Let $f : \mathbb { R } \rightarrow [ 0 , \infty )$ be a continuous function such that $$f ( x + y ) = f ( x ) f ( y )$$ for all $x , y \in \mathbb { R }$. Suppose that $f$ is differentiable at $x = 1$ and $$\left. \frac { d f ( x ) } { d x } \right| _ { x = 1 } = 2$$ Then, the value of $f ( 1 ) \log _ { e } f ( 1 )$ is
(A) $e$.
(B) 2.
(C) $\log _ { e } 2$.
(D) 1 .

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function with $f ( 0 ) = 1$ and satisfying the equation
$$f ( x + y ) = f ( x ) f ^ { \prime } ( y ) + f ^ { \prime } ( x ) f ( y ) \text { for all } x , y \in \mathbb { R }$$
Then, the value of $\log _ { e } ( f ( 4 ) )$ is $\_\_\_\_$ .

Let $f ( x ) = a ^ { x }$ $( a > 0 )$ be written as $f ( x ) = f _ { 1 } ( x ) + f _ { 2 } ( x )$, where $f _ { 1 } ( x )$ is an even function and $f _ { 2 } ( x )$ is an odd function. Then $f _ { 1 } ( x + y ) + f _ { 1 } ( x - y )$ equals:
(1) $2 f _ { 1 } ( x ) f _ { 1 } ( y )$
(2) $2 f _ { 1 } ( x + y ) f _ { 1 } ( x - y )$
(3) $2 f _ { 1 } ( x ) f _ { 2 } ( y )$
(4) $2 f _ { 1 } ( x + y ) f _ { 2 } ( x - y )$

Let $\sum _ { k = 1 } ^ { 10 } f ( a + k ) = 16 \left( 2 ^ { 10 } - 1 \right)$, where the function $f$ satisfies $f ( x + y ) = f ( x ) f ( y )$ for all natural numbers $x , y$ and $f ( 1 ) = 2$. Then the natural number ' $a$ ' is:
(1) 3
(2) 16
(3) 4
(4) 2

3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.

Computer Science and Computer Science \& Philosophy applicants should turn to page 14.
The function $f ( x )$ is defined for all real numbers and has the following properties, valid for all $x$ and $y$ :
(A) $\quad f ( x + y ) = f ( x ) f ( y )$.
(B) $\quad \mathrm { d } f / \mathrm { d } x = f ( x )$.
(C) $\quad f ( x ) > 0$.
Throughout this question, these should be the only properties of $f$ that you use; no marks will be awarded for any use of the exponential function.
Let $a = f ( 1 )$.
(i) Show that $f ( 0 ) = 1$.
(ii) Let
$$I = \int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x$$
Show that $I = a - 1$.
(iii) The trapezium rule with $n$ steps is used to produce an estimate $I _ { n }$ for the integral $I$. Show that
$$I _ { n } = \frac { 1 } { 2 n } \left( \frac { b + 1 } { b - 1 } \right) ( a - 1 )$$
where $b = f ( 1 / n )$.
(iv) Given that $I _ { n } \geqslant I$ for all $n$, show that
$$a \leqslant \left( 1 + \frac { 2 } { 2 n - 1 } \right) ^ { n }$$