| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation and evaluate integral |
| Difficulty | Standard +0.8 This question requires proving a non-trivial trigonometric identity involving double angles and compound angles, then applying it to solve an equation and adapt the technique to integration. Part (a) demands algebraic manipulation through multiple steps (expanding sin 2θ, simplifying with reciprocal identities, factoring to reach compound angle form), part (b) applies the result, and part (c) requires recognizing the pattern with halved angles and using substitution—all requiring solid technique and insight beyond routine exercises. |
| Spec | 1.05j 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 intervals1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express left-hand side in terms of \(\sin\theta\) and \(\cos\theta\) | M1 | |
| Obtain \(2\cos\theta - 2\sin\theta\) | A1 | |
| Attempt to express \(a\cos\theta + b\sin\theta\) in \(R\cos(\theta + \beta)\) form | M1 | |
| Confirm \(R = \sqrt{8}\) | A1 | AG |
| Carry out necessary trigonometry and confirm \(\frac{1}{4}\pi\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out correct process to find \(\theta\) from \(\cos\left(\theta + \frac{1}{4}\pi\right) = \frac{1}{\sqrt{8}}\) | M1 | |
| Obtain 0.424 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express integrand as \(\sqrt{8}\cos\left(\frac{1}{2}x + \frac{1}{4}\pi\right)\) or as \(2\cos\frac{1}{2}x - 2\sin\frac{1}{2}x\) | B1 | |
| Integrate to obtain \(k\sin\left(\frac{1}{2}x + \frac{1}{4}\pi\right)\) or \(k_1\sin\frac{1}{2}x + k_2\cos\frac{1}{2}x\) | M1 | |
| Obtain correct \(2\sqrt{8}\sin\left(\frac{1}{2}x + \frac{1}{4}\pi\right)\) or \(4\sin\frac{1}{2}x + 4\cos\frac{1}{2}x\) | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express left-hand side in terms of $\sin\theta$ and $\cos\theta$ | M1 | |
| Obtain $2\cos\theta - 2\sin\theta$ | A1 | |
| Attempt to express $a\cos\theta + b\sin\theta$ in $R\cos(\theta + \beta)$ form | M1 | |
| Confirm $R = \sqrt{8}$ | A1 | AG |
| Carry out necessary trigonometry and confirm $\frac{1}{4}\pi$ | A1 | AG |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out correct process to find $\theta$ from $\cos\left(\theta + \frac{1}{4}\pi\right) = \frac{1}{\sqrt{8}}$ | M1 | |
| Obtain 0.424 | A1 | |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express integrand as $\sqrt{8}\cos\left(\frac{1}{2}x + \frac{1}{4}\pi\right)$ or as $2\cos\frac{1}{2}x - 2\sin\frac{1}{2}x$ | B1 | |
| Integrate to obtain $k\sin\left(\frac{1}{2}x + \frac{1}{4}\pi\right)$ or $k_1\sin\frac{1}{2}x + k_2\cos\frac{1}{2}x$ | M1 | |
| Obtain correct $2\sqrt{8}\sin\left(\frac{1}{2}x + \frac{1}{4}\pi\right)$ or $4\sin\frac{1}{2}x + 4\cos\frac{1}{2}x$ | A1 | |
---6
\begin{enumerate}[label=(\alph*)]
\item Prove that
$$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) \equiv \sqrt { 8 } \cos \left( \theta + \frac { 1 } { 4 } \pi \right)$$
\item Solve the equation
$$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) = 1$$
for $0 < \theta < \frac { 1 } { 2 } \pi$. Give the answer correct to 3 significant figures.
\item Find $\int \sin x \left( \operatorname { cosec } \frac { 1 } { 2 } x - \sec \frac { 1 } { 2 } x \right) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q6 [10]}}