Use the Maclaurin series for \(\ln ( 1 + x )\) and \(\ln ( 1 - x )\) to obtain the first three non-zero terms in the Maclaurin series for \(\ln \left( \frac { 1 + x } { 1 - x } \right)\). State the range of validity of this series.
Find the value of \(x\) for which \(\frac { 1 + x } { 1 - x } = 3\). Hence find an approximation to \(\ln 3\), giving your answer to three decimal places.
A curve has polar equation \(r = \frac { a } { 1 + \sin \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant. The points on the curve have cartesian coordinates \(x\) and \(y\).
By plotting suitable points, or otherwise, sketch the curve.
Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve.
Obtain the characteristic equation for the matrix \(\mathbf { M }\) where
$$\mathbf { M } = \left( \begin{array} { r r r }
3 & 1 & - 2
0 & - 1 & 0
2 & 0 & 1
\end{array} \right)$$
Hence or otherwise obtain the value of \(\operatorname { det } ( \mathbf { M } )\).
Show that - 1 is an eigenvalue of \(\mathbf { M }\), and show that the other two eigenvalues are not real.
Find an eigenvector corresponding to the eigenvalue - 1 .
Hence or otherwise write down the solution to the following system of equations.
$$\begin{aligned}
3 x + y - 2 z & = - 0.1
- y & = 0.6
2 x + z & = 0.1
\end{aligned}$$
State the Cayley-Hamilton theorem and use it to show that
$$\mathbf { M } ^ { 3 } = 3 \mathbf { M } ^ { 2 } - 3 \mathbf { M } - 7 \mathbf { I }$$
Obtain an expression for \(\mathbf { M } ^ { - 1 }\) in terms of \(\mathbf { M } ^ { 2 } , \mathbf { M }\) and \(\mathbf { I }\).
Find the numerical values of the elements of \(\mathbf { M } ^ { - 1 }\), showing your working.