Question 4 - A-Level Maths

CAIE FP1 2013 November — Question 4 7 marks

Exam Board	CAIE
Module	FP1 (Further Pure Mathematics 1)
Year	2013
Session	November
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Parametric differentiation
Type	Find second derivative d²y/dx²
Difficulty	Standard +0.3 This is a standard Further Maths parametric differentiation question requiring the chain rule formula dy/dx = (dy/dθ)/(dx/dθ), followed by finding the second derivative. The trigonometric differentiation is straightforward, and identifying where dx/dθ = 0 is routine. While it's Further Maths content, it follows a well-practiced procedure with no novel insight required, making it slightly easier than average.
Spec	1.03g Parametric equations: of curves and conversion to cartesian 1.07s Parametric and implicit differentiation

4 A curve has parametric equations $$x = 2 \theta - \sin 2 \theta , \quad y = 1 - \cos 2 \theta , \quad \text { for } - 3 \pi \leqslant \theta \leqslant 3 \pi$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta$$ except for certain values of $\theta$, which should be stated. Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $\theta = \frac { 1 } { 4 } \pi$.

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Question 4:

Answer	Marks	Guidance
Working/Answer	Marks	Guidance
$\dot{x} = 2 - 2\cos 2\theta$, $\dot{y} = 2\sin 2\theta$	B1	Differentiates each equation wrt $\theta$
$\frac{dy}{dx} = \frac{4\sin\theta\cos\theta}{4\sin^2\theta} = \cot\theta$	M1A1	(4) AG
Except for $\theta = k\pi$ ($k$ an integer)	A1	States exceptions
$\frac{d^2y}{dx^2} = \frac{d}{d\theta}\cot\theta \times \frac{1}{\dot{x}} = -\text{cosec}^2\theta \times \frac{1}{4\sin^2\theta}$	M1A1	Differentiates wrt $x$
$= -1$ when $\theta = \frac{1}{4}\pi$	A1	(3)

Total: [7]

## Question 4:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $\dot{x} = 2 - 2\cos 2\theta$, $\dot{y} = 2\sin 2\theta$ | B1 | Differentiates each equation wrt $\theta$ |
| $\frac{dy}{dx} = \frac{4\sin\theta\cos\theta}{4\sin^2\theta} = \cot\theta$ | M1A1 | **(4)** AG |
| Except for $\theta = k\pi$ ($k$ an integer) | A1 | States exceptions |
| $\frac{d^2y}{dx^2} = \frac{d}{d\theta}\cot\theta \times \frac{1}{\dot{x}} = -\text{cosec}^2\theta \times \frac{1}{4\sin^2\theta}$ | M1A1 | Differentiates wrt $x$ |
| $= -1$ when $\theta = \frac{1}{4}\pi$ | A1 | **(3)** |

**Total: [7]**

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4 A curve has parametric equations

$$x = 2 \theta - \sin 2 \theta , \quad y = 1 - \cos 2 \theta , \quad \text { for } - 3 \pi \leqslant \theta \leqslant 3 \pi$$

Show that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta$$

except for certain values of $\theta$, which should be stated.

Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $\theta = \frac { 1 } { 4 } \pi$.

\hfill \mbox{\textit{CAIE FP1 2013 Q4 [7]}}

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Q1 6 Q2 6 Q3 7 Q4 7 Q5 8 Q6 8 Q7 9 Q8 11 Q9 11 Q10 13 Q11 EITHER Q11 OR