| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.8 This question requires proving a non-standard identity involving reciprocal functions and double angles (requiring multiple substitutions and algebraic manipulation), then applying it to solve an equation and find an exact value. While systematic, it demands fluency with reciprocal trig functions, double angle formulas, and multi-step algebraic reasoning beyond routine exercises, placing it moderately above average difficulty. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(\cosec\theta = \frac{1}{\sin\theta}\) and \(\sec\theta = \frac{1}{\cos\theta}\) | B1 | |
| Attempt to simplify left-hand side | M1 | |
| Confirm given right-hand side \(4\cos 2\theta\) with no errors seen | A1 | [3] |
| (ii)(a) State or imply \(\cos 2\theta = \frac{3}{4}\) | B1 | |
| Attempt correct process to find at least one angle | M1 | |
| Obtain \(20.7°\) | A1 | |
| Obtain \(159.3°\) and no others in range | A1 | [4] |
| (ii)(b) Recognise as \(\frac{4\cos 30°}{\sin^2 30°}\) | B1 | |
| Obtain \(8\sqrt{3}\) | B1 | [2] |
**(i)** Use $\cosec\theta = \frac{1}{\sin\theta}$ and $\sec\theta = \frac{1}{\cos\theta}$ | B1 |
Attempt to simplify left-hand side | M1 |
Confirm given right-hand side $4\cos 2\theta$ with no errors seen | A1 | [3]
**(ii)(a)** State or imply $\cos 2\theta = \frac{3}{4}$ | B1 |
Attempt correct process to find at least one angle | M1 |
Obtain $20.7°$ | A1 |
Obtain $159.3°$ and no others in range | A1 | [4]
**(ii)(b)** Recognise as $\frac{4\cos 30°}{\sin^2 30°}$ | B1 |
Obtain $8\sqrt{3}$ | B1 | [2]8 (i) Prove that $\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) \equiv 4 \cos 2 \theta$.\\
(ii) Hence
\begin{enumerate}[label=(\alph*)]
\item solve for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$ the equation $\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) = 3$,
\item find the exact value of $\operatorname { cosec } ^ { 2 } 15 ^ { \circ } - \sec ^ { 2 } 15 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2011 Q8 [9]}}