6 Fig. 6 shows part of the curve \(\sin 2 y = x - 1\). P is the point with coordinates \(\left( 1.5 , \frac { 1 } { 12 } \pi \right)\) on the curve.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414a6b7f-cd96-4fa0-9521-ebe500bab375-2_458_691_1610_687}
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\caption{Fig. 6}
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- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).
Hence find the exact gradient of the curve \(\sin 2 y = x - 1\) at the point P .
The part of the curve shown is the image of the curve \(y = \arcsin x\) under a sequence of two geometrical transformations.
- Find \(y\) in terms of \(x\) for the curve \(\sin 2 y = x - 1\).
Hence describe fully the sequence of transformations.
\(7 \quad\) You are given that \(n\) is a positive integer.
By expressing \(x ^ { 2 n } - 1\) as a product of two factors, prove that \(2 ^ { 2 n } - 1\) is divisible by 3 .
Section B (36 marks)