| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard harmonic form question with straightforward algebraic manipulation in part (i), routine R-α conversion in part (ii), and standard equation solving in part (iii). The initial simplification requires recognizing sin 2θ = 2sin θ cos θ and converting sec/cosec to cos/sin, which are well-practiced techniques. While multi-step, each component is a textbook exercise with no novel insight required, making it slightly easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}[label=(\roman*)]
\item Express $\sin 2\theta (3 \sec \theta + 4 \cosec \theta)$ in the form $a \sin \theta + b \cos \theta$, where $a$ and $b$ are integers. [3]
\item Hence express $\sin 2\theta (3 \sec \theta + 4 \cosec \theta)$ in the form $R \sin(\theta + \alpha)$ where $R > 0$ and $0° < \alpha < 90°$. [3]
\item Using the result of part (ii), solve the equation $\sin 2\theta (3 \sec \theta + 4 \cosec \theta) = 7$ for $0° \leq \theta \leq 360°$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q7 [10]}}