Show that the curve with Cartesian equation
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 5 } { 2 } } = 4 x y \left( x ^ { 2 } - y ^ { 2 } \right)$$
has polar equation \(r = \sin 4 \theta\).
The curve \(C\) has polar equation \(r = \sin 4 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
Sketch \(C\) and state the equation of the line of symmetry.
Find the exact value of the area of the region enclosed by \(C\).
Using the identity \(\sin 4 \theta \equiv 4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta\), find the maximum distance of \(C\) from the line \(\theta = \frac { 1 } { 2 } \pi\). Give your answer correct to 2 decimal places.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
7 (a) Show that the curve with Cartesian equation
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 5 } { 2 } } = 4 x y \left( x ^ { 2 } - y ^ { 2 } \right)$$
has polar equation $r = \sin 4 \theta$.\\
The curve $C$ has polar equation $r = \sin 4 \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$.\\
(b) Sketch $C$ and state the equation of the line of symmetry.\\
(c) Find the exact value of the area of the region enclosed by $C$.\\
(d) Using the identity $\sin 4 \theta \equiv 4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta$, find the maximum distance of $C$ from the line $\theta = \frac { 1 } { 2 } \pi$. Give your answer correct to 2 decimal places.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q7}}