CAIE P3 2016 June — Question 3 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2016
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicReciprocal Trig & Identities
TypeSolve equation with reciprocal functions
DifficultyStandard +0.3 This question requires converting reciprocal trig functions to standard form and using the Pythagorean identity, leading to a quadratic in cos θ. While it involves multiple steps (rewriting cosec and cot, clearing fractions, using sin²θ + cos²θ = 1, solving quadratic), these are all standard techniques for P3 level with no novel insight required. The algebraic manipulation is straightforward once the approach is identified.
Spec1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals
3 By expressing the equation \(\operatorname { cosec } \theta = 3 \sin \theta + \cot \theta\) in terms of \(\cos \theta\) only, solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
Question 3:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Correctly restate equation in terms of \(\sin\theta\) and \(\cos\theta\)B1
Using Pythagoras obtain a horizontal equation in \(\cos\theta\)M1
Reduce to correct quadratic in \(\cos\theta\), e.g. \(3\cos^2\theta - \cos\theta - 2 = 0\)A1
Solve 3-term quadratic for \(\cos\theta\)M1
Obtain \(\theta = 131.8°\) onlyA1
Total[5] [Ignore answers outside given interval] 
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correctly restate equation in terms of $\sin\theta$ and $\cos\theta$ | B1 | |
| Using Pythagoras obtain a horizontal equation in $\cos\theta$ | M1 | |
| Reduce to correct quadratic in $\cos\theta$, e.g. $3\cos^2\theta - \cos\theta - 2 = 0$ | A1 | |
| Solve 3-term quadratic for $\cos\theta$ | M1 | |
| Obtain $\theta = 131.8°$ only | A1 | |
| **Total** | **[5]** | [Ignore answers outside given interval] |

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3 By expressing the equation $\operatorname { cosec } \theta = 3 \sin \theta + \cot \theta$ in terms of $\cos \theta$ only, solve the equation for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.

\hfill \mbox{\textit{CAIE P3 2016 Q3 [5]}}
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