FP2 June 2017 Q6
6.
Figure 1 shows a sketch of a curve with polar equation \[r = 6 + a\sin\theta\] where \(0 \lt a \lt 6\) and \(0 \leqslant \theta \lt 2\pi\)
The area enclosed by the curve is \(\dfrac{97\pi}{2}\)
Find the value of the constant \(a\). (8)
| Scheme | Marks |
|---|---|
| \(r = 6 + a\sin\theta\) | |
| \(A = \dfrac{1}{2}\displaystyle\int (6 + a\sin\theta)^2\,\mathrm{d}\theta\) Use of \(\dfrac{1}{2}\displaystyle\int r^2(\mathrm{d}\theta)\) Limits not needed. Can be gained if \(\dfrac{1}{2}\) appears later | B1 |
| \((6 + a\sin\theta)^2 = 36 + 12a\sin\theta + a^2\sin^2\theta\) | |
| \((6 + a\sin\theta)^2 = 36 + 12a\sin\theta + a^2\left(\dfrac{1 - \cos 2\theta}{2}\right)\) M1: Squares (\(36 + k\sin^2\theta\), where \(k = a^2\) or \(a\) as min) and attempts to change : \(\sin^2\theta\) to an expression in \(\cos 2\theta\) A1: Correct expression | M1A1 |
| \(\left(\dfrac{1}{2}\right)\left[36\theta - 12a\cos\theta + \dfrac{a^2}{2}\theta - \dfrac{a^2}{4}\sin 2\theta\right]\) dM1: Attempt to integrate \(\cos 2\theta \to \pm\dfrac{1}{2}\sin 2\theta\) Limits not needed A1: Correct integration limits not needed | dM1A1 |
| \(= 36\pi + \dfrac{\pi a^2}{2}\) Correct area obtained from correct integration and correct limits. No need to simplify but trig functions must be evaluated. | A1 |
| \(36\pi + \dfrac{\pi a^2}{2} = \dfrac{97\pi}{2} \Rightarrow a = \ldots\) Set their area \(= \dfrac{97\pi}{2}\) and attempt to solve for \(a\) (depends on both M marks above) If \(\dfrac{1}{2}\) omitted from the initial formula and area set \(= 97\pi\), give the B1 by implication as well as this mark. | ddM1 |
| \(a = 5\) cao and cso \(a = \pm5\) or \(a = -5\) scores A0 | A1cso |
| (8 marks) |
Notes
Alternatives: Splitting the area and so using 2 integrals with different limits.
Marks the same as the main scheme.
1 Limits 0 to \(\pi\) (area above initial line) and limits \(\pi\) to \(2\pi\) (area below initial line) and add the two results.
2 Limits 0 to \(\dfrac{\pi}{2}\) and \(\dfrac{3\pi}{2}\) to \(2\pi\) Twice the sum of the results needed.