad6-3a31-43db-af23-48285e63cd42-07_451_464_1539_1462}\\
\includegraphics[max width=\textwidth, alt={}, center]{197f6ad6-3a31-43db-af23-48285e63cd42-07_451_460_2030_461}\\
\includegraphics[max width=\textwidth, alt={}, center]{197f6ad6-3a31-43db-af23-48285e63cd42-07_446_451_2033_1078}