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 \(x ^ { 4 }\).
Hence express \(\arctan x\) in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\).
Find, in exact form, the value of the following integral.
$$\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$$
A curve has polar equation \(r = \frac { a } { \sqrt { \theta } }\) where \(a > 0\).
Sketch the curve for \(\frac { \pi } { 4 } \leqslant \theta \leqslant 2 \pi\).
State what happens to \(r\) as \(\theta\) tends to zero.
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.
(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 \(n\),
$$\frac { C } { S } = \cot \left( \frac { 1 } { 2 } n \theta \right)$$
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.
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.
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.
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)
Given that \(y = \cosh x\), use the definition of \(\cosh x\) in terms of exponential functions to prove that
$$x = \pm \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right) .$$
Solve the equation
$$\cosh x + \cosh 2 x = 5$$
giving the roots in an exact logarithmic form.
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.
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