CAIE A-Level Further Mathematics 1.5 Polar Coordinates

$r^{5}=4 x y\left(x^{2}-y^{2}\right)$

B1

Uses $r^{2}=x^{2}+y^{2}$.

$r^{5}=4 r^{4} \cos \theta \sin \theta\left(\cos ^{2} \theta-\sin ^{2} \theta\right)$

B1

Uses $x=r \cos \theta$ and $y=r \sin \theta$.

$r=2 \sin 2 \theta \cos 2 \theta$

M1

Applies at least one double angle formula.

$r=\sin 4 \theta$

A1

Applies both double angle formulae, AG.

4

B1

Initial line drawn and one loop in the first quadrant.

B1

Correct shape at extremities.

$\theta=\frac{1}{8} \pi$

B1

States the equation of the line of symmetry.

3

$\frac{1}{2} \int_{0}^{\frac{1}{4} \pi} \sin ^{2} 4 \theta \mathrm{~d} \theta$

M1

Uses $\frac{1}{2} \int r^{2} \mathrm{~d} \theta$ with correct limits.

$=\frac{1}{4} \int_{0}^{\frac{1}{4} \pi} 1-\cos 8 \theta \mathrm{~d} \theta=\frac{1}{4}\left[\theta-\frac{1}{8} \sin 8 \theta\right]_{0}^{\frac{1}{4} \pi}$

M1 A1

Applies double angle formula and integrates.

$=\frac{1}{16} \pi$

A1

4