Answer only one of the following two alternatives.
EITHER
Let \(\omega=\cos \frac{1}{5} \pi+\mathrm{i} \sin \frac{1}{5} \pi\). Show that \(\omega^{5}+1=0\) and deduce that
\(\omega^{4}-\omega^{3}+\omega^{2}-\omega=-1\)

Show further that
\(\omega-\omega^{4}=2 \cos \frac{1}{5} \pi \quad \text { and } \quad \omega^{3}-\omega^{2}=2 \cos \frac{3}{5} \pi .\)

Hence find the values of
\(\cos \frac{1}{5} \pi+\cos \frac{3}{5} \pi \quad \text { and } \quad \cos \frac{1}{5} \pi \cos \frac{3}{5} \pi\)

Find a quadratic equation having roots \(\cos \frac{1}{5} \pi\) and \(\cos \frac{3}{5} \pi\) and deduce the exact value of \(\cos \frac{1}{5} \pi\).

OR

Given that
\(x^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+4 x(1+x) \frac{\mathrm{d} y}{\mathrm{~d} x}+2\left(1+4 x+2 x^{2}\right) y=8 x^{2}\)
and that \(x^{2} y=z\), show that
\(\frac{\mathrm{d}^{2} z}{\mathrm{~d} x^{2}}+4 \frac{\mathrm{~d} z}{\mathrm{~d} x}+4 z=8 x^{2} .\)

Find the general solution for \(y\) in terms of \(x\).

Describe the behaviour of \(y\) as \(x \rightarrow \infty\).