Question 3 - A-Level Maths

CAIE P3 2012 June — Question 3 5 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2012
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Parametric differentiation
Type	Show dy/dx simplifies to given form
Difficulty	Moderate -0.3 This is a straightforward parametric differentiation question requiring standard application of dy/dx = (dy/dθ)/(dx/dθ). The derivatives involve routine trigonometric differentiation (chain rule for sin 2θ and cos 2θ), followed by algebraic simplification using the double angle formula cos 2θ = 1 - 2sin²θ. While it requires careful manipulation, it's a standard textbook exercise with no novel insight needed, making it slightly easier than average.
Spec	1.03g Parametric equations: of curves and conversion to cartesian 1.07s Parametric and implicit differentiation

3 The parametric equations of a curve are $$x = \sin 2 \theta - \theta , \quad y = \cos 2 \theta + 2 \sin \theta$$ Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \cos \theta } { 1 + 2 \sin \theta }$.

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Answer	Marks	Guidance
Obtain $\frac{dy}{d\theta} = 2\cos 2\theta - 1$ or $\frac{dy}{d\theta} = -2\sin 2\theta + 2\cos\theta$, or equivalent	B1
Use $\frac{dy}{dx} = \frac{dy}{d\theta} \div \frac{dx}{d\theta}$	M1
Obtain $\frac{dy}{dx} = \frac{-2\sin 2\theta + 2\cos\theta}{2\cos 2\theta - 1}$, or equivalent	A1
At any stage use correct double angle formulae throughout	M1
Obtain the given answer following full and correct working	A1	[5]

Obtain $\frac{dy}{d\theta} = 2\cos 2\theta - 1$ or $\frac{dy}{d\theta} = -2\sin 2\theta + 2\cos\theta$, or equivalent | B1 |
Use $\frac{dy}{dx} = \frac{dy}{d\theta} \div \frac{dx}{d\theta}$ | M1 |
Obtain $\frac{dy}{dx} = \frac{-2\sin 2\theta + 2\cos\theta}{2\cos 2\theta - 1}$, or equivalent | A1 |
At any stage use correct double angle formulae throughout | M1 |
Obtain the given answer following full and correct working | A1 | [5]

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3 The parametric equations of a curve are

$$x = \sin 2 \theta - \theta , \quad y = \cos 2 \theta + 2 \sin \theta$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \cos \theta } { 1 + 2 \sin \theta }$.

\hfill \mbox{\textit{CAIE P3 2012 Q3 [5]}}

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Q1 4 Q2 4 Q3 5 Q4 6 Q5 8 Q6 8 Q7 9 Q8 10 Q9 10 Q10 11

Answer	Marks	Guidance
Obtain \(\frac{dy}{d\theta} = 2\cos 2\theta - 1\) or \(\frac{dy}{d\theta} = -2\sin 2\theta + 2\cos\theta\), or equivalent	B1
Use \(\frac{dy}{dx} = \frac{dy}{d\theta} \div \frac{dx}{d\theta}\)	M1
Obtain \(\frac{dy}{dx} = \frac{-2\sin 2\theta + 2\cos\theta}{2\cos 2\theta - 1}\), or equivalent	A1
At any stage use correct double angle formulae throughout	M1
Obtain the given answer following full and correct working	A1	[5]