5 The curve \(C\) has polar equation \(r = 1 + 2 \cos \theta\). Sketch the curve for \(- \frac { 2 } { 3 } \pi \leqslant \theta < \frac { 2 } { 3 } \pi\).
Find the area bounded by \(C\) and the half-lines \(\theta = - \frac { 1 } { 3 } \pi , \theta = \frac { 1 } { 3 } \pi\).
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Question 5:
Answer Marks
Guidance
Working/Answer Marks
Guidance
Correct shape and orientation B1
Sketches graph
Passing through \((0,0)\) and \((3,0)\) B1
Area \(= 2 \times \frac{1}{2}\int_0^{\pi/3}(1+2\cos\theta)^2\,d\theta\) M1
Uses area of sector formula
\(= \int_0^{\pi/3}(1+4\cos\theta+4\cos^2\theta)\,d\theta\) —
\(= \int_0^{\pi/3}(3+4\cos\theta+2\cos 2\theta)\,d\theta\) M1
Uses double angle formula and integrates
\(= \left[3\theta + 4\sin\theta + \sin 2\theta\right]_0^{\pi/3}\) A1
\(= \left[\pi + \frac{5}{2}\sqrt{3}\right]\) A1
Obtains result
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## Question 5:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Correct shape and orientation | B1 | Sketches graph |
| Passing through $(0,0)$ and $(3,0)$ | B1 | |
| Area $= 2 \times \frac{1}{2}\int_0^{\pi/3}(1+2\cos\theta)^2\,d\theta$ | M1 | Uses area of sector formula |
| $= \int_0^{\pi/3}(1+4\cos\theta+4\cos^2\theta)\,d\theta$ | — | |
| $= \int_0^{\pi/3}(3+4\cos\theta+2\cos 2\theta)\,d\theta$ | M1 | Uses double angle formula and integrates |
| $= \left[3\theta + 4\sin\theta + \sin 2\theta\right]_0^{\pi/3}$ | A1 | |
| $= \left[\pi + \frac{5}{2}\sqrt{3}\right]$ | A1 | Obtains result |
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5 The curve $C$ has polar equation $r = 1 + 2 \cos \theta$. Sketch the curve for $- \frac { 2 } { 3 } \pi \leqslant \theta < \frac { 2 } { 3 } \pi$.
Find the area bounded by $C$ and the half-lines $\theta = - \frac { 1 } { 3 } \pi , \theta = \frac { 1 } { 3 } \pi$.
\hfill \mbox{\textit{CAIE FP1 2012 Q5 [6]}}