CAIE Further Paper 2 2022 June — Question 1 5 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2022
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolar coordinates
TypeArc length of polar curve
DifficultyStandard +0.8 This is a Further Maths polar arc length question requiring the formula s = ∫√(r² + (dr/dθ)²)dθ, differentiation of an exponential, and algebraic manipulation to make α the subject. While the integration is straightforward due to the exponential form, it requires confident handling of the arc length formula and rearrangement, placing it moderately above average difficulty.
Spec4.09c Area enclosed: by polar curve
1 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { \frac { 3 } { 4 } \theta }\) for \(0 \leqslant \theta \leqslant \alpha\).
Given that the length of \(C\) is \(s\), find \(\alpha\) in terms of \(s\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(s = \int_0^{\alpha} \sqrt{e^{\frac{3}{2}\theta} + \frac{9}{16}e^{\frac{3}{2}\theta}} \, d\theta = \frac{5}{4}\int_0^{\alpha} e^{\frac{3}{4}\theta} \, d\theta\)M1 A1 Forms \(s\) correctly
\(s = \frac{5}{3}\left[e^{\frac{3}{4}\theta}\right]_0^{\alpha} = \frac{5}{3}\left(e^{\frac{3}{4}\alpha} - 1\right)\)M1 A1 Integrates
\(\alpha = \frac{4}{3}\ln\left(1 + \frac{3}{5}s\right)\)A1
Total5 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $s = \int_0^{\alpha} \sqrt{e^{\frac{3}{2}\theta} + \frac{9}{16}e^{\frac{3}{2}\theta}} \, d\theta = \frac{5}{4}\int_0^{\alpha} e^{\frac{3}{4}\theta} \, d\theta$ | M1 A1 | Forms $s$ correctly |
| $s = \frac{5}{3}\left[e^{\frac{3}{4}\theta}\right]_0^{\alpha} = \frac{5}{3}\left(e^{\frac{3}{4}\alpha} - 1\right)$ | M1 A1 | Integrates |
| $\alpha = \frac{4}{3}\ln\left(1 + \frac{3}{5}s\right)$ | A1 | |
| **Total** | **5** | |

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1 The curve $C$ has polar equation $r = \mathrm { e } ^ { \frac { 3 } { 4 } \theta }$ for $0 \leqslant \theta \leqslant \alpha$.\\
Given that the length of $C$ is $s$, find $\alpha$ in terms of $s$.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2022 Q1 [5]}}
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Q1 5 Q2 8 Q3 8 Q4 10 Q5 10 Q6 10 Q7 11 Q8 13