FP2 June 2012 Q2

2. The curve \(C\) has polar equation \[r = 1 + 2\cos\theta, \quad 0 \leqslant \theta \leqslant \frac{\pi}{2}\]

At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the initial line.

Given that \(O\) is the pole, find the exact length of the line \(OP\). (7)

Mark scheme

Scheme	Marks
\(y = r\sin\theta = \sin\theta + 2\sin\theta\cos\theta\)	B1
\(\dfrac{\mathrm{d}y}{\mathrm{d}\theta} = \cos\theta + 2\cos 2\theta\)	M1
\(4\cos^2\theta + \cos\theta - 2 = 0\)	A1oe
\(\cos\theta = \dfrac{-1 \pm \sqrt{1 + 32}}{8}\)	M1 A1
\(OP = r = 1 + \dfrac{-1 + \sqrt{1 + 32}}{4} = \dfrac{3 + \sqrt{33}}{4}\)	M1 A1
	(7)
	(7 marks)

B1 for \(\sin\theta + 2\sin\theta\cos\theta\) or \(\sin\theta(1 + 2\cos\theta)\)

1^st M1 for use of Product Rule or Chain Rule (require 2 or condone \(\tfrac{1}{2}\))

1^st A1 equation required

2^nd M1 Valid attempt at solving 3 term quadratic (usual rules) to give \(\cos\theta = \ldots\)

2^nd A1 for exact or 3 dp or better (\(-0.843\ldots\) and \(0.593\ldots\))

3^rd M1 for \(1 + 2x\) ‘their \(\cos\theta\)’

3^rd A1 for any form A0 if negative solution not discounted.