FP2 June 2009 Q4

Figure 1 shows a sketch of the curve with polar equation \[r = a + 3\cos\theta, \quad a > 0, \quad 0 \leqslant \theta < 2\pi\]

The area enclosed by the curve is \(\dfrac{107}{2}\pi\).

Find the value of \(a\). (8)

Mark scheme

Scheme	Marks
\(A = \dfrac{1}{2}\displaystyle\int_0^{2\pi}(a + 3\cos\theta)^2\,\mathrm{d}\theta\) Applies \(\frac{1}{2}\int_0^{2\pi} r^2(\mathrm{d}\theta)\) with correct limits. Ignore \(\mathrm{d}\theta\).	B1
\((a + 3\cos\theta)^2 = a^2 + 6a\cos\theta + 9\cos^2\theta\)
\(= \underline{a^2 + 6a\cos\theta + 9\left(\dfrac{1 + \cos 2\theta}{2}\right)}\) \(\cos^2\theta = \frac{\pm 1 \pm \cos 2\theta}{2}\) Correct underlined expression.	M1 A1
\(A = \dfrac{1}{2}\displaystyle\int_0^{2\pi}\left(a^2 + 6a\cos\theta + \frac{9}{2} + \frac{9}{2}\cos 2\theta\right)\mathrm{d}\theta\)
\(= \left(\dfrac{1}{2}\right)\left[a^2\theta + 6a\sin\theta + \dfrac{9}{2}\theta + \dfrac{9}{4}\sin 2\theta\right]_0^{2\pi}\) Integrated expression with at least 3 out of 4 terms of the form \(\pm A\theta \pm B\sin\theta \pm C\theta \pm D\sin 2\theta\). Ignore the \(\frac{1}{2}\). Ignore limits. \(a^2\theta + 6a\sin\theta +\) correct ft integration. Ignore the \(\frac{1}{2}\). Ignore limits.	M1* A1 ft
\(= \dfrac{1}{2}\left[(2\pi a^2 + 0 + 9\pi + 0) - (0)\right]\)
\(= \pi a^2 + \dfrac{9\pi}{2}\) \(\pi a^2 + \frac{9\pi}{2}\)	A1
Hence, \(\pi a^2 + \dfrac{9\pi}{2} = \dfrac{107}{2}\pi\) Integrated expression equal to \(\frac{107}{2}\pi\).	dM1*
\(a^2 + \dfrac{9}{2} = \dfrac{107}{2}\) \(a^2 = 49\)
As \(a > 0\), \(a = 7\) \(a = 7\)	A1 cso
	(8 marks)

Some candidates may achieve \(a = 7\) from incorrect working. Such candidates will not get full marks