Two unknowns from two conditions

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), and that the remainder is 35 when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) and show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 3 x ^ { 2 } + b x + 12$$ where \(a\) and \(b\) are constants. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\). It is also given that the remainder is 18 when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values,
   (a) show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root,
   (b) solve the equation \(\mathrm { p } ( \sec y ) = 0\) for \(- 180 ^ { \circ } < y < 180 ^ { \circ }\).