5 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.

1. It is given that \(a = \frac { 2 } { 3 }\) and \(b = - \frac { 1 } { 2 }\). Show that \(\left( \frac { d x } { d t } \right) ^ { 2 } + \left( \frac { d y } { d t } \right) ^ { 2 } = \frac { 25 } { 4 } \left( t ^ { 2 } + t ^ { - 2 } \right) ^ { 2 }\) and find the exact length of \(C\).
2. It is given instead that \(\mathrm { a } = \mathrm { b } = 0\). Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) when \(t = 1\).
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