CAIE P3 2021 November — Question 5 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2021
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicAddition & Double Angle Formulae
TypeSolve equation with sin2x/cos2x by substitution
DifficultyStandard +0.3 This is a straightforward double angle equation requiring substitution of cos 2θ = 1 - 2sin²θ, leading to a quadratic in sin θ. The steps are standard: substitute, rearrange to quadratic form, solve, and find angles in the given range. Slightly easier than average due to the routine nature of the technique.
Spec1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals

5 Solve the equation \(\sin \theta = 3 \cos 2 \theta + 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

Question 5:
AnswerMarks Guidance
AnswerMark Guidance
Use double angle formula and obtain an equation in \(\sin\theta\)M1
Reduce to \(6\sin^2\theta + \sin\theta - 5 = 0\), or 3-term equivalentA1
Solve a 3-term quadratic in \(\sin\theta\) and calculate \(\theta\)M1
Obtain answer, e.g. \(56.4°\)A1
Obtain second and third answers, e.g. \(123.6°\) and \(270°\) and no others in the given intervalA1 Ignore answers outside the interval. Treat answers in radians as a misread.
## Question 5:

| Answer | Mark | Guidance |
|--------|------|----------|
| Use double angle formula and obtain an equation in $\sin\theta$ | M1 | |
| Reduce to $6\sin^2\theta + \sin\theta - 5 = 0$, or 3-term equivalent | A1 | |
| Solve a 3-term quadratic in $\sin\theta$ and calculate $\theta$ | M1 | |
| Obtain answer, e.g. $56.4°$ | A1 | |
| Obtain second and third answers, e.g. $123.6°$ and $270°$ and no others in the given interval | A1 | Ignore answers outside the interval. Treat answers in radians as a misread. |

---
5 Solve the equation $\sin \theta = 3 \cos 2 \theta + 2$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.\\

\hfill \mbox{\textit{CAIE P3 2021 Q5 [5]}}