The curve \(C\) has parametric equations
\[x=3 t+2 t^{-1}+a t^{3}, \quad y=4 t-\frac{3}{2} t^{-1}+b t^{3}, \quad \text { for } 1 \leqslant t \leqslant 2\]
where \(a\) and \(b\) are constants.
(a) It is given that \(a=\frac{2}{3}\) and \(b=-\frac{1}{2}\).

Show that \(\left(\frac{\mathrm{d} x}{\mathrm{~d} t}\right)^{2}+\left(\frac{\mathrm{d} y}{\mathrm{~d} t}\right)^{2}=\frac{25}{4}\left(t^{2}+t^{-2}\right)^{2}\) and find the exact length of \(C\).

(b) It is given instead that \(a=b=0\).

Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) when \(t=1\).