Hint
Angles are in radians in derivatives and area integrals. Maximise cos θ over a full turn.
Worked solution
(a)
\(\displaystyle r_{max}=9(1+\frac{1}{2})=\frac{27}{2}\)
Answer: \(\displaystyle \frac{27}{2}\).
Answer: \(\displaystyle \frac{27}{2}\)
(b)
\(\displaystyle \frac{1}{2}\int_{0}^{2\pi } r^{2}\, d\theta = \pi a^{2}(1+ \frac{b^{2}}{2})\); coefficient \(\displaystyle \frac{729}{8}\).
Answer: \(\displaystyle \frac{729}{8}\).
Answer: \(\displaystyle \frac{729}{8}\)
(c)
\(\displaystyle \frac{dy}{dx}= \frac{r'\sin \theta + r\cos \theta }{r'\cos \theta - r\sin \theta }= b= \frac{1}{2}\)
Answer: \(\displaystyle \frac{1}{2}\).
Answer: \(\displaystyle \frac{1}{2}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Maximise cos θ over a full turn.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Area integrates \(\displaystyle \frac{r^2}{2}\), not r.
(b) · 2 marks
- 1 mark: Square r and integrate the constant, cosine and cosine-square terms.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: The polar radius is not generally the Cartesian x-coordinate.
(c) · 2 marks
- 1 mark: Differentiate the Cartesian parametrisation by θ.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Area integrates \(\displaystyle \frac{r^2}{2}\), not r.
Total: 6 marks.