CAIE FP1 2014 November — Question 2 6 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2014
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicParametric integration
TypeParametric arc length calculation
DifficultyChallenging +1.2 This is a parametric arc length problem requiring differentiation of exponential-trigonometric products, application of the arc length formula, and integration. While it involves multiple steps and Further Maths content (making it inherently harder than typical A-level), the derivatives follow standard product rule patterns, and the resulting integral simplifies nicely to √2∫e^t dt, which is straightforward. It's a standard textbook exercise for FP1 parametric integration rather than requiring novel insight.
Spec4.08a Maclaurin series: find series for function
2 A curve \(C\) has parametric equations $$x = \mathrm { e } ^ { t } \cos t , \quad y = \mathrm { e } ^ { t } \sin t , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$ Find the arc length of \(C\).
Question 2:
AnswerMarks Guidance
Working/AnswerMarks Guidance
\(\dot{x} = e^t\cos t - e^t\sin t \qquad \dot{y} = e^t\sin t + e^t\cos t\)M1A1
\(\dot{x}^2+\dot{y}^2 = 2e^{2t}(\cos^2 t+\sin^2 t)=2e^{2t}\)B1
\(s=\int_0^{\frac{1}{2}\pi}\sqrt{2}\,e^t\,\mathrm{d}t\)M1
\(=\left[\sqrt{2}\,e^t\right]_0^{\frac{1}{2}\pi}=\sqrt{2}\!\left(e^{\frac{1}{2}\pi}-1\right) \quad(=5.39)\)A1A1 (6) 
## Question 2:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\dot{x} = e^t\cos t - e^t\sin t \qquad \dot{y} = e^t\sin t + e^t\cos t$ | M1A1 | |
| $\dot{x}^2+\dot{y}^2 = 2e^{2t}(\cos^2 t+\sin^2 t)=2e^{2t}$ | B1 | |
| $s=\int_0^{\frac{1}{2}\pi}\sqrt{2}\,e^t\,\mathrm{d}t$ | M1 | |
| $=\left[\sqrt{2}\,e^t\right]_0^{\frac{1}{2}\pi}=\sqrt{2}\!\left(e^{\frac{1}{2}\pi}-1\right) \quad(=5.39)$ | A1A1 (6) | |

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2 A curve $C$ has parametric equations

$$x = \mathrm { e } ^ { t } \cos t , \quad y = \mathrm { e } ^ { t } \sin t , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$

Find the arc length of $C$.

\hfill \mbox{\textit{CAIE FP1 2014 Q2 [6]}}
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Q1 5 Q2 6 Q3 7 Q4 7 Q5 8 Q6 9 Q7 10 Q8 11 Q9 11 Q10 12 Q11 EITHER Q11 OR