Question 5 - A-Level Maths

CAIE P3 2009 November — Question 5 8 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2009
Session	November
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Standard Integrals and Reverse Chain Rule
Type	Use trig identity before definite integration
Difficulty	Standard +0.3 This is a two-part question requiring a standard trigonometric identity proof using double angle formulas, followed by a straightforward definite integral using the proven result. The proof involves routine manipulation of cos 2θ and cos 4θ formulas, and the integration is direct substitution with simple limits. Slightly easier than average due to the guided structure and standard techniques.
Spec	1.05l Double angle formulae: and compound angle formulae 1.05p Proof involving trig: functions and identities 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

Prove the identity $\cos 4 \theta - 4 \cos 2 \theta + 3 \equiv 8 \sin ^ { 4 } \theta$.
Using this result find, in simplified form, the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta$$

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(i) EITHER:

Answer	Marks
Use double angle formulae correctly to express LHS in terms of trig functions of $2\theta$	M1
Use trig formulae correctly to express LHS in terms of $\sin\theta$, converting at least two terms	M1
Obtain expression in any correct form in terms of $\sin\theta$	A1
Obtain given answer correctly	A1

OR:

Answer	Marks
Use double angle formulae correctly to express RHS in terms of trig functions of $2\theta$	M1
Use trig formulae correctly to express RHS in terms of $\cos 4\theta$ and $\cos 2\theta$	M1
Obtain expression in any correct form in terms of $\cos 4\theta$ and $\cos 2\theta$	A1
Obtain given answer correctly	A1
[4]

(ii)

Answer	Marks
State indefinite integral $\frac{1}{4}\sin 4\theta - \frac{4}{9}\sin 2\theta + 3\theta$, or equivalent	B2
(award B1 if there is just one incorrect term)
Use limits correctly, having attempted to use the identity	M1
Obtain answer $\frac{1}{12}(2\pi - \sqrt{3})$, or any simplified exact equivalent	A1
[4]

**(i) EITHER:**

| Use double angle formulae correctly to express LHS in terms of trig functions of $2\theta$ | M1 |
| Use trig formulae correctly to express LHS in terms of $\sin\theta$, converting at least two terms | M1 |
| Obtain expression in any correct form in terms of $\sin\theta$ | A1 |
| Obtain given answer correctly | A1 |

**OR:**

| Use double angle formulae correctly to express RHS in terms of trig functions of $2\theta$ | M1 |
| Use trig formulae correctly to express RHS in terms of $\cos 4\theta$ and $\cos 2\theta$ | M1 |
| Obtain expression in any correct form in terms of $\cos 4\theta$ and $\cos 2\theta$ | A1 |
| Obtain given answer correctly | A1 |
| [4] |

**(ii)**

| State indefinite integral $\frac{1}{4}\sin 4\theta - \frac{4}{9}\sin 2\theta + 3\theta$, or equivalent | B2 |
| (award B1 if there is just one incorrect term) | |
| Use limits correctly, having attempted to use the identity | M1 |
| Obtain answer $\frac{1}{12}(2\pi - \sqrt{3})$, or any simplified exact equivalent | A1 |
| [4] |

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5 (i) Prove the identity $\cos 4 \theta - 4 \cos 2 \theta + 3 \equiv 8 \sin ^ { 4 } \theta$.\\
(ii) Using this result find, in simplified form, the exact value of

$$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta$$

\hfill \mbox{\textit{CAIE P3 2009 Q5 [8]}}

This paper (10 questions)

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

Answer	Marks
State indefinite integral \(\frac{1}{4}\sin 4\theta - \frac{4}{9}\sin 2\theta + 3\theta\), or equivalent	B2
(award B1 if there is just one incorrect term)
Use limits correctly, having attempted to use the identity	M1
Obtain answer \(\frac{1}{12}(2\pi - \sqrt{3})\), or any simplified exact equivalent	A1
[4]