By sketching a suitable pair of graphs, show that the equation
$$\mathrm { e } ^ { 2 x } = 14 - x ^ { 2 }$$
has exactly two real roots.
Show by calculation that the positive root lies between 1.2 and 1.3.
Show that this root also satisfies the equation
$$x = \frac { 1 } { 2 } \ln \left( 14 - x ^ { 2 } \right) .$$
Use an iteration process based on the equation in part (iii), with a suitable starting value, to find the root correct to 2 decimal places. Give the result of each step of the process to 4 decimal places.
Express \(4 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.