A2 June 2023 Paper 1 Q7

OCR MEICurrent spec6 marksPolar Coordinates

7 The diagram below shows the curve with polar equation \(r = a(1 - 2\sin\theta)\) for \(0 \leqslant \theta \leqslant 2\pi\), where \(a\) is a positive constant.

Polar curve: a large outer loop with a small dashed inner loop inside it, both passing through the pole O; the initial line theta = 0 from O crosses the curve at A; B is the lowest point of the inner loop and C the lowest point of the outer loop

The curve crosses the initial line at A, and the points B and C are the lowest points on the two loops.

(a) Find the values of \(r\) and \(\theta\) at the points A, B and C. [3]

(b) Find the set of values of \(\theta\) for the points on the inner loop (shown in the diagram with a broken line). [3]

Scheme	Marks	AO
A: \(r = a,\ \theta = 0\)	B1	1.1
B: \(r = -a,\ \theta = \pi/2\)	B1	1.1
C: \(r = 3a,\ \theta = 3\pi/2\)	B1	1.1
	[3]

Mark scheme (b)

Scheme	Marks	AO
\(\sin\theta \gt \dfrac{1}{2}\)	M1	3.1a
\(\dfrac{1}{6}\pi, \dfrac{5}{6}\pi\)	A1	1.1
\(\dfrac{1}{6}\pi \lt \theta \lt \dfrac{5}{6}\pi\)	A1	1.1
	[3]

Notes

M1: Allow \(\sin\theta = \frac{1}{2}\) (or \(\lt \frac{1}{2}\) or \(\leqslant \frac{1}{2}\) or \(\geqslant \frac{1}{2}\))

A1: both

A1: condone \(\leqslant\) for \(\lt\)