| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Standard +0.8 This is a multi-part Further Maths polar coordinates question requiring: finding tangent at pole (solving r=0 with tan θ), identifying maximum r, sketching an unfamiliar curve (r = √3 + tan θ), and computing area with the polar integral formula. While the individual techniques are standard for FP2, the curve itself is non-standard and the multi-step nature with 4 parts requiring different skills makes this moderately challenging for Further Maths students. |
| Spec | 4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt to solve \(r=0\), \(\tan \theta = - \sqrt{3}\) | M1 | Allow \(\pm \sqrt{3}\) |
| Get \(\theta = -\frac{1}{3}\pi\) only | A1 | Allow \(-60°\) |
| (ii) \(r = \sqrt{3} + 1\) when \(\theta = \frac{1}{4}\pi\) | B1, B1 | AEF for \(r\), \(45°\) for \(\theta\) |
| (iii) | B1 | Correct \(r\) at correct end-values of \(\theta\); Ignore extra \(\theta\) used |
| [Graph showing correct shape] | B1 | Correct shape with \(r\) not decreasing |
| (iv) Formula with correct \(r\) used | M1 | \(l^2\) may be implied |
| Replace \(\tan^2 \theta = \sec^2 \theta - 1\) | B1 | |
| Attempt to integrate their expression | M1 | Must be 3 different terms leading to any 2 of \(a\theta + b\ln(\sec\theta/\cos\theta) + c\tan\theta\) |
| Get \(\theta + \sqrt{3}\ln\sec\theta + \frac{1}{2}\tan\theta\) | A1 | Condone answer x2 if \(\frac{1}{2}\) seen elsewhere |
| Correct limits to \(\frac{1}{4}\pi + \sqrt{3}\ln 2 + \frac{1}{2}\) | A1 | cao; AEF |
**(i)** Attempt to solve $r=0$, $\tan \theta = - \sqrt{3}$ | M1 | Allow $\pm \sqrt{3}$
Get $\theta = -\frac{1}{3}\pi$ only | A1 | Allow $-60°$
**(ii)** $r = \sqrt{3} + 1$ when $\theta = \frac{1}{4}\pi$ | B1, B1 | AEF for $r$, $45°$ for $\theta$
**(iii)** | B1 | Correct $r$ at correct end-values of $\theta$; Ignore extra $\theta$ used
[Graph showing correct shape] | B1 | Correct shape with $r$ not decreasing
**(iv)** Formula with correct $r$ used | M1 | $l^2$ may be implied
Replace $\tan^2 \theta = \sec^2 \theta - 1$ | B1 |
Attempt to integrate their expression | M1 | Must be 3 different terms leading to any 2 of $a\theta + b\ln(\sec\theta/\cos\theta) + c\tan\theta$
Get $\theta + \sqrt{3}\ln\sec\theta + \frac{1}{2}\tan\theta$ | A1 | Condone answer x2 if $\frac{1}{2}$ seen elsewhere
Correct limits to $\frac{1}{4}\pi + \sqrt{3}\ln 2 + \frac{1}{2}$ | A1 | cao; AEF7 The equation of a curve, in polar coordinates, is
$$r = \sqrt { 3 } + \tan \theta , \quad \text { for } - \frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$$
(i) Find the equation of the tangent at the pole.\\
(ii) State the greatest value of $r$ and the corresponding value of $\theta$.\\
(iii) Sketch the curve.\\
(iv) Find the exact area of the region enclosed by the curve and the lines $\theta = 0$ and $\theta = \frac { 1 } { 4 } \pi$.
\hfill \mbox{\textit{OCR FP2 2006 Q7 [11]}}