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\includegraphics[max width=\textwidth, alt={}, center]{ce0d5faa-9428-4afd-829d-7634c5bd150d-12_446_613_274_760} The diagram shows the curve with parametric equations $$x = k \tan t , \quad y = 3 \sin 2 t - 4 \sin t ,$$ for \(0 < t < \frac { 1 } { 2 } \pi\). It is given that \(k\) is a positive constant. The curve crosses the \(x\)-axis at the point \(P\).

1. Find the value of \(\cos t\) at \(P\), giving your answer as an exact fraction.
2. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(k\) and \(\cos t\).
3. Given that the normal to the curve at \(P\) has gradient \(\frac { 9 } { 10 }\), find the value of \(k\), giving your answer as an exact fraction.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.