June 2024 Paper 3 Q12

OCR MEICurrent spec9 marksDifferentiation Parametric Equations

12 The diagram shows the curve with parametric equations

\(x = \sin 2\theta + 2,\ y = 2\cos\theta + \cos 2\theta\), for \(0 \leqslant \theta \lt 2\pi\).

Curve shaped like a large loop above the x-axis crossing itself below the x-axis, with two small loops at the bottom, symmetrical about a vertical line

(a) In this question you must show detailed reasoning.
Determine the exact coordinates of all the stationary points on the curve. [8]

(b) Write down the equation of the line of symmetry of the curve. [1]

Mark scheme (a)

Scheme	Marks	AO
DR \(\dfrac{\mathrm{d}y}{\mathrm{d}\theta} = -2\sin\theta - 2\sin 2\theta\)	M1	3.1a
\(\sin\theta + 2\sin\theta\cos\theta = 0\)	M1	3.1a
\(\boldsymbol{\sin\theta(1 + 2\cos\theta) = 0}\)	*M1	1.1
\(\sin\theta = 0\) or \(\cos\theta = -\tfrac{1}{2}\)	DM1	1.1
\(\theta = 0\), \(\dfrac{2\pi}{3}\), \(\pi\), \(\dfrac{4\pi}{3}\)	A1 A1	1.1 1.1
\((2, 3)\) \((2, -1)\)	A1	3.2a
\(\left(2 - \dfrac{\sqrt{3}}{2},\ -1.5\right)\)
\(\left(2 + \dfrac{\sqrt{3}}{2},\ -1.5\right)\)	A1	3.2a
	[8]

Notes

M1: For differentiation of \(y\) wrt. \(\theta\), may be as part of \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\)

M1: For \(\dfrac{\mathrm{d}y}{\mathrm{d}\theta} = 0\) or \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) and use of sine or cosine double angle formula.
If \(\dfrac{\mathrm{d}x}{\mathrm{d}\theta}\) is used incorrectly then give M0

*M1: All on one side and factorised.
If they have divided through by \(\sin\theta\) then \(\boldsymbol{1 = -2\cos\theta}\) or \(\boldsymbol{1 + 2\cos\theta = 0}\) will earn this mark

DM1: Dependent on the previous method mark
At least one trig value
Condone if divides through by sin.

A1 A1: www
A marks dependent on M4
At least 2 correct values
All four values correct with no extras

A1: www
At least 2 correct coordinates
oe, must be exact form

A1: www
All four coordinates correct with no extras
oe, must be exact form

Additional guidance

The first M1 is for differentiating y wrt \(\theta\). It might be seen on it’s own as \(\mathrm{d}y/\mathrm{d}\theta\) or as the numerator of \(\mathrm{d}y/\mathrm{d}x\). We are ignoring \(\mathrm{d}x/\mathrm{d}\theta\) unless they try to use it incorrectly in their subsequent work.

The second M1 is for putting their expression equal to 0 AND using either \(\sin 2\theta = 2\sin\theta\cos\theta\) or \(\cos 2\theta\) = one of the correct forms. But if \(\mathrm{d}x/\mathrm{d}\theta\) is used incorrectly here then give M0.

The third M1 is for getting terms on one side and factorised. However, some divide through by sin at this point – they can still get the M1 and also the fourth M1. The 3^rd and 4^th M1s can be implied (BOD) by seeing 2 correct solutions \(\sin\theta = 0\) and \(\cos\theta = -\frac{1}{2}\).

The 4^th M1 is dependent on the previous M1. (We have seen some candidates obtain \(\sin\theta = 0\) (and possibly \(\cos\theta = -\frac{1}{2}\)) from completely wrong working but this does not score).

The A marks are dependent on getting M4 and are for finding exact values of \(\theta\) and the coordinates. Note that the A marks are all www marks (without wrong working), so dividing through by \(\sin\theta\) will cost all the A marks. Allow equivalent correct forms (e.g. \(\frac{4-\sqrt{3}}{2}\) for \(2 - \frac{\sqrt{3}}{2}\), \(\frac{4+\sqrt{3}}{2}\) for \(2 + \frac{\sqrt{3}}{2}\)) for the x-values.

Scheme	Marks	AO
\(x = 2\)	B1	2.2a
	[1]