CAIE P1 2014 November — Question 3 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2014
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard trigonometric equations
TypeRational trig expressions
DifficultyStandard +0.8 This question requires substituting sin²θ = 1 - cos²θ, multiplying through by the denominator to clear fractions, and solving a resulting quadratic in cos θ. While the algebraic manipulation is moderately involved and the rational expression adds complexity beyond standard trig equations, the solution path is relatively standard once the identity is applied. The restricted domain simplifies the final step.
Spec1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals
3 Solve the equation \(\frac { 13 \sin ^ { 2 } \theta } { 2 + \cos \theta } + \cos \theta = 2\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
Question 3:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(13\sin^2\theta + 2\cos\theta + \cos^2\theta = 4 + 2\cos\theta\)M1 Attempt to multiply by \(2 + \cos\theta\)
\(13\sin^2\theta + 1 - \sin^2\theta = 4 \rightarrow \sin^2\theta = \frac{1}{4}\)M1 Use of \(s^2 + c^2\) appropriately
or \(13 - 13\cos^2\theta + \cos^2\theta = 4 \rightarrow \cos^2\theta = \frac{3}{4}\)A1A1✓ SC both answers correct in radians, A1 only; Ft on \(180°\) — their first value of \(\theta\)
\(30°,\ 150°\)[4] 
## Question 3:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $13\sin^2\theta + 2\cos\theta + \cos^2\theta = 4 + 2\cos\theta$ | **M1** | Attempt to multiply by $2 + \cos\theta$ |
| $13\sin^2\theta + 1 - \sin^2\theta = 4 \rightarrow \sin^2\theta = \frac{1}{4}$ | **M1** | Use of $s^2 + c^2$ appropriately |
| or $13 - 13\cos^2\theta + \cos^2\theta = 4 \rightarrow \cos^2\theta = \frac{3}{4}$ | **A1A1**✓ | **SC** both answers correct in radians, **A1** only; Ft on $180°$ — their first value of $\theta$ |
| $30°,\ 150°$ | **[4]** | |

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3 Solve the equation $\frac { 13 \sin ^ { 2 } \theta } { 2 + \cos \theta } + \cos \theta = 2$ for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.

\hfill \mbox{\textit{CAIE P1 2014 Q3 [4]}}
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