| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Double angle with reciprocal functions |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard identities and techniques. Part (i) requires routine manipulation of reciprocal trig functions and double angle formulas. Part (ii)(a) is a simple equation solve after the identity is established, and part (ii)(b) is a direct substitution integral with standard limits. All steps are textbook-standard 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=11.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(\cosec 2\theta = \frac{1}{\sin 2\theta}\) | B1 | |
| Express left-hand side in terms of \(\sin \theta\) and \(\cos \theta\) | M1 | |
| Obtain given answer \(\sec^2 \theta\) correctly | A1 | [3] |
| (ii) (a) State or imply \(\cos \theta = \frac{1}{\sqrt{5}}\) or \(\tan \theta = 2\) at least | B1 | |
| Obtain 1.11 or awrt 1.11, allow 0.353π | B1 | |
| Obtain 2.03 or awrt 2.03, allow 0.648π and no other values between 0 and π | B1 | [3] |
| (b) State integrand as \(\sec^2 2x\) | B1 | |
| Integrate to obtain expression of form \(k \tan mx\) | M1 | |
| Obtain correct \(\frac{1}{2}\tan 2x\) | A1 | |
| Obtain \(\frac{1}{3}\sqrt{3}\) or exact equivalent | A1 | [4] |
(i) State or imply $\cosec 2\theta = \frac{1}{\sin 2\theta}$ | B1 |
Express left-hand side in terms of $\sin \theta$ and $\cos \theta$ | M1 |
Obtain given answer $\sec^2 \theta$ correctly | A1 | [3]
(ii) (a) State or imply $\cos \theta = \frac{1}{\sqrt{5}}$ or $\tan \theta = 2$ at least | B1 |
Obtain 1.11 or awrt 1.11, allow 0.353π | B1 |
Obtain 2.03 or awrt 2.03, allow 0.648π and no other values between 0 and π | B1 | [3]
(b) State integrand as $\sec^2 2x$ | B1 |
Integrate to obtain expression of form $k \tan mx$ | M1 |
Obtain correct $\frac{1}{2}\tan 2x$ | A1 |
Obtain $\frac{1}{3}\sqrt{3}$ or exact equivalent | A1 | [4]
---6 (i) Prove that $2 \operatorname { cosec } 2 \theta \tan \theta \equiv \sec ^ { 2 } \theta$.\\
(ii) Hence
\begin{enumerate}[label=(\alph*)]
\item solve the equation $2 \operatorname { cosec } 2 \theta \tan \theta = 5$ for $0 < \theta < \pi$,
\item find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 2 \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2015 Q6 [10]}}