| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Sketch polar curve |
| Difficulty | Challenging +1.2 This is a multi-part polar curve question requiring finding tangent angles, proving symmetry algebraically, and sketching. While it involves Further Maths content (polar coordinates), the techniques are fairly standard: finding where r=0, using trigonometric identities for symmetry, and identifying max/min values. The symmetry proof is routine algebraic manipulation. More challenging than average A-level but not requiring deep insight. |
| Spec | 1.05m Geometric proofs: of trig sum and double angle formulae4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
| Answer | Marks |
|---|---|
| (i) Show at least correct \(\cos \theta \cos 60 + \sin \theta \sin 60\) or \(\cos \theta \cos 60 - \sin \theta \sin 60\) | B1 |
| Attempt expansion of both with exact numerical values attempted | M1 |
| Obtain \(\frac{1}{3}\sqrt{3}\sin \theta + \frac{2}{3}\cos \theta\) | A1 |
| (ii) Attempt correct process for finding \(R\) | M1 |
| Attempt recognisable process for finding \(a\) | M1 |
| Obtain \(\sqrt{7}\sin(\theta + 70.9)\) | A1 |
| (iii) Attempt correct process to find any value of \(\theta +\) their \(a\) | M1 |
| Obtain any correct value for \(\theta + 70.9\) | A1 |
| Attempt correct process to find \(\theta +\) their \(a\) in 3rd quadrant | M1 |
| Obtain \(131\) | A1 |
| | |
|---|---|
| (i) Show at least correct $\cos \theta \cos 60 + \sin \theta \sin 60$ or $\cos \theta \cos 60 - \sin \theta \sin 60$ | B1 |
| Attempt expansion of both with exact numerical values attempted | M1 |
| Obtain $\frac{1}{3}\sqrt{3}\sin \theta + \frac{2}{3}\cos \theta$ | A1 |
| (ii) Attempt correct process for finding $R$ | M1 |
| Attempt recognisable process for finding $a$ | M1 |
| Obtain $\sqrt{7}\sin(\theta + 70.9)$ | A1 |
| (iii) Attempt correct process to find any value of $\theta +$ their $a$ | M1 |
| Obtain any correct value for $\theta + 70.9$ | A1 |
| Attempt correct process to find $\theta +$ their $a$ in 3rd quadrant | M1 |
| Obtain $131$ | A1 |8 The equation of a curve, in polar coordinates, is
$$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
\begin{enumerate}[label=(\roman*)]
\item \\
\includegraphics[max width=\textwidth, alt={}, center]{63a316f6-1c18-4224-930f-0b58112c9f71-3_268_796_1567_717}
The diagram shows the part of the curve for which $0 \leqslant \theta \leqslant \alpha$, where $\theta = \alpha$ is the equation of the tangent to the curve at $O$. Find $\alpha$ in terms of $\pi$.
\item (a) If $\mathrm { f } ( \theta ) = 1 - \sin 2 \theta$, show that $\mathrm { f } \left( \frac { 1 } { 2 } ( 2 k + 1 ) \pi - \theta \right) = \mathrm { f } ( \theta )$ for all $\theta$, where $k$ is an integer.\\
(b) Hence state the equations of the lines of symmetry of the curve
$$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
\item Sketch the curve with equation
$$r = 1 - \sin 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$
State the maximum value of $r$ and the corresponding values of $\theta$.
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2008 Q8 [11]}}