CAIE FP1 2011 June — Question 5 8 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2011
SessionJune
Marks8
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Mark schemeDownload PDF ↗
TopicPolar coordinates
TypeArea enclosed by polar curve
DifficultyStandard +0.8 This question requires understanding of polar curves including the four-petalled rose shape of r=2cos(2θ), identifying the domain for one loop (e.g., -π/4 to π/4), and applying the polar area formula with integration of cos²(2θ). While the integration itself is standard using double-angle formulas, the conceptual understanding of polar curves and determining correct limits makes this moderately challenging for Further Maths students.
Spec4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve
5 The curve \(C\) has polar equation \(r = 2 \cos 2 \theta\). Sketch the curve for \(0 \leqslant \theta < 2 \pi\). Find the exact area of one loop of the curve.
Question 5:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Position and through pole and \((2,0)\)B1B1 Right-hand loop
Position and through pole and \((2,\pi)\)B1B1 Left-hand loop; Deduct 1 mark for extra loops \((r<0)\); Part mark: 4
\(A = \frac{1}{2}\int 4\cos^2\theta\, d\theta\, d\theta\)M1 Uses \(A = \frac{1}{2}\int r^2\, d\theta\)
\(= \int(\cos 4\theta + 1)\,d\theta\) (LNR)M1 Uses double angle formula
\(= \left[\sin 4\theta + \theta\right]\) (LNR)A1 Integrates
e.g. \(\left[\sin 4\theta + \theta\right]_{-\pi/4}^{\pi/4} = \frac{\pi}{2}\)A1 Inserts any appropriate limits which legitimately give the result; Part mark: 4 
## Question 5:

| Answer/Working | Marks | Guidance |
|---|---|---|
| Position and through pole and $(2,0)$ | B1B1 | Right-hand loop |
| Position and through pole and $(2,\pi)$ | B1B1 | Left-hand loop; Deduct 1 mark for extra loops $(r<0)$; Part mark: 4 |
| $A = \frac{1}{2}\int 4\cos^2\theta\, d\theta\, d\theta$ | M1 | Uses $A = \frac{1}{2}\int r^2\, d\theta$ |
| $= \int(\cos 4\theta + 1)\,d\theta$ (LNR) | M1 | Uses double angle formula |
| $= \left[\sin 4\theta + \theta\right]$ (LNR) | A1 | Integrates |
| e.g. $\left[\sin 4\theta + \theta\right]_{-\pi/4}^{\pi/4} = \frac{\pi}{2}$ | A1 | Inserts any appropriate limits which legitimately give the result; Part mark: 4 |

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5 The curve $C$ has polar equation $r = 2 \cos 2 \theta$. Sketch the curve for $0 \leqslant \theta < 2 \pi$.

Find the exact area of one loop of the curve.

\hfill \mbox{\textit{CAIE FP1 2011 Q5 [8]}}
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