| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Sketch polar curve |
| Difficulty | Challenging +1.2 This is a multi-part polar coordinates question requiring standard techniques: finding maximum r by differentiation, sketching from a table of values, and applying the polar area formula. While it involves several steps and Further Maths content, each part uses routine methods without requiring novel insight or particularly complex manipulation. |
| Spec | 4.09a 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 |
| \(2(c(c) - (1+s)s) = 0\) | M1 | Finds \(\frac{dr}{d\theta}\) and sets equal to 0 |
| \(1 - s - 2s^2 = 0\) | A1 | |
| \(\sin\theta = -1\), \(\sin\theta = \frac{1}{2}\) leading to \(\theta = \frac{1}{6}\pi\) | M1 A1 | Solves quadratic in \(\sin\theta\) |
| \(\left(\frac{3}{2}\sqrt{3}, \frac{1}{6}\pi\right)\) | A1 | Allow \((2.60, \frac{1}{6}\pi)\) 3sf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Correct curve shape: tangential to \(\theta = \frac{\pi}{2}\) at pole, \(r\) strictly increasing to \(\frac{\pi}{6}\) then decreasing] | B1 | Correct shape. Tangential to \(\theta = \frac{\pi}{2}\) at pole, \(r\) strictly increasing to \(\frac{\pi}{6}\) then decreasing |
| [Correct position: all in first quadrant] | B1 | Correct position. All in first quadrant |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}\int_0^{\frac{1}{2}\pi} 4c^2(1+s)^2\, d\theta = 2\int_0^{\frac{1}{2}\pi} c^2 + 2sc^2 + s^2c^2\, d\theta\) | M1 A1 | Uses \(\frac{1}{2}\int r^2\, d\theta\) with correct limits |
| \(2\int_0^{\frac{1}{2}\pi} \frac{1}{2}(\cos 2\theta +1) + 2\sin\theta\cos^2\theta + \frac{1}{4}\sin^2 2\theta\, d\theta\) | M1 | Uses double angle formulae on first and last terms |
| \(\int_0^{\frac{1}{2}\pi} \cos 2\theta + 1 + 4\sin\theta\cos^2\theta + \frac{1}{4} - \frac{1}{4}\cos 4\theta\, d\theta\) | A1 | Obtains integrable form |
| \(\left[\frac{5}{4}\theta + \frac{1}{2}\sin 2\theta - \frac{4}{3}\cos^3\theta - \frac{1}{16}\sin 4\theta\right]_0^{\frac{1}{2}\pi}\) | M1 | Integrates |
| \(\frac{5}{8}\pi + \frac{4}{3}\) | A1 | |
| 6 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2(c(c) - (1+s)s) = 0$ | M1 | Finds $\frac{dr}{d\theta}$ and sets equal to 0 |
| $1 - s - 2s^2 = 0$ | A1 | |
| $\sin\theta = -1$, $\sin\theta = \frac{1}{2}$ leading to $\theta = \frac{1}{6}\pi$ | M1 A1 | Solves quadratic in $\sin\theta$ |
| $\left(\frac{3}{2}\sqrt{3}, \frac{1}{6}\pi\right)$ | A1 | Allow $(2.60, \frac{1}{6}\pi)$ 3sf |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Correct curve shape: tangential to $\theta = \frac{\pi}{2}$ at pole, $r$ strictly increasing to $\frac{\pi}{6}$ then decreasing] | B1 | Correct shape. Tangential to $\theta = \frac{\pi}{2}$ at pole, $r$ strictly increasing to $\frac{\pi}{6}$ then decreasing |
| [Correct position: all in first quadrant] | B1 | Correct position. All in first quadrant |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}\int_0^{\frac{1}{2}\pi} 4c^2(1+s)^2\, d\theta = 2\int_0^{\frac{1}{2}\pi} c^2 + 2sc^2 + s^2c^2\, d\theta$ | M1 A1 | Uses $\frac{1}{2}\int r^2\, d\theta$ with correct limits |
| $2\int_0^{\frac{1}{2}\pi} \frac{1}{2}(\cos 2\theta +1) + 2\sin\theta\cos^2\theta + \frac{1}{4}\sin^2 2\theta\, d\theta$ | M1 | Uses double angle formulae on first and last terms |
| $\int_0^{\frac{1}{2}\pi} \cos 2\theta + 1 + 4\sin\theta\cos^2\theta + \frac{1}{4} - \frac{1}{4}\cos 4\theta\, d\theta$ | A1 | Obtains integrable form |
| $\left[\frac{5}{4}\theta + \frac{1}{2}\sin 2\theta - \frac{4}{3}\cos^3\theta - \frac{1}{16}\sin 4\theta\right]_0^{\frac{1}{2}\pi}$ | M1 | Integrates |
| $\frac{5}{8}\pi + \frac{4}{3}$ | A1 | |
| | **6** | |
---6 The curve $C$ has polar equation $r = 2 \cos \theta ( 1 + \sin \theta )$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
\begin{enumerate}[label=(\alph*)]
\item Find the polar coordinates of the point on $C$ that is furthest from the pole.
\item Sketch C.
\item Find the area of the region bounded by $C$ and the initial line, giving your answer in exact form.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2021 Q6 [13]}}