Question 7 - A-Level Maths

CAIE P2 2007 November — Question 7 8 marks

Exam Board	CAIE
Module	P2 (Pure Mathematics 2)
Year	2007
Session	November
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Addition & Double Angle Formulae
Type	Prove identity with double/compound angles
Difficulty	Standard +0.3 Part (i) is a straightforward algebraic expansion followed by direct application of standard double angle formulae (cos 2x = 1 - 2sin²x and sin 2x = 2sin x cos x). Part (ii) requires integrating the simplified form using standard results, with careful evaluation at π/4. This is a routine multi-step question testing standard techniques with no novel insight required, making it slightly easier than average.
Spec	1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1 1.05l Double angle formulae: and compound angle formulae 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits

Prove the identity $$( \cos x + 3 \sin x ) ^ { 2 } \equiv 5 - 4 \cos 2 x + 3 \sin 2 x$$
Using the identity, or otherwise, find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \cos x + 3 \sin x ) ^ { 2 } d x$$

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(i)

Answer	Marks	Guidance
Expand and use $\sin 2A$ formula	M1
Use $\cos 2A$ formula at least once	M1
Obtain any correct expression in terms of $\cos 2x$ and $\sin 2x$ only – can be implied	A1
Obtain given answer correctly	A1	[4]

(ii)

Answer	Marks	Guidance
State indefinite integral $5x - 2\sin 2x - \frac{3}{2}\cos 2x$	B2
[Award B1 if one error in one term]
Substitute limits correctly – must be correct limits	M1
Obtain answer $\frac{1}{4}(5\pi - 2)$, or exact simplified equivalent	A1	[4]

**(i)**

Expand and use $\sin 2A$ formula | M1 |
Use $\cos 2A$ formula at least once | M1 |
Obtain any correct expression in terms of $\cos 2x$ and $\sin 2x$ only – can be implied | A1 |
Obtain given answer correctly | A1 | [4]

**(ii)**

State indefinite integral $5x - 2\sin 2x - \frac{3}{2}\cos 2x$ | B2 |
[Award B1 if one error in one term] | |
Substitute limits correctly – must be correct limits | M1 |
Obtain answer $\frac{1}{4}(5\pi - 2)$, or exact simplified equivalent | A1 | [4]

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(i) Prove the identity

$$( \cos x + 3 \sin x ) ^ { 2 } \equiv 5 - 4 \cos 2 x + 3 \sin 2 x$$

(ii) Using the identity, or otherwise, find the exact value of

$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \cos x + 3 \sin x ) ^ { 2 } d x$$

\hfill \mbox{\textit{CAIE P2 2007 Q7 [8]}}

This paper (8 questions)

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Q1 4 Q2 5 Q3 5 Q4 5 Q5 6 Q6 7 Q7 8 Q8 10

Answer	Marks	Guidance
Expand and use \(\sin 2A\) formula	M1
Use \(\cos 2A\) formula at least once	M1
Obtain any correct expression in terms of \(\cos 2x\) and \(\sin 2x\) only – can be implied	A1
Obtain given answer correctly	A1	[4]

Answer	Marks	Guidance
State indefinite integral \(5x - 2\sin 2x - \frac{3}{2}\cos 2x\)	B2
[Award B1 if one error in one term]
Substitute limits correctly – must be correct limits	M1
Obtain answer \(\frac{1}{4}(5\pi - 2)\), or exact simplified equivalent	A1	[4]