Edexcel C34 2015 January — Question 2 5 marks

Exam BoardEdexcel
ModuleC34 (Core Mathematics 3 & 4)
Year2015
SessionJanuary
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard trigonometric equations
TypeDouble angle equations requiring identity expansion and factorisation
DifficultyStandard +0.3 This is a standard double angle equation requiring the identity cos 2θ = 1 - 2sin²θ to convert to a quadratic in sin θ, then solving 2sin²θ - 13sin θ + 3 = 0. It's slightly above average due to the algebraic manipulation and needing to find multiple solutions in the given range, but follows a well-practiced technique with no novel insight required.
Spec1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals
2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$2 \cos 2 \theta = 5 - 13 \sin \theta$$ Give your answers in radians to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Question 2:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(2\cos 2\theta = 5 - 13\sin\theta \Rightarrow 4\sin^2\theta - 13\sin\theta + 3 = 0\)M1A1 Uses \(\cos 2\theta = 1-2\sin^2\theta\); A1 for \(\pm(4\sin^2\theta - 13\sin\theta + 3)=0\)
\((4\sin\theta - 1)(\sin\theta - 3) = 0\)
\(\sin\theta = \frac{1}{4}\)M1 Solves 3TQ by factorisation, formula or completing the square; must reach \(\sin\theta = \ldots\)
\(\theta =\) awrt \(0.253\), \(2.889\) (3dp)A1, A1 cso First A1: either value awrt \(0.25\), \(2.89\) (2dp); Second A1: both correct (3dp), no extra solutions in range 
## Question 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\cos 2\theta = 5 - 13\sin\theta \Rightarrow 4\sin^2\theta - 13\sin\theta + 3 = 0$ | M1A1 | Uses $\cos 2\theta = 1-2\sin^2\theta$; A1 for $\pm(4\sin^2\theta - 13\sin\theta + 3)=0$ |
| $(4\sin\theta - 1)(\sin\theta - 3) = 0$ | | |
| $\sin\theta = \frac{1}{4}$ | M1 | Solves 3TQ by factorisation, formula or completing the square; must reach $\sin\theta = \ldots$ |
| $\theta =$ awrt $0.253$, $2.889$ (3dp) | A1, A1 cso | First A1: either value awrt $0.25$, $2.89$ (2dp); Second A1: both correct (3dp), no extra solutions in range |

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2. Solve, for $0 \leqslant \theta < 2 \pi$,

$$2 \cos 2 \theta = 5 - 13 \sin \theta$$

Give your answers in radians to 3 decimal places.\\
(Solutions based entirely on graphical or numerical methods are not acceptable.)\\

\hfill \mbox{\textit{Edexcel C34 2015 Q2 [5]}}
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