| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2022 |
| Session | November |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Convert Cartesian to polar equation |
| Difficulty | Standard +0.8 This is a multi-part Further Maths polar coordinates question requiring conversion using standard substitutions (part a), sketching with domain restrictions (part b), area integration with r² formula (part c), and optimization requiring calculus of y = r sin θ (part d). While the conversion and area are standard Further Maths techniques, the final optimization requires insight into maximizing a composite function, elevating it above routine exercises. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian4.09a 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 |
| \(x^2 + y^2 = r^2\), \(x = r\cos\theta\), \(y = r\sin\theta\) | B1 | Used. |
| \(r^4 = 36r^2(\cos^2\theta - \sin^2\theta) = 36r^2\cos 2\theta\) | M1 | Substitutes and applies \(\cos 2\theta = \cos^2\theta - \sin^2\theta\) |
| \(r^2 = 36\cos 2\theta\) | A1 | AG. |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Closed curve, correct position and symmetrical about initial line | B1 | Closed curve. Correct position and symmetrical about initial line. |
| Single correct loop shown | B1 | Single correct loop. |
| \(6\) | B1 | States maximum distance or labels sketch. |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(18\int_{-\pi/4}^{\pi/4} \cos 2\theta \, d\theta = 9[\sin 2\theta]_{-\pi/4}^{\pi/4}\) | M1 | Forms \(\frac{1}{2}\int r^2 \, d\theta\) |
| \(18\) | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = 6\cos^{\frac{1}{2}} 2\theta \sin\theta\) | B1 | |
| \(\cos^{\frac{1}{2}} 2\theta \cos\theta - \cos^{-\frac{1}{2}} 2\theta \sin 2\theta \sin\theta = 0\) | M1 A1 | Sets \(\frac{dy}{d\theta} = 0\) |
| \(\cos 2\theta\cos\theta - \sin 2\theta\sin\theta = 0\) leading to \(1 = \dfrac{2\tan^2\theta}{1-\tan^2\theta}\) | M1 | Applies suitable trigonometric identity. |
| \(\theta = \pm\frac{1}{6}\pi\) | A1 | |
| \(\dfrac{3}{2}\sqrt{2}\) | A1 | |
| Total: 6 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 + y^2 = r^2$, $x = r\cos\theta$, $y = r\sin\theta$ | B1 | Used. |
| $r^4 = 36r^2(\cos^2\theta - \sin^2\theta) = 36r^2\cos 2\theta$ | M1 | Substitutes and applies $\cos 2\theta = \cos^2\theta - \sin^2\theta$ |
| $r^2 = 36\cos 2\theta$ | A1 | AG. |
| **Total: 3** | | |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Closed curve, correct position and symmetrical about initial line | B1 | Closed curve. Correct position and symmetrical about initial line. |
| Single correct loop shown | B1 | Single correct loop. |
| $6$ | B1 | States maximum distance or labels sketch. |
| **Total: 3** | | |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $18\int_{-\pi/4}^{\pi/4} \cos 2\theta \, d\theta = 9[\sin 2\theta]_{-\pi/4}^{\pi/4}$ | M1 | Forms $\frac{1}{2}\int r^2 \, d\theta$ |
| $18$ | A1 | |
| **Total: 2** | | |
## Question 6(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = 6\cos^{\frac{1}{2}} 2\theta \sin\theta$ | B1 | |
| $\cos^{\frac{1}{2}} 2\theta \cos\theta - \cos^{-\frac{1}{2}} 2\theta \sin 2\theta \sin\theta = 0$ | M1 A1 | Sets $\frac{dy}{d\theta} = 0$ |
| $\cos 2\theta\cos\theta - \sin 2\theta\sin\theta = 0$ leading to $1 = \dfrac{2\tan^2\theta}{1-\tan^2\theta}$ | M1 | Applies suitable trigonometric identity. |
| $\theta = \pm\frac{1}{6}\pi$ | A1 | |
| $\dfrac{3}{2}\sqrt{2}$ | A1 | |
| **Total: 6** | | |6
\begin{enumerate}[label=(\alph*)]
\item Show that the curve with Cartesian equation
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 36 \left( x ^ { 2 } - y ^ { 2 } \right)$$
has polar equation $r ^ { 2 } = 36 \cos 2 \theta$.\\
The curve $C$ has polar equation $r ^ { 2 } = 36 \cos 2 \theta$, for $- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$.
\item Sketch $C$ and state the maximum distance of a point on $C$ from the pole.
\item Find the area of the region enclosed by $C$.
\item Find the maximum distance of a point on $C$ from the initial line, giving the answer in exact form.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q6 [14]}}