CAIE P3 2010 June — Question 2 6 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2010
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard trigonometric equations
TypeDouble angle equations requiring identity expansion and factorisation
DifficultyStandard +0.3 This is a straightforward double angle equation requiring substitution of cos 2θ = 1 - 2sin²θ to form a quadratic in sin θ, then solving for θ in a given interval. It's slightly easier than average as it follows a standard technique with no conceptual surprises, though it does require careful algebraic manipulation and finding multiple solutions.
Spec1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals

2 Solve the equation $$\sin \theta = 2 \cos 2 \theta + 1$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

Question 2:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Use correct \(\cos 2A\) formula and obtain an equation in \(\sin\theta\)M1
Obtain \(4\sin^2\theta + \sin\theta - 3 = 0\), or equivalentA1
Make reasonable attempt to solve a 3-term quadratic in \(\sin\theta\)M1
Obtain answer \(48.6°\)A1
Obtain answer \(131.4°\) and no others in the given rangeA1\(\sqrt{}\)
Obtain answer \(270°\) and no others in the given rangeA1 [6]
*Note: Treat giving of answers in radians as a misread. Ignore answers outside the given range.*
## Question 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct $\cos 2A$ formula and obtain an equation in $\sin\theta$ | M1 | |
| Obtain $4\sin^2\theta + \sin\theta - 3 = 0$, or equivalent | A1 | |
| Make reasonable attempt to solve a 3-term quadratic in $\sin\theta$ | M1 | |
| Obtain answer $48.6°$ | A1 | |
| Obtain answer $131.4°$ and no others in the given range | A1$\sqrt{}$ | |
| Obtain answer $270°$ and no others in the given range | A1 | [6] |

*Note: Treat giving of answers in radians as a misread. Ignore answers outside the given range.*

---
2 Solve the equation

$$\sin \theta = 2 \cos 2 \theta + 1$$

giving all solutions in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

\hfill \mbox{\textit{CAIE P3 2010 Q2 [6]}}