| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2020 |
| Session | November |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Convert Cartesian to polar equation |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths polar coordinates question requiring standard conversions (x=r cos θ, y=r sin θ), sketching, area integration using the standard formula, and optimization using a given identity. While it involves several techniques and is from Further Maths, each part follows well-established procedures without requiring novel insight—the conversion is algebraic manipulation, the area uses the memorized formula ½∫r²dθ, and part (d) provides the key identity needed. Slightly above average difficulty due to the algebraic complexity and multi-step nature, but not exceptionally challenging. |
| Spec | 1.05l Double angle formulae: and compound angle formulae4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(r^5 = 4xy(x^2 - y^2)\) | B1 | Uses \(r^2 = x^2 + y^2\) |
| \(r^5 = 4r^4\cos\theta\sin\theta(\cos^2\theta - \sin^2\theta)\) | B1 | Uses \(x = r\cos\theta\) and \(y = r\sin\theta\) |
| \(r = 2\sin 2\theta\cos 2\theta\) | M1 | Applies at least one double angle formula |
| \(r = \sin 4\theta\) | A1 | Applies both double angle formulae, AG |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Initial line drawn and one loop in first quadrant] | B1 | Initial line drawn and one loop in first quadrant |
| [Correct shape at extremities] | B1 | Correct shape at extremities |
| \(\theta = \frac{1}{8}\pi\) | B1 | States the equation of the line of symmetry |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}\int_0^{\frac{1}{4}\pi}\sin^2 4\theta\, \mathrm{d}\theta\) | M1 | Uses \(\frac{1}{2}\int r^2\,\mathrm{d}\theta\) with correct limits |
| \(= \frac{1}{4}\int_0^{\frac{1}{4}\pi} 1 - \cos 8\theta\,\mathrm{d}\theta = \frac{1}{4}\left[\theta - \frac{1}{8}\sin 8\theta\right]_0^{\frac{1}{4}\pi}\) | M1 A1 | Applies double angle formula and integrates |
| \(= \frac{1}{16}\pi\) | A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = 4(\sin\theta\cos^4\theta - \sin^3\theta\cos^2\theta)\) | B1 | Uses \(x = r\cos\theta\) |
| \(-4\sin^2\theta\cos^3\theta + \cos^5\theta + 2\sin^4\theta\cos\theta - 3\sin^2\theta\cos^3\theta = 0\) | M1 | Differentiates and sets equal to 0 |
| \(\cos\theta(2\sin^4\theta - 7\sin^2\theta\cos^2\theta + \cos^4\theta) = 0\) | A1 | |
| \(\cos\theta = 0\) or \(2\tan^4\theta - 7\tan^2\theta + 1 = 0\) | M1 | Forms quadratic in \(\tan^2\theta\); must see consideration of \(\cos\theta = 0\) |
| \(\tan^2\theta = \frac{1}{4}(7 \pm \sqrt{41}) \Rightarrow \theta = \pm 0.369, \pm 1.071\) | B1 | Allow use of decimals |
| \(\theta = 0.369 \Rightarrow x = 0.93\) (or \(\theta = -0.369 \Rightarrow \ | x\ | = 0.93\)) |
| Total: 6 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r^5 = 4xy(x^2 - y^2)$ | B1 | Uses $r^2 = x^2 + y^2$ |
| $r^5 = 4r^4\cos\theta\sin\theta(\cos^2\theta - \sin^2\theta)$ | B1 | Uses $x = r\cos\theta$ and $y = r\sin\theta$ |
| $r = 2\sin 2\theta\cos 2\theta$ | M1 | Applies at least one double angle formula |
| $r = \sin 4\theta$ | A1 | Applies both double angle formulae, AG |
| **Total: 4** | | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Initial line drawn and one loop in first quadrant] | B1 | Initial line drawn and one loop in first quadrant |
| [Correct shape at extremities] | B1 | Correct shape at extremities |
| $\theta = \frac{1}{8}\pi$ | B1 | States the equation of the line of symmetry |
| **Total: 3** | | |
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}\int_0^{\frac{1}{4}\pi}\sin^2 4\theta\, \mathrm{d}\theta$ | M1 | Uses $\frac{1}{2}\int r^2\,\mathrm{d}\theta$ with correct limits |
| $= \frac{1}{4}\int_0^{\frac{1}{4}\pi} 1 - \cos 8\theta\,\mathrm{d}\theta = \frac{1}{4}\left[\theta - \frac{1}{8}\sin 8\theta\right]_0^{\frac{1}{4}\pi}$ | M1 A1 | Applies double angle formula and integrates |
| $= \frac{1}{16}\pi$ | A1 | |
| **Total: 4** | | |
## Question 7(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = 4(\sin\theta\cos^4\theta - \sin^3\theta\cos^2\theta)$ | B1 | Uses $x = r\cos\theta$ |
| $-4\sin^2\theta\cos^3\theta + \cos^5\theta + 2\sin^4\theta\cos\theta - 3\sin^2\theta\cos^3\theta = 0$ | M1 | Differentiates and sets equal to 0 |
| $\cos\theta(2\sin^4\theta - 7\sin^2\theta\cos^2\theta + \cos^4\theta) = 0$ | A1 | |
| $\cos\theta = 0$ or $2\tan^4\theta - 7\tan^2\theta + 1 = 0$ | M1 | Forms quadratic in $\tan^2\theta$; must see consideration of $\cos\theta = 0$ |
| $\tan^2\theta = \frac{1}{4}(7 \pm \sqrt{41}) \Rightarrow \theta = \pm 0.369, \pm 1.071$ | B1 | Allow use of decimals |
| $\theta = 0.369 \Rightarrow x = 0.93$ (or $\theta = -0.369 \Rightarrow \|x\| = 0.93$) | A1 | Substituting $\theta = \pm 0.369$ gives maximum value of $\|x\|$ |
| **Total: 6** | | |7
\begin{enumerate}[label=(\alph*)]
\item Show that the curve with Cartesian equation
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 5 } { 2 } } = 4 x y \left( x ^ { 2 } - y ^ { 2 } \right)$$
has polar equation $r = \sin 4 \theta$.\\
The curve $C$ has polar equation $r = \sin 4 \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$.
\item Sketch $C$ and state the equation of the line of symmetry.
\item Find the exact value of the area of the region enclosed by $C$.
\item Using the identity $\sin 4 \theta \equiv 4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta$, find the maximum distance of $C$ from the line $\theta = \frac { 1 } { 2 } \pi$. Give your answer correct to 2 decimal places.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2020 Q7 [17]}}