| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Standard +0.8 This question requires sketching an unfamiliar polar curve (not a standard form like cardioid or rose), then calculating area bounded by the curve and initial line. The area calculation involves setting up and evaluating ∫(1/2)r²dθ with r = 1/(1+θ), requiring integration of 1/(1+θ)² over [0,2π]. While the integration itself is straightforward, identifying the correct region and setup requires solid understanding of polar coordinates beyond routine exercises. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
**Question 3(i)**
[Sketch of decreasing spiral] **B3**
- **B1** Starts at $(1, 0)$
- **B1** Decreasing spiral
- **B1** All (essentially) correct
**Question 3(ii)**
Area $= \dfrac{1}{2}\displaystyle\int_0^{2\pi} \dfrac{1}{(1+\theta)^2}\,\text{d}\theta$ **M1** Attempt to integrate $k(1+\theta)^{-2}$
$= \dfrac{1}{2}\left[\dfrac{-1}{1+\theta}\right]_0^{2\pi}$ **A1** Correct integration
$= \dfrac{1}{2}\!\left(1 - \dfrac{1}{1+2\pi}\right)$ or $\dfrac{\pi}{1+2\pi}$ **A1** Correct answer
**Total: 6 marks**3 (i) Sketch the curve with polar equation $r = \frac { 1 } { 1 + \theta } , 0 \leqslant \theta \leqslant 2 \pi$.\\
(ii) Find, in terms of $\pi$, the area of the region enclosed by the curve and the part of the initial line between the endpoints of the curve.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2017 Q3 [6]}}