show that \(\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\),
obtain the Maclaurin series for \(\mathrm { f } ( x )\) as far as the term in \(x ^ { 3 }\).
A curve has polar equation \(r = \theta + \sin \theta , \theta \geqslant 0\).
By considering \(\frac { \mathrm { d } r } { \mathrm {~d} \theta }\) show that \(r\) increases as \(\theta\) increases.
Sketch the curve for \(0 \leqslant \theta \leqslant 4 \pi\).
You are given that \(\sin \theta \approx \theta\) for small \(\theta\). Find in terms of \(\alpha\) the approximate area bounded by the curve and the lines \(\theta = 0\) and \(\theta = \alpha\), where \(\alpha\) is small.
The infinite series \(C\) and \(S\) are defined as follows.
$$\begin{gathered}
C = a \cos \theta + a ^ { 2 } \cos 2 \theta + a ^ { 3 } \cos 3 \theta + \ldots
S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots
\end{gathered}$$
where \(a\) is a real number and \(| a | < 1\).
By considering \(C + \mathrm { j } S\), show that
$$S = \frac { a \sin \theta } { 1 - 2 a \cos \theta + a ^ { 2 } }$$
Find a corresponding expression for \(C\).
P is one vertex of a regular hexagon in an Argand diagram. The centre of the hexagon is at the origin. P corresponds to the complex number \(\sqrt { 3 } + \mathrm { j }\).
Find, in the form \(x + \mathrm { j } y\), the complex numbers corresponding to the other vertices of the hexagon.
The six complex numbers corresponding to the vertices of the hexagon are squared to form the vertices of a new figure. Find, in the form \(x + \mathrm { j } y\), the vertices of the new figure. Find the area of the new figure.
Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { l l }
6 & - 3
4 & - 1
\end{array} \right)$$
Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
The \(3 \times 3\) matrix \(\mathbf { B }\) has characteristic equation
$$\lambda ^ { 3 } - 4 \lambda ^ { 2 } - 3 \lambda - 10 = 0$$
Show that 5 is an eigenvalue of \(\mathbf { B }\). Show that \(\mathbf { B }\) has no other real eigenvalues.
An eigenvector corresponding to the eigenvalue 5 is \(\left( \begin{array} { r } - 2 1 4 \end{array} \right)\).
Evaluate \(\mathbf { B } \left( \begin{array} { r } - 2 1 4 \end{array} \right)\) and \(\mathbf { B } ^ { 2 } \left( \begin{array} { r } 4 - 2 - 8 \end{array} \right)\).
Solve the equation \(\mathbf { B } \left( \begin{array} { l } x y z \end{array} \right) = \left( \begin{array} { r } - 20 10 40 \end{array} \right)\) for \(x , y , z\).
Show that \(\mathbf { B } ^ { 4 } = 19 \mathbf { B } ^ { 2 } + 22 \mathbf { B } + 40 \mathbf { I }\).
Given that \(\sinh y = x\), show that
$$y = \ln \left( x + \sqrt { 1 + x ^ { 2 } } \right)$$
Differentiate (*) to show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$$
Find \(\int \frac { 1 } { \sqrt { 25 + 4 x ^ { 2 } } } \mathrm {~d} x\), expressing your answer in logarithmic form.
Use integration by substitution with \(2 x = 5 \sinh u\) to show that
$$\int \sqrt { 25 + 4 x ^ { 2 } } \mathrm {~d} x = \frac { 25 } { 4 } \left( \ln \left( \frac { 2 x } { 5 } + \sqrt { 1 + \frac { 4 x ^ { 2 } } { 25 } } \right) + \frac { 2 x } { 5 } \sqrt { 1 + \frac { 4 x ^ { 2 } } { 25 } } \right) + c$$
where \(c\) is an arbitrary constant.
\section*{OCR}