| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Standard +0.3 Part (i) is a straightforward trigonometric identity proof using standard definitions and the double angle formula—routine for P3 level. Part (ii) requires recognizing that the integral of cosec 2θ can be evaluated using the result from (i), involving a substitution and logarithmic integration, which is standard technique but requires careful execution. Overall slightly easier than average due to the guided structure and use of standard methods. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.06f Laws of logarithms: addition, subtraction, power rules1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use Pythagoras | M1 | |
| Use the \(\sin 2A\) formula | M1 | |
| Obtain the given result | A1 | [3] |
| (ii) Integrate and obtain a \(k\ln\sin\theta\) or \(m\ln\cos\theta\) term, or obtain integral of the form \(p\ln\tan\theta\) | M1* | |
| Obtain indefinite integral \(\frac{1}{2}\ln\sin\theta - \frac{1}{2}\ln\cos\theta\), or equivalent, or \(\frac{1}{2}\ln\tan\theta\) | A1 | |
| Substitute limits correctly | M1(dep)* | |
| Obtain the given answer correctly having shown appropriate working | A1 | [4] |
**(i)** Use Pythagoras | M1 |
Use the $\sin 2A$ formula | M1 |
Obtain the given result | A1 | [3]
**(ii)** Integrate and obtain a $k\ln\sin\theta$ or $m\ln\cos\theta$ term, or obtain integral of the form $p\ln\tan\theta$ | M1* |
Obtain indefinite integral $\frac{1}{2}\ln\sin\theta - \frac{1}{2}\ln\cos\theta$, or equivalent, or $\frac{1}{2}\ln\tan\theta$ | A1 |
Substitute limits correctly | M1(dep)* |
Obtain the given answer correctly having shown appropriate working | A1 | [4]\begin{enumerate}[label=(\roman*)]
\item Prove that $\cot \theta + \tan \theta = 2\cosec 2\theta$. [3]
\item Hence show that $\int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} \cosec 2\theta \, d\theta = \frac{1}{2}\ln 3$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q5 [7]}}