CAIE P3 2014 November — Question 2 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2014
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicParametric differentiation
TypeShow dy/dx simplifies to given form
DifficultyStandard +0.3 This is a straightforward parametric differentiation question requiring the chain rule (dy/dx = dy/dθ ÷ dx/dθ) and standard trigonometric differentiation. While it involves some algebraic manipulation with trigonometric identities to reach the given form, the technique is routine and the question provides the target expression to verify, making it slightly easier than average.
Spec1.03g Parametric equations: of curves and conversion to cartesian1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07s Parametric and implicit differentiation
2 A curve is defined for \(0 < \theta < \frac { 1 } { 2 } \pi\) by the parametric equations $$x = \tan \theta , \quad y = 2 \cos ^ { 2 } \theta \sin \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \cos ^ { 5 } \theta - 4 \cos ^ { 3 } \theta\).
AnswerMarks Guidance
Use correct product rule or correct chain rule to differentiate \(y\)M1
Use \(\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}\)M*1
Obtain \(\frac{-4\cos\theta\sin^2\theta + 2\cos^3\theta}{\sec^2\theta}\) or equivalentA1
Express \(\frac{dy}{dx}\) in terms of \(\cos\theta\)DM*1
Confirm given answer \(6\cos^3\theta - 4\cos^3\theta\) legitimatelyA1 [5] 
Use correct product rule or correct chain rule to differentiate $y$ | M1 |
Use $\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$ | M*1 |
Obtain $\frac{-4\cos\theta\sin^2\theta + 2\cos^3\theta}{\sec^2\theta}$ or equivalent | A1 |
Express $\frac{dy}{dx}$ in terms of $\cos\theta$ | DM*1 |
Confirm given answer $6\cos^3\theta - 4\cos^3\theta$ legitimately | A1 | [5]
2 A curve is defined for $0 < \theta < \frac { 1 } { 2 } \pi$ by the parametric equations

$$x = \tan \theta , \quad y = 2 \cos ^ { 2 } \theta \sin \theta$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 \cos ^ { 5 } \theta - 4 \cos ^ { 3 } \theta$.

\hfill \mbox{\textit{CAIE P3 2014 Q2 [5]}}
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