WJEC Further Unit 4 Specimen — Question 5 8 marks

Exam BoardWJEC
ModuleFurther Unit 4 (Further Unit 4)
SessionSpecimen
Marks8
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Mark schemeDownload PDF ↗
TopicStandard trigonometric equations
TypeFactorization method
DifficultyChallenging +1.2 This is a Further Maths trigonometric equation requiring sum-to-product identities and systematic solving. While it involves multiple steps (combining cos θ + cos 5θ first, then factoring), the techniques are standard for Further Maths students. The 8-mark allocation reflects moderate length rather than exceptional difficulty, and the interval restriction simplifies the solution space.
Spec1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals
Find all the roots of the equation $$\cos \theta + \cos 3\theta + \cos 5\theta = 0$$ lying in the interval \([0, \pi]\). Give all the roots in radians in terms of \(\pi\). [8]
Rewrite the equation in the form:
AnswerMarks Guidance
\(\cos 3\theta + 2\cos 2\theta \cos 3\theta = 0\)M1 AO1
\(\cos 3\theta(1 + 2\cos 2\theta) = 0\)A1 AO1
Either \(\cos 3\theta = 0\):
AnswerMarks Guidance
\(3\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}\)M1 AO1
\(\theta = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}\)A1 AO1
Or \(\cos 2\theta = -\frac{1}{2}\):
AnswerMarks Guidance
\(2\theta = \frac{2\pi}{3}, \frac{4\pi}{3}\)M1 AO1
\(\theta = \frac{\pi}{3}, \frac{2\pi}{3}\)A1 AO1
Total: [8]
Rewrite the equation in the form:
$\cos 3\theta + 2\cos 2\theta \cos 3\theta = 0$ | M1 | AO1

$\cos 3\theta(1 + 2\cos 2\theta) = 0$ | A1 | AO1

Either $\cos 3\theta = 0$:

$3\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}$ | M1 | AO1

$\theta = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}$ | A1 | AO1

Or $\cos 2\theta = -\frac{1}{2}$:

$2\theta = \frac{2\pi}{3}, \frac{4\pi}{3}$ | M1 | AO1

$\theta = \frac{\pi}{3}, \frac{2\pi}{3}$ | A1 | AO1

**Total: [8]**

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Find all the roots of the equation
$$\cos \theta + \cos 3\theta + \cos 5\theta = 0$$
lying in the interval $[0, \pi]$. Give all the roots in radians in terms of $\pi$. [8]

\hfill \mbox{\textit{WJEC Further Unit 4  Q5 [8]}}
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Q1 7 Q2 6 Q3 5 Q4 9 Q5 8 Q6 7 Q7 10 Q8 10 Q9 14 Q10 11 Q11 17 Q12 16