| Exam Board | CAIE |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Area of region with line boundary |
| Difficulty | Challenging +1.2 This is a multi-part polar coordinates question requiring sketching, finding intersections by solving a trigonometric equation, and computing area using the polar area formula. While it involves several steps and integration, the techniques are standard for Further Maths polar coordinates: the intersection requires solving 3 + 2sin(θ) = 2/sin(θ), leading to a quadratic in sin(θ), and the area calculation splits into curve and line segments using standard formulas. No novel geometric insight is required, making it moderately above average difficulty. |
| 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 |
|---|---|---|
| Correct symmetrical shape, closed loop (diagram showing limaçon) | B1 | Correct symmetrical shape, closed loop |
| Answer | Marks | Guidance |
|---|---|---|
| Line \(l\) parallel to initial line and correct side of pole | B1 | |
| \(2 = 3\sin\theta + 2\sin^2\theta\) | M1 | Forms quadratic in \(\sin\theta\). Or in \(r\): \(r = 3 + \dfrac{4}{r}\) |
| \(\sin\theta = \tfrac{1}{2}\) | M1 | Solves for \(\sin\theta\) |
| \(\left(4, \tfrac{1}{6}\pi\right)\) | A1 | SC1 For finding both angles correctly |
| \(\left(4, \tfrac{5}{6}\pi\right)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(2 \times \tfrac{1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{6}}(3 + 2\sin\theta)^2\, d\theta\) | M1 | Finds the part of the required area enclosed by the curved outer edge and two line segments from the pole. Limits must be correct. |
| \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{6}} 9 + 12\sin\theta + 2(1-\cos 2\theta)\, d\theta\) | M1 | Uses double angle formula and integrates |
| \(\left[11\theta - 12\cos\theta - \sin 2\theta\right]_{-\frac{\pi}{2}}^{\frac{\pi}{6}} = \tfrac{22}{3}\pi - \tfrac{13}{2}\sqrt{3}\) | A1 A1 | |
| \(\tfrac{22}{3}\pi - \tfrac{13}{2}\sqrt{3} + \left(4\cos\tfrac{\pi}{6}\right)\times 2\) | M1 | Adds area of triangle |
| \(\tfrac{22}{3}\pi - \tfrac{5}{2}\sqrt{3}\) | A1 |
## Question 5(a):
| Correct symmetrical shape, closed loop (diagram showing limaçon) | B1 | Correct symmetrical shape, closed loop |
|---|---|---|
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## Question 5(b):
| Line $l$ parallel to initial line and correct side of pole | B1 | |
|---|---|---|
| $2 = 3\sin\theta + 2\sin^2\theta$ | M1 | Forms quadratic in $\sin\theta$. Or in $r$: $r = 3 + \dfrac{4}{r}$ |
| $\sin\theta = \tfrac{1}{2}$ | M1 | Solves for $\sin\theta$ |
| $\left(4, \tfrac{1}{6}\pi\right)$ | A1 | SC1 For finding both angles correctly |
| $\left(4, \tfrac{5}{6}\pi\right)$ | A1 | |
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## Question 5(c):
| $2 \times \tfrac{1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{6}}(3 + 2\sin\theta)^2\, d\theta$ | M1 | Finds the part of the required area enclosed by the curved outer edge and two line segments from the pole. Limits must be correct. |
|---|---|---|
| $\int_{-\frac{\pi}{2}}^{\frac{\pi}{6}} 9 + 12\sin\theta + 2(1-\cos 2\theta)\, d\theta$ | M1 | Uses double angle formula and integrates |
| $\left[11\theta - 12\cos\theta - \sin 2\theta\right]_{-\frac{\pi}{2}}^{\frac{\pi}{6}} = \tfrac{22}{3}\pi - \tfrac{13}{2}\sqrt{3}$ | A1 A1 | |
| $\tfrac{22}{3}\pi - \tfrac{13}{2}\sqrt{3} + \left(4\cos\tfrac{\pi}{6}\right)\times 2$ | M1 | Adds area of triangle |
| $\tfrac{22}{3}\pi - \tfrac{5}{2}\sqrt{3}$ | A1 | |5 The curve $C$ has polar equation $r = 3 + 2 \sin \theta$, for $- \pi < \theta \leqslant \pi$.
\begin{enumerate}[label=(\alph*)]
\item The diagram shows part of $C$. Sketch the rest of $C$ on the diagram.\\
\begin{tikzpicture}[>=stealth, thick, scale=0.75]
% --- Polar curve r = 3 + 2*sin(theta), theta from -90 to 90 ---
\draw[thick] plot[domain=-90:90, samples=200, smooth]
({(3 + 2*sin(\x))*cos(\x)}, {(3 + 2*sin(\x))*sin(\x)});
% --- Half-line from O in the theta=0 direction ---
\draw[->] (0, 0) -- (4.2, 0) node[right] {$\theta = 0$};
\draw[opacity=0] (-4, 0) -- (0,0);
% --- Origin label ---
\node[left] at (0, 0) {$O$};
\end{tikzpicture}
The straight line $l$ has polar equation $r \sin \theta = 2$.
\item Add $l$ to the diagram in part (a) and find the polar coordinates of the points of intersection of $C$ and $l$.
\item The region $R$ is enclosed by $C$ and $l$, and contains the pole.
Find the area of $R$, giving your answer in exact form.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 1 2021 Q5 [12]}}