By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - ( \sqrt { } 2 ) \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
Hence solve the equation
$$\cos \left( x + 45 ^ { \circ } \right) - ( \sqrt { } 2 ) \sin x = 2$$
for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
Express \(\frac { 1 } { x ^ { 2 } ( 2 x + 1 ) }\) in the form \(\frac { A } { x ^ { 2 } } + \frac { B } { x } + \frac { C } { 2 x + 1 }\).
The variables \(x\) and \(y\) satisfy the differential equation
$$y = x ^ { 2 } ( 2 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$
and \(y = 1\) when \(x = 1\). Solve the differential equation and find the exact value of \(y\) when \(x = 2\). Give your value of \(y\) in a form not involving logarithms.
(a) The complex number \(w\) is such that \(\operatorname { Re } w > 0\) and \(w + 3 w ^ { * } = \mathrm { i } w ^ { 2 }\), where \(w ^ { * }\) denotes the complex conjugate of \(w\). Find \(w\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
(b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z - 2 i | \leqslant 2\) and \(0 \leqslant \arg ( z + 2 ) \leqslant \frac { 1 } { 4 } \pi\). Calculate the greatest value of \(| z |\) for points in this region, giving your answer correct to 2 decimal places.