Answer only one of the following two alternatives.

EITHER

State the fifth roots of unity in the form \(\cos \theta+\mathrm{i} \sin \theta\), where \(-\pi\lt \theta \leqslant \pi\).

Simplify
\(\left(x-\left[\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right]\right)\left(x-\left[\cos \frac{2}{5} \pi-i \sin \frac{2}{5} \pi\right]\right) .\)

Hence find the real factors of
\(x^{5}-1\)

Express the six roots of the equation
\(x^{6}-x^{3}+1=0\)
as three conjugate pairs, in the form \(\cos \theta \pm \mathrm{i} \sin \theta\).

Hence find the real factors of
\(x^{6}-x^{3}+1\)

OR
Given that
\(y^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}-6 y^{2} \frac{\mathrm{~d} y}{\mathrm{~d} x}+2 y\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^{2}+3 y^{3}=25 \mathrm{e}^{-2 x}\)
and that \(v=y^{3}\), show that
\(\frac{\mathrm{d}^{2} v}{\mathrm{~d} x^{2}}-6 \frac{\mathrm{~d} v}{\mathrm{~d} x}+9 v=75 \mathrm{e}^{-2 x}\)

Find the particular solution for \(y\) in terms of \(x\), given that when \(x=0, y=2\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=1\).