| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Challenging +1.2 This is a Further Maths polar coordinates question requiring sketching and area calculation. While the absolute value adds a twist requiring identification of where the expression is negative, the actual integration is straightforward once set up. The topic itself (polar areas) is standard Further Maths content, and this is a typical two-part question with moderate algebraic manipulation but no deep conceptual insight required. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
**Question 10** [6+4 marks]
**(i)**
$\frac{1}{2}+\sin\theta = 0$ when $\theta = \frac{7}{6}\pi,\ \frac{11}{6}\pi$ M1A1
Symmetry in $y$-axis B1
$\left(\frac{1}{2}, 0\right)$ on initial line B1
Correct upper portion B1
Correct lower portion B1
**[6]**
**(ii)**
$A = \left(\frac{1}{2}\right)\int_0^{2\pi}\left(\frac{1}{2}+\sin\theta\right)^2\,\mathrm{d}\theta$ M1 (Penalise incorrect multiples with final A0)
$= \frac{1}{2}\int_0^{2\pi}\left(\frac{1}{4}+\sin\theta+\frac{1}{2}-\frac{1}{2}\cos 2\theta\right)\mathrm{d}\theta$ M1 (Double-angle formula)
$= \frac{1}{2}\left[\frac{3}{4}\theta - \cos\theta - \frac{1}{4}\sin 2\theta\right]_0^{2\pi}$ A1 (correctly integrated 3 suitable terms)
$= \frac{3}{4}\pi$ A1
**[4]**10 (i) Sketch the curve with polar equation $r = \left| \frac { 1 } { 2 } + \sin \theta \right|$, for $0 \leqslant \theta < 2 \pi$.\\
(ii) Find in an exact form the total area enclosed by the curve.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q10 [10]}}