2019 CJC P2 Q5
2019 CJC P2 Q5
14 marks
Prelims
A curve \(C\) has parametric equations \(x=1-\sin \theta \), \(y=\theta +\cos \theta \), where \(-\pi \le \theta \le \frac{\pi }{2}\).
- Sketch the graph of \(C\), stating the exact coordinates of the end points.[2]
- Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(\theta \). What can be said about the tangent to \(C\) as \(\theta \to -\frac{\pi }{2}\)?[3]
- A point \(P\) on \(C\) has parameter \(p\), where \(0 < p < \frac{\pi }{2}\). Show that the normal to \(C\) at \(P\) crosses the \(y\)-axis at point \(Q\) with coordinates \(\left( 0,p \right)\).[3]
- Show that the area of region bounded by \(C\), the normal to \(C\) at point \(P\) and the \(y\)-axis is given by \(a\pi +bp+c\cos p\), where \(a\), \(b\) and \(c\) are to be determined.[6]
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Answer:(i) Endpoints \(\left(0,\frac\pi2\right)\) and \((1,-\pi-1)\); sketch in typed solution. (ii) \(\frac{\mathrm dy}{\mathrm dx}=\frac{\sin\theta-1}{\cos\theta}\); vertical tangent \(x=2\) at \(\theta=-\frac\pi2\). (iii) \(Q=(0,p)\). (iv) \(A=\frac{3\pi}{4}-\frac{3p}{2}-\frac{3\cos p}{2}\), with \(a=\frac34\) and \(b=c=-\frac32\).
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