Question 1 is a multiple choice question for which marks are given solely for the correct answers. Answer Question 1 on the grid on Page 2. Write your answers to Questions $\mathbf { 2 , 3 , 4 , 5 }$ in the space provided, continuing on the back of this booklet if necessary.

THE USE OF CALCULATORS OR FORMULA SHEETS IS PROHIBITED.

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  \item For each part of the question on pages 3-7 you will be given four possible answers, just one of which is correct. Indicate for each part $\mathbf { A } - \mathbf { J }$ which answer (a), (b), (c), or (d) you think is correct with a tick $( \checkmark )$ in the corresponding column in the table below. You may use the spaces between the parts for rough working.
\end{enumerate}

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A. The number of solutions of the equation

$$x ^ { 3 } + a x ^ { 2 } - x - 2 = 0$$

for which $x > 0$ is\\
(a) 1\\
(b) 2\\
(c) 3\\
(d) dependent on the value of $a$.\\
B. Of the following three alleged algebraic identities, at least one is wrong.\\
(i) $y z ( z - y ) + z x ( x - z ) + x y ( y - x ) = ( z - y ) ( x - z ) ( y - x )$\\
(ii) $y z ( z - y ) + z x ( x - z ) + x y ( y - x ) = ( z - y ) ( z - x ) ( y - x )$\\
(iii) $y z ( z + y ) + z x ( z + x ) + x y ( y + x ) = ( z + y ) ( z + x ) ( y + x )$.

Which of the following statements is correct?\\
(a) Only identity (i) is right\\
(b) Only identity (ii) is right\\
(c) Identities (ii) and (iii) are right\\
(d) All these identities are wrong.\\
C. A child is presented with the following lettered tiles: M A M M A L. The number of different "words" he can make using all six tiles is\\
(a) 6\\
(b) 30\\
(c) 60\\
(d) 120 .\\
D. Let $f ( x )$ be the function $\mathrm { e } ^ { \mathrm { e } ^ { e ^ { x } } }$. The value of $f ^ { \prime } ( x )$ when $x = \ln 3$ is which of the following?\\
(a) $3 e ^ { e ^ { 3 } }$\\
(b) $3 e ^ { e ^ { 3 } + 3 }$\\
(c) $e ^ { 3 e + e ^ { 3 } }$\\
(d) $9 e ^ { e ^ { 3 } + 1 }$.\\
E. Which of the following integrals has the greatest value?\\
(a) $\int _ { 0 } ^ { \pi / 2 } \sin ^ { 2 } x \cos x d x$\\
(b) $\int _ { 0 } ^ { \pi } \sin ^ { 2 } x \cos x d x$\\
(c) $\int _ { 0 } ^ { \pi / 2 } \sin x \cos ^ { 2 } x d x$\\
(d) $\int _ { 0 } ^ { \pi / 2 } \sin 2 x \cos x d x$.\\
F. Observe that $2 ^ { 3 } = 8,2 ^ { 5 } = 32,3 ^ { 2 } = 9$ and $3 ^ { 3 } = 27$. From these facts, we can deduce that $\log _ { 2 } 3$, the logarithm of 3 to base 2 , is\\
(a) between $1 \frac { 1 } { 3 }$ and $1 \frac { 1 } { 2 }$\\
(b) between $1 \frac { 1 } { 2 }$ and $1 \frac { 2 } { 3 }$\\
(c) between $1 \frac { 2 } { 3 }$ and 2\\
(d) none of the above.\\
G. The figure shows a regular hexagon with its circumscribed and inscribed circles. What is the ratio of the area of the two circles?\\
\includegraphics[max width=\textwidth, alt={}, center]{455b2f35-67b1-4743-8350-741ca5c51568-06_457_456_494_771}\\
(a) $4 : 3$\\
(b) $6 : 5$\\
(c) $7 : 5$\\
(d) $\sqrt { 3 } : 2$\\
H. Aris, Boris, Clarice and Doris have to decide who will do the washing up. They decide to throw a fair 6 -sided die: if it lands showing a 5 or 6 , Aris will wash up; otherwise they throw again. The second time, if the result is a 5 or 6 , Boris will wash up; otherwise they throw one last time. The final time, if the result is a 5 or 6 , Clarice washes up, and otherwise it's Doris. (Of course, this is not a fair procedure!) Of the four, who is second most likely to do the washing up?\\
(a) Aris\\
(b) Boris\\
(c) Clarice\\
(d) Doris.\\
I. The fixed positive integers $a , b , c , d$ are such that exactly two of the following four statements are valid:\\
(i) $a \leqslant b < c \leqslant d$\\
(ii) $a + b = c + d$\\
(iii) $a = c$ and $b = d$\\
(iv) $a d = b c$.

You are also told that (ii) and (iv) is not the pair of valid statements. Which of the following must be the pair of valid statements?\\
(a) (i) and (ii)\\
(b) (i) and (iii)\\
(c) (i) and (iv)\\
(d) (iii) and (iv).\\
J. Just one of the following expressions is equal to $\sin 5 \alpha$ for all values of $\alpha$. Which one is it?\\
(a) $5 \sin \alpha - 20 \sin ^ { 3 } \alpha + 16 \sin ^ { 5 } \alpha$\\
(b) $5 \sin \alpha - 20 \sin ^ { 3 } \alpha + 14 \sin ^ { 5 } \alpha$\\
(c) $5 \sin \alpha - 10 \sin ^ { 2 } \alpha + 10 \sin ^ { 3 } \alpha - 5 \sin ^ { 4 } \alpha + \sin ^ { 5 } \alpha$\\
(d) $\sin \alpha - 5 \sin ^ { 2 } \alpha + 10 \sin ^ { 3 } \alpha - 10 \sin ^ { 4 } \alpha + 5 \sin ^ { 5 } \alpha$.\\