CAIE FP1 2013 June — Question 1 4 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2013
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolar coordinates
TypeArea enclosed by polar curve
DifficultyStandard +0.3 This is a standard polar area question requiring direct application of the formula A = ½∫r²dθ with a straightforward cardioid curve. The integration involves expanding (1+cosθ)² and using standard trigonometric identities, making it slightly easier than average but still requiring competent technique with polar coordinates—a Further Maths topic.
Spec4.09c Area enclosed: by polar curve
1 Find the area of the region enclosed by the curve with polar equation \(r = 2 ( 1 + \cos \theta )\), for \(0 \leqslant \theta < 2 \pi\).
Question 1:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(A = \frac{1}{2}\int_0^{2\pi} 4(1 + 2\cos\theta + \cos^2\theta)\,d\theta\)M1 Use of \(\frac{1}{2}\int r^2\,d\theta\)
\(= \int_0^{2\pi}(3 + 4\cos\theta + \cos 2\theta)\,d\theta\)M1 Use of double angle formula and attempt to integrate
\(= \left[3\theta + 4\sin\theta + \frac{\sin 2\theta}{2}\right]_0^{2\pi}\)A1 Integrates correctly
\(= 6\pi\) (CWO) Accept 18.8A1 Finds value
Total: [4]
## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $A = \frac{1}{2}\int_0^{2\pi} 4(1 + 2\cos\theta + \cos^2\theta)\,d\theta$ | M1 | Use of $\frac{1}{2}\int r^2\,d\theta$ |
| $= \int_0^{2\pi}(3 + 4\cos\theta + \cos 2\theta)\,d\theta$ | M1 | Use of double angle formula and attempt to integrate |
| $= \left[3\theta + 4\sin\theta + \frac{\sin 2\theta}{2}\right]_0^{2\pi}$ | A1 | Integrates correctly |
| $= 6\pi$ (CWO) Accept 18.8 | A1 | Finds value |

**Total: [4]**

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1 Find the area of the region enclosed by the curve with polar equation $r = 2 ( 1 + \cos \theta )$, for $0 \leqslant \theta < 2 \pi$.

\hfill \mbox{\textit{CAIE FP1 2013 Q1 [4]}}
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