| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Challenging +1.2 This is a multi-part polar coordinates question requiring standard techniques: part (i) uses the property that opposite points have θ differing by π; part (ii) is a routine polar area integral; part (iii) involves standard polar-to-Cartesian conversion. While it requires multiple steps and careful algebra, all techniques are textbook methods for Further Maths polar coordinates with no novel insight required. |
| Spec | 4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
**(i)**
$P = (2+\cos\alpha, \alpha) \Rightarrow Q = (2+\cos(\pi+\alpha), \pi+\alpha)$ **M1** (or $\alpha - \pi$)
$= (2-\cos\alpha, \pi+\alpha)$ **A1**
so $PQ = 4$ **A1** [3]
**(ii)**
$A = 2\times\int_0^{\pi}\frac{1}{2}(4+4\cos\theta+\cos^2\theta)\,d\theta$ **M1** (incl. attempt at $r^2$)
$= \int_0^{\pi}\left(\frac{9}{2}+4\cos\theta+\frac{1}{2}\cos 2\theta\right)d\theta$ **M1** Use of double-angle formula
$= \left[\frac{9}{2}\theta+4\sin\theta+\frac{1}{4}\sin 2\theta\right]_0^{\pi} = \frac{9}{2}\pi$ **A1** From fully correct working [3]
**(iii)**
$r = 2+\cos\theta \Rightarrow r^2 = 2r + r\cos\theta$
$\Rightarrow x^2+y^2 = 2\sqrt{x^2+y^2}+x$ **M1** Use of the standard results
$\Rightarrow (x^2+y^2-x)^2 = 4(x^2+y^2)$ **A1** (Legitimately: ANSWER GIVEN)
Allow both marks to those who verify in reverse and overlook $\pm$ when square-rooting
$x = y = 0$ excluded (or "the origin", etc.) **B1** [3]7 A curve $C$ has polar equation $r = 2 + \cos \theta$ for $- \pi < \theta \leqslant \pi$.\\
(i) The point $P$ on $C$ corresponds to $\theta = \alpha$, and the point $Q$ on $C$ is such that $P O Q$ is a straight line, where $O$ is the pole. Show that the length $P Q$ is independent of $\alpha$.\\
(ii) Find, in an exact form, the area of the region enclosed by $C$.\\
(iii) Show that $\left( x ^ { 2 } + y ^ { 2 } - x \right) ^ { 2 } = 4 \left( x ^ { 2 } + y ^ { 2 } \right)$ is a cartesian equation for $C$. Identify the coordinates of the point which is included in this cartesian equation but is not on $C$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2010 Q7 [9]}}