Find stationary/turning points

7 Fig. 7a shows the curve with the parametric equations $$x = 2 \cos \theta , \quad y = \sin 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 } .$$ The curve meets the \(x\)-axis at O and P . Q and R are turning points on the curve. The scales on the axes are the same. \begin{figure}[h] \end{figure}

1. State, with their coordinates, the points on the curve for which \(\theta = - \frac { \pi } { 2 } , \theta = 0\) and \(\theta = \frac { \pi } { 2 }\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Hence find the gradient of the curve when \(\theta = \frac { \pi } { 2 }\), and verify that the two tangents to the curve at the origin meet at right angles.
3. Find the exact coordinates of the turning point Q . When the curve is rotated about the \(x\)-axis, it forms a paperweight shape, as shown in Fig. 7b. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9001b0d0-8d06-43f4-8831-23c0d6aef59d-3_321_385_1758_831} \captionsetup{labelformat=empty} \caption{Fig. 7b}
   \end{figure}
4. Express \(\sin ^ { 2 } \theta\) in terms of \(x\). Hence show that the cartesian equation of the curve is \(y ^ { 2 } = x ^ { 2 } \left( 1 - \frac { 1 } { 4 } x ^ { 2 } \right)\).
5. Find the volume of the paperweight shape.

6 A curve has cartesian equation \(y ^ { 2 } - x ^ { 2 } = 4\).

1. Verify that $$x = t - \frac { 1 } { t } , \quad y = t + \frac { 1 } { t } ,$$ are parametric equations of the curve.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - 1 ) ( t + 1 ) } { t ^ { 2 } + 1 }\). Hence find the coordinates of the stationary points of the curve. Section B (36 marks)

5
\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-05_339_869_262_244} The diagram shows the curve \(C\) with parametric equations
\(x = \frac { 3 } { t } , y = t ^ { 3 } \mathrm { e } ^ { - 2 t }\), where \(t > 0\).
The maximum point on \(C\) is denoted by \(P\).

1. Determine the exact coordinates of \(P\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 6\).
2. Show that the area of \(R\) is given by $$\int _ { a } ^ { b } 3 t \mathrm { e } ^ { - 2 t } \mathrm {~d} t ,$$ where \(a\) and \(b\) are constants to be determined.
3. Hence determine the exact area of \(R\).

12 The diagram shows the curve with parametric equations
\(x = \sin 2 \theta + 2 , y = 2 \cos \theta + \cos 2 \theta\), for \(0 \leqslant \theta < 2 \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-10_771_673_397_239}

1. In this question you must show detailed reasoning. Determine the exact coordinates of all the stationary points on the curve.
2. Write down the equation of the line of symmetry of the curve.