OR

A curve \(C\) has parametric equations

\(x=1-3t^2,\qquad y=t(1-3t^2),\qquad 0\leqslant t\leqslant \frac{1}{\sqrt3}.\)

Show that

\(\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2=(1+9t^2)^2.\)

Hence find (i) the arc length of \(C\), and (ii) the surface area generated when \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis.

Use the fact that \(t=\dfrac{y}{x}\) to find a cartesian equation of \(C\). Hence show that the polar equation of \(C\) is \(r=\sec\theta(1-3\tan^2\theta)\), and state the domain of \(\theta\).

Find the area of the region enclosed between \(C\) and the initial line.