1
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Given that $\mathrm { f } ( x ) = \arctan x$, write down an expression for $\mathrm { f } ^ { \prime } ( x )$. Assuming that $x$ is small, use a binomial expansion to express $\mathrm { f } ^ { \prime } ( x )$ in ascending powers of $x$ as far as the term in $x ^ { 4 }$.
\item Hence express $\arctan x$ in ascending powers of $x$ as far as the term in $x ^ { 5 }$.
\end{enumerate}\item Find, in exact form, the value of the following integral.

$$\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$$
\item A curve has polar equation $r = \frac { a } { \sqrt { \theta } }$ where $a > 0$.
\begin{enumerate}[label=(\roman*)]
\item Sketch the curve for $\frac { \pi } { 4 } \leqslant \theta \leqslant 2 \pi$.
\item State what happens to $r$ as $\theta$ tends to zero.
\item Find the area of the region enclosed by the part of the curve sketched in part (i) and the lines $\theta = \frac { \pi } { 4 }$ and $\theta = 2 \pi$. Give your answer in an exact simplified form.\\
(a) (i) Express $2 \sin \frac { 1 } { 2 } \theta \left( \sin \frac { 1 } { 2 } \theta - \mathrm { j } \cos \frac { 1 } { 2 } \theta \right)$ in terms of $z$ where $z = \cos \theta + \mathrm { j } \sin \theta$.\\
(ii) The series $C$ and $S$ are defined as follows.

$$\begin{aligned}
C & = 1 - \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \cos n \theta \\
S & = - \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \sin n \theta
\end{aligned}$$

Show that

$$C + \mathrm { j } S = \left\{ - 2 \mathrm { j } \sin \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { j } \sin \frac { 1 } { 2 } \theta \right) \right\} ^ { n } .$$

Hence show that, for even values of $n$,

$$\frac { C } { S } = \cot \left( \frac { 1 } { 2 } n \theta \right)$$

(b) Write the complex number $z = \sqrt { 6 } + \mathrm { j } \sqrt { 2 }$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, expressing $r$ and $\theta$ as simply as possible. Hence find the cube roots of $z$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$.

Show the points representing $z$ and its cube roots on an Argand diagram.
\begin{enumerate}[label=(\roman*)]
\item Find the eigenvalues and eigenvectors of the matrix $\mathbf { M }$, where

$$\mathbf { M } = \left( \begin{array} { l l } 
\frac { 1 } { 2 } & \frac { 1 } { 2 } \\
\frac { 2 } { 3 } & \frac { 1 } { 3 }
\end{array} \right)$$

Hence express $\mathbf { M }$ in the form $\mathbf { P D P } ^ { - 1 }$ where $\mathbf { D }$ is a diagonal matrix.
\item Write down an equation for $\mathbf { M } ^ { n }$ in terms of the matrices $\mathbf { P }$ and $\mathbf { D }$.

Hence obtain expressions for the elements of $\mathbf { M } ^ { n }$.\\
Show that $\mathbf { M } ^ { n }$ tends to a limit as $n$ tends to infinity. Find that limit.
\item Express $\mathbf { M } ^ { - 1 }$ in terms of the matrices $\mathbf { P }$ and $\mathbf { D }$. Hence determine whether or not $\left( \mathbf { M } ^ { - 1 } \right) ^ { n }$ tends to a limit as $n$ tends to infinity.

Section B (18 marks)\\
(i) Given that $y = \cosh x$, use the definition of $\cosh x$ in terms of exponential functions to prove that

$$x = \pm \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right) .$$

(ii) Solve the equation

$$\cosh x + \cosh 2 x = 5$$

giving the roots in an exact logarithmic form.\\
(iii) Sketch the curve with equation $y = \cosh x + \cosh 2 x$. Show on your sketch the line $y = 5$.

Find the area of the finite region bounded by the curve and the line $y = 5$. Give your answer in an exact form that does not involve hyperbolic functions.

\section*{END OF QUESTION PAPER}
\end{enumerate}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI FP2 2016 Q1}}