| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 7 |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Standard +0.8 This is a standard polar coordinates question requiring application of memorized formulas for area (½∫r²dθ) and arc length (∫√(r² + (dr/dθ)²)dθ). While the integration involves polynomial expressions requiring careful algebraic manipulation, the question is conceptually straightforward with no novel problem-solving required. It's moderately harder than average due to being a Further Maths topic and requiring accurate execution of two multi-step integrations. |
| Spec | 4.08f Integrate using partial fractions4.09c Area enclosed: by polar curve |
**(i)** $A = \frac{1}{2}\int r^2\,d\theta$, attempted: M1
$r^2 = \theta^4 + 4\theta^3 + 4\theta^2$: B1
$\frac{1}{2}\left[\frac{\theta^5}{5} + \theta^4 + \frac{4\theta^3}{3}\right]_0^3$, correct integration: A1
$= 82.8$: A1 **[4]**
**(ii)** $\frac{dr}{d\theta} = 2\theta + 2$: B1
Attempt at $r^2 + \left(\frac{dr}{d\theta}\right)^2$: M1
$= \theta^4 + 4\theta^3 + 4\theta^2 + 4\theta^2 + 8\theta + 4$: A1
Attempt to complete the square on this: M1
$= (\theta^2 + 2\theta + 2)^2$: A1
Use of $L = \int\sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta$ with their expression: M1
$= \left[\frac{\theta^3}{3} + \theta^2 + 2\theta\right]_0^3 = 24$ **cao** from correct integration: A1 **[7]**8 The curve $C$ has polar equation $r = \theta ^ { 2 } + 2 \theta$ for $0 \leq \theta \leq 3$.\\
(i) Find the area of the region enclosed by $C$ and the half-lines $\theta = 0$ and $\theta = 3$.\\
(ii) Determine the length of $C$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q8 [7]}}