11.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-30_686_707_205_680}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation
$$y = 12 \sin x$$
where \(x\) is measured in radians.
The point \(P\) shown in Figure 4 is a maximum point on \(C _ { 1 }\)
- Find the coordinates of \(P\).
The curve \(C _ { 2 }\) has equation
$$y = 12 \sin x + k$$
where \(k\) is a constant.
Given that the maximum value of \(y\) on \(C _ { 2 }\) is 3 - find the coordinates of the minimum point on \(C _ { 2 }\) which has the smallest positive \(x\) coordinate.
The curve \(C _ { 3 }\) has equation
$$y = 12 \sin ( x + B )$$
where \(B\) is a positive constant.
Given that \(\left( \frac { \pi } { 4 } , A \right)\), where \(A\) is a constant, is the minimum point on \(C _ { 3 }\) which has the smallest positive \(x\) coordinate, - find
- the value of \(A\),
- the smallest possible value of \(B\).