| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Standard +0.8 This is a Further Maths polar coordinates question requiring sketching of an unfamiliar curve involving exponential decay, then applying the polar area formula with integration by parts. The algebraic manipulation to reach the specific form (ln 2 - 13/32) requires careful execution across multiple steps, making it moderately challenging but still within standard FM1 scope. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks |
|---|---|
| Approximately correct shape and location for \(0 \le \theta < 2\pi\) | B1 |
| Shows initial line to be tangential to C at the pole | B1 |
| Asymptotic approach to circle \(r = a\) | B1 |
| [3] | |
| (ii) \(A = (a^2/2) \int_{\ln2}^{\ln4} (1 - 2e^{-\theta} + e^{-2\theta})d\theta\) | M1A1 |
| \(= (a^2/2)[\theta + 2e^{-\theta} - (1/2)e^{-2\theta}]_{\ln2}^{\ln4}\) | A1 |
| \(= ... = (a^2/2)(\ln 2 - 13/32)\) (AG) | A1 |
| [4] |
**(i) Sketch of C:**
Approximately correct shape and location for $0 \le \theta < 2\pi$ | B1 |
Shows initial line to be tangential to C at the pole | B1 |
Asymptotic approach to circle $r = a$ | B1 |
| [3] |
**(ii)** $A = (a^2/2) \int_{\ln2}^{\ln4} (1 - 2e^{-\theta} + e^{-2\theta})d\theta$ | M1A1 |
$= (a^2/2)[\theta + 2e^{-\theta} - (1/2)e^{-2\theta}]_{\ln2}^{\ln4}$ | A1 |
$= ... = (a^2/2)(\ln 2 - 13/32)$ (AG) | A1 |
| [4] |2 The curve $C$ has polar equation
$$r = a \left( 1 - \mathrm { e } ^ { - \theta } \right)$$
where $a$ is a positive constant and $0 \leqslant \theta < 2 \pi$.\\
(i) Draw a sketch of $C$.\\
(ii) Show that the area of the region bounded by $C$ and the lines $\theta = \ln 2$ and $\theta = \ln 4$ is
$$\frac { 1 } { 2 } a ^ { 2 } \left( \ln 2 - \frac { 13 } { 32 } \right)$$
\hfill \mbox{\textit{CAIE FP1 2010 Q2 [7]}}