A2 June 2025 Paper 1 Q6

OCR MEICurrent spec9 marksPolar Coordinates

6 The figure below shows the curve with cartesian equation \((x^2 + y^2)^2 = xy\).

Curve (x squared + y squared) squared = xy: two loops through the origin O, one in the first quadrant and one in the third quadrant

(a) Show that the polar equation of the curve is \(r^2 = a\sin b\theta\), where \(a\) and \(b\) are positive constants to be determined. [3]

(b) Determine the exact maximum value of \(r\). [2]

Mark scheme (a)

Scheme	Marks	AO
\(x^2 + y^2 = r^2,\; x = r\cos\theta,\; y = r\sin\theta\)	M1	1.1
\(\Rightarrow r^4 = r^2\cos\theta\sin\theta\)	A1	1.1
\(\Rightarrow r^2 = \tfrac{1}{2}\sin 2\theta\) [so \(a = \tfrac{1}{2}\) and \(b = 2\)]	A1	1.1
	[3]

Notes

M1: substituting for all \(x\) and \(y\) terms correctly. Implied by a correct unsimplified equation for \(r^4\) or \((r^2)^2\)

A1: RHS could be unsimplified; \(r^4\) could be left as \((r^2)^2\)

Mark scheme (b)

Scheme	Marks	AO
maximum when \(\sin 2\theta = 1\)	B1FT	3.1a
\(\Rightarrow r = \frac{1}{\sqrt{2}}\)	B1FT	1.1
	[2]

Notes

B1FT: or \(r^2\) maximum \(= \frac{1}{2}\) or correct differentiation of their \(r^2\) (or \(r\)), equating to zero and getting \(\cos 2\theta = 0\), or \(\theta = \frac{\pi}{4}\). soi by correct value for their \(r\). FT from part (a)

B1FT: o.e. must be exact. www. FT from part (a)

Mark scheme (c)

Scheme	Marks	AO
\(A = \dfrac{1}{4}\displaystyle\int_0^{\frac{\pi}{2}} \sin 2\theta\,\mathrm{d}\theta\)	M1	1.1
\(= \left[-\dfrac{1}{8}\cos 2\theta\right]_0^{\frac{\pi}{2}}\)	A1FT	1.1
\(= -\dfrac{1}{8}\left(\cos 2\left(\dfrac{\pi}{2}\right) - \cos 0\right)\)	M1	1.1
\(= \dfrac{1}{4}\)	A1	1.1
	[4]

Notes

M1: \(\frac{1}{2}\int r^2\,\mathrm{d}\theta\) used with their \(r^2\) provided it is in the form \(a\sin b\theta\). Must be seen. Condone missing \(\mathrm{d}\theta\) and missing or incorrect limits

A1FT: FT their integral (i.e. \(-\frac{a}{2b}\cos b\theta\)). Must be seen.
Note \(\int \sin\theta\cos\theta\,\mathrm{d}\theta = \frac{1}{2}\sin^2\theta\) or \(-\frac{1}{2}\cos^2\theta\),

M1: substituting correct limits after attempt to integrate, soi by e.g. correct answer or \(\frac{1}{4}\left(\frac{1}{2} + \frac{1}{2}\right)\) if correct integral found. Accept correct alternative limits, e.g. \(\pi\) to \(\frac{3\pi}{2}\)
Requires an initial integral of the form \(k\int a\sin b\theta\,\mathrm{d}\theta\)

A1: cao www