WJEC Further Unit 4 Specimen — Question 3 5 marks

Exam BoardWJEC
ModuleFurther Unit 4 (Further Unit 4)
SessionSpecimen
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolar coordinates
TypeArea of region with line boundary
DifficultyStandard +0.8 This is a Further Maths polar coordinates question requiring application of the polar area formula A = ½∫r²dθ, followed by integration of a trigonometric expression involving cos²θ. While the setup is standard, the algebraic manipulation (using double angle formula) and careful arithmetic to reach the exact form requires solid technique and accuracy across multiple steps, placing it moderately above average difficulty.
Spec4.09c Area enclosed: by polar curve
The curve \(C\) has polar equation \(r = 3(2 + \cos \theta)\), \(0 \leq \theta \leq \pi\). Determine the area enclosed between \(C\) and the initial line. Give your answer in the form \(\frac{a}{b}\pi\), where \(a\) and \(b\) are positive integers whose values are to be found. [5]
AnswerMarks Guidance
\(\text{Area} = \frac{1}{2}\int_0^{\pi} r^2 d\theta\)M1 AO1
\(= \frac{9}{2}\int_0^{\pi}(4 + 4\cos\theta + \cos^2\theta)d\theta\)A1 AO1
\(= \frac{9}{2}\left(\frac{9}{2} + 4\cos\theta + \frac{\cos 2\theta}{2}\right)\)A1 AO1
\(= \frac{9}{2}\left[\frac{9}{2}\theta + 4\sin\theta + \frac{\sin 2\theta}{4}\right]_0^{\pi}\)A1 AO1
\(= \frac{81\pi}{4}\)A1 AO1
Total: [5]
$\text{Area} = \frac{1}{2}\int_0^{\pi} r^2 d\theta$ | M1 | AO1

$= \frac{9}{2}\int_0^{\pi}(4 + 4\cos\theta + \cos^2\theta)d\theta$ | A1 | AO1

$= \frac{9}{2}\left(\frac{9}{2} + 4\cos\theta + \frac{\cos 2\theta}{2}\right)$ | A1 | AO1

$= \frac{9}{2}\left[\frac{9}{2}\theta + 4\sin\theta + \frac{\sin 2\theta}{4}\right]_0^{\pi}$ | A1 | AO1

$= \frac{81\pi}{4}$ | A1 | AO1

**Total: [5]**

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The curve $C$ has polar equation $r = 3(2 + \cos \theta)$, $0 \leq \theta \leq \pi$. Determine the area enclosed between $C$ and the initial line. Give your answer in the form $\frac{a}{b}\pi$, where $a$ and $b$ are positive integers whose values are to be found. [5]

\hfill \mbox{\textit{WJEC Further Unit 4  Q3 [5]}}
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