\begin{enumerate} \item Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y = x\) to obtain \(y\) as a function of \(x\). \item (a) Simplify the expression \(\frac { ( x + 3 ) ( x + 9 ) } { x - 1 } - ( 3 x - 5 )\), giving your answer in the form \(\frac { a ( x + b ) ( x + c ) } { x - 1 }\), where \(a , b\) and \(c\) are integers.
(b) Hence, or otherwise, solve the inequality \(\frac { ( x + 3 ) ( x + 9 ) } { x - 1 } > 3 x - 5 \quad\) (4)(Total \(\mathbf { 8 }\) marks) \item (a) Find the general solution of the differential equation \(3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = x ^ { 2 }\)
(b) Find the particular solution for which, at \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\).(6)(Total 14 marks) \item The diagram above shows the curve \(C _ { 1 }\) which has polar equation \(\boldsymbol { r } = \boldsymbol { a } ( \mathbf { 3 } + \mathbf { 2 } \boldsymbol { \operatorname { c o s } } \boldsymbol { \theta } ) , 0 \leq \theta < 2 \pi\) and the circle \(C _ { 2 }\) with equation \(\boldsymbol { r } = \mathbf { 4 } \boldsymbol { a } , 0 \leq \theta < 2 \pi\), where \(a\) is a positive constant.