| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Integration using reciprocal identities |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard trigonometric identity manipulation and basic integration. Part (i) is routine algebraic manipulation of trig identities, part (ii)(a) is direct substitution, and part (ii)(b) uses the proven identity to simplify an integral into a standard form (sin 2θ). All techniques are standard P2/C3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Express left-hand side as a single fraction | M1 | |
| Use \(\sin 2\theta = 2\sin \theta \cos \theta\) at some point | B1 | |
| Complete proof with no errors seen (AG) | A1 | [3] |
| (ii) (a) State \(\frac{2}{\sin \frac{1}{4}\pi}\) or equivalent | B1 | |
| Obtain \(2\sqrt{2}\) or exact equivalent (dependent on first B1) | B1 | [2] |
| (ii) (b) State or imply \(k\sin 2\theta\) for any \(k\) | B1 | |
| Integrate to obtain \(-\frac{1}{2}\cos 2\theta\) | B1 | |
| Substitute both limits correctly to obtain \(3\) | B1 | [3] |
**(i)** Express left-hand side as a single fraction | M1 |
Use $\sin 2\theta = 2\sin \theta \cos \theta$ at some point | B1 |
Complete proof with no errors seen (AG) | A1 | [3]
**(ii) (a)** State $\frac{2}{\sin \frac{1}{4}\pi}$ or equivalent | B1 |
Obtain $2\sqrt{2}$ or exact equivalent (dependent on first B1) | B1 | [2]
**(ii) (b)** State or imply $k\sin 2\theta$ for any $k$ | B1 |
Integrate to obtain $-\frac{1}{2}\cos 2\theta$ | B1 |
Substitute both limits correctly to obtain $3$ | B1 | [3](i) Prove that $\tan \theta + \cot \theta \equiv \frac { 2 } { \sin 2 \theta }$.\\
(ii) Hence
\begin{enumerate}[label=(\alph*)]
\item find the exact value of $\tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 8 } \pi$,
\item evaluate $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 6 } { \tan \theta + \cot \theta } \mathrm { d } \theta$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2014 Q5 [8]}}