Question 5 - A-Level Maths

CAIE P3 2016 June — Question 5 8 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2016
Session	June
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Addition & Double Angle Formulae
Type	Prove identity then solve equation
Difficulty	Standard +0.3 This is a standard two-part question on double angle formulae. Part (i) requires systematic application of cos 2θ and cos 4θ identities—routine but multi-step. Part (ii) is straightforward substitution leading to sin⁴θ = 0, giving θ = 0°, 90°, 180°, 270°, 360°. Slightly above average due to the algebraic manipulation required, but follows predictable patterns for P3 trigonometric identities.
Spec	1.05l Double angle formulae: and compound angle formulae 1.05o Trigonometric equations: solve in given intervals 1.05p Proof involving trig: functions and identities

Prove the identity $\cos 4 \theta - 4 \cos 2 \theta \equiv 8 \sin ^ { 4 } \theta - 3$.
Hence solve the equation $$\cos 4 \theta = 4 \cos 2 \theta + 3$$ for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

Show mark scheme Show mark scheme source

Answer	Marks	Guidance
(i) EITHER: Express $\cos 4\theta$ in terms of $\cos 2\theta$ and/or $\sin 2\theta$	B1
Use correct double angle formulae to express LHS in terms of $\sin \theta$ and/or $\cos \theta$	M1
Obtain a correct expression in terms of $\sin \theta$ alone	A1
Reduce correctly to the given form	A1
OR: Use correct double angle formula to express RHS in terms of $\cos 2\theta$	M1
Express $\cos^2 2\theta$ in terms of $\cos 4\theta$ and $\cos 2\theta$	B1
Obtain a correct expression in terms of $\cos 4\theta$ and $\cos 2\theta$	A1
Reduce correctly to the given form	A1	[4]
(ii) Use the identity and carry out a method for finding a root	M1
Obtain answer $68.5°$	A1
Obtain a second answer, e.g. $291.5°$	A1♦
Obtain the remaining answers, e.g. $111.5°$ and $248.5°$, and no others in the given interval	A1♦	[Ignore answers outside the given interval. Treat answers in radians as a misread.]

**(i) EITHER:** Express $\cos 4\theta$ in terms of $\cos 2\theta$ and/or $\sin 2\theta$ | B1 |
Use correct double angle formulae to express LHS in terms of $\sin \theta$ and/or $\cos \theta$ | M1 |
Obtain a correct expression in terms of $\sin \theta$ alone | A1 |
Reduce correctly to the given form | A1 |

**OR:** Use correct double angle formula to express RHS in terms of $\cos 2\theta$ | M1 |
Express $\cos^2 2\theta$ in terms of $\cos 4\theta$ and $\cos 2\theta$ | B1 |
Obtain a correct expression in terms of $\cos 4\theta$ and $\cos 2\theta$ | A1 |
Reduce correctly to the given form | A1 | [4]

**(ii)** Use the identity and carry out a method for finding a root | M1 |
Obtain answer $68.5°$ | A1 |
Obtain a second answer, e.g. $291.5°$ | A1♦ |
Obtain the remaining answers, e.g. $111.5°$ and $248.5°$, and no others in the given interval | A1♦ | [Ignore answers outside the given interval. Treat answers in radians as a misread.] | [4]

Show LaTeX source

5 (i) Prove the identity $\cos 4 \theta - 4 \cos 2 \theta \equiv 8 \sin ^ { 4 } \theta - 3$.\\
(ii) Hence solve the equation

$$\cos 4 \theta = 4 \cos 2 \theta + 3$$

for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

\hfill \mbox{\textit{CAIE P3 2016 Q5 [8]}}

This paper (10 questions)

View full paper

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

Answer	Marks	Guidance
(i) EITHER: Express \(\cos 4\theta\) in terms of \(\cos 2\theta\) and/or \(\sin 2\theta\)	B1
Use correct double angle formulae to express LHS in terms of \(\sin \theta\) and/or \(\cos \theta\)	M1
Obtain a correct expression in terms of \(\sin \theta\) alone	A1
Reduce correctly to the given form	A1
OR: Use correct double angle formula to express RHS in terms of \(\cos 2\theta\)	M1
Express \(\cos^2 2\theta\) in terms of \(\cos 4\theta\) and \(\cos 2\theta\)	B1
Obtain a correct expression in terms of \(\cos 4\theta\) and \(\cos 2\theta\)	A1
Reduce correctly to the given form	A1	[4]
(ii) Use the identity and carry out a method for finding a root	M1
Obtain answer \(68.5°\)	A1
Obtain a second answer, e.g. \(291.5°\)	A1♦
Obtain the remaining answers, e.g. \(111.5°\) and \(248.5°\), and no others in the given interval	A1♦	[Ignore answers outside the given interval. Treat answers in radians as a misread.]