2013 NJC P1 P8

The curve \(C\) has parametric equations

\(x={{t}^{3}}-t,y={{\mathrm{e}}^{t}},\mathrm{for }t\ge -1.\)

1. Sketch \(C\), labelling all axial intercept(s).[2]
2. Find the equation of the tangent to the curve when \(t=1\), leaving your answer in terms of \(e\).[3]
3. The normal to the curve \(C\) at point \(P\left( {{p}^{3}}-p,{{\mathrm{e}}^{p}} \right)\) is parallel to the \(x\)-axis and intersects the \(y\)-axis at point \(S\left( 0,s \right)\), where\(s<1\). Find the exact value of \(p\).[2]
4. Find the area of the region bounded by the curve \(C\) and the \(y\)-axis.[3]