2. (a) You are given that $$\frac { 1 } { ( x - 1 ) ( x - 2 ) } = \frac { A } { x - 2 } + \frac { B } { x - 1 }$$ where $A$ and $B$ are constants. Find the values of $A$ and $B$. (b) Simplify $$\frac { 1 } { ( x - 1 ) ^ { n + 1 } ( x - 2 ) } - \frac { 1 } { ( x - 1 ) ^ { n } ( x - 2 ) }$$ (c) You are given that $$\frac { 1 } { ( x - 1 ) ^ { n } ( x - 2 ) } = \frac { A _ { 0 } } { x - 2 } + \sum _ { i = 1 } ^ { n } \frac { A _ { i } } { ( x - 1 ) ^ { i } }$$ where $A _ { 0 } , A _ { 1 } , A _ { 2 } , \ldots$ are constants. Using your answers to (a) and (b), or otherwise, find the values of these constants.
3. Let $P$ and $Q$ be the points with co-ordinates $( 7,1 )$ and $( 11,2 )$. (a) The mirror image of the point $P$ in the $x$-axis is the point $R$ with coordinates $( 7 , - 1 )$. Mark the points $P , Q$ and $R$ on the grid provided. (b) Consider paths from $P$ to $Q$ each of which consists of two straight line segments $P X$ and $X Q$ where $X$ is a point on the $x$-axis. Find the length of the shortest such path, giving clear reasoning for your answer. (You may refer to the diagram to help your explanation, if you wish.) (c) Sketch in the line $\ell$ with equation $y = x$. Find the co-ordinates of $S$, the mirror image in the line $\ell$ of the point $Q$, and mark in the point $S$. (d) Consider paths from $P$ to $Q$ each of which consists of three straight line segments $P Y , Y Z$ and $Z Q$, where $Y$ is on the $x$-axis and $Z$ is on the line $\ell$. Find the length of the shortest such path, giving clear reasoning for your answer. [Figure]
4. (a) Find $\frac { d y } { d x }$ for each of the functions $$\begin{aligned}
& y = \sin ( \ln x ) \\
& y = x \sin ( \ln x ) \\
& y = x \cos ( \ln x )
\end{aligned}$$ (b) Sketch the following curves using the axes provided on the next page: (i) $y = \ln x$, for $1 \leqslant x \leqslant e ^ { \pi }$, (ii) $y = \sin ( \ln x )$, for $1 \leqslant x \leqslant e ^ { \pi }$. (c) Evaluate $$\int _ { 1 } ^ { e ^ { \pi } } \sin ( \ln x ) d x$$ [Figure]
5. With an unlimited supply of black pebbles and white pebbles, there are 4 ways in which you can put two of them in a row: $B B , B W , W B$ and $W W$. (a) Write down the 8 different ways in which you can put three of the pebbles in a row. In how many different ways can you put $N$ of the pebbles in a row? Suppose now that you are not allowed to put black pebbles next to each other: with two pebbles there are now only 3 ways of putting them in a row, because $B B$ is forbidden. (b) Write down the 5 different ways that are still allowed for three pebbles. Now let $r _ { N }$ be the number of possible arrangements for $N$ pebbles in a row, still under the no-two-black-together restriction, so that $r _ { 2 } = 3$ and $r _ { 3 } = 5$. (c) Show that for $N \geqslant 4$ we have $r _ { N } = r _ { N - 1 } + r _ { N - 2 }$. [Hint: Consider separately the two possible cases for the colour of the last pebble.] Finally, suppose that we impose the further restriction that the first pebble and the last pebble cannot both be black. For $N$ pebbles call the number of such arrangements $w _ { N }$, so that for example $w _ { 3 } = 4$ (although $r _ { 3 } = 5$, the arrangement $B W B$ is now forbidden). (d) When $N \geqslant 5$, write down a formula for $w _ { N }$ in terms of the numbers $r _ { i }$, and explain why it is correct.