| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Derive triple angle then evaluate integral |
| Difficulty | Standard +0.3 This is a structured multi-part question that guides students through deriving a triple angle formula then applying it. Part (a) is routine expansion using addition formulae, part (b) is direct substitution requiring recognition that the expression equals cos(5π/6)/2, and part (c) requires reversing the identity to integrate. While it spans multiple techniques, each step is straightforward with clear signposting, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05p Proof involving trig: functions and identities1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\cos 2\theta\cos\theta - \sin 2\theta\sin\theta\) | B1 | |
| Attempt correct relevant identities to express in terms of \(\cos\theta\) only | M1 | M0 if moving terms from side to side |
| Confirm \(4\cos^3\theta - 3\cos\theta\) with sufficient detail | A1 | AG |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use identity with \(\theta = \frac{5}{18}\pi\) | M1 | |
| Obtain \(\frac{1}{2}\cos\frac{5}{6}\pi\) and hence \(-\frac{1}{4}\sqrt{3}\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express integrand in form \(k_1(\cos 3x + 3\cos x) + k_2(\cos 9x + 3\cos 3x)\) | M1 | |
| Obtain correct integrand \(9\cos x - \cos 9x\) | A1 | OE (allow unsimplified) |
| Integrate to obtain form \(k_3\sin x + k_4\sin 9x\) | M1 | |
| Obtain correct \(9\sin x - \frac{1}{9}\sin 9x\) | A1 | Now simplified; condone missing \(\ldots + c\) |
| 4 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\cos 2\theta\cos\theta - \sin 2\theta\sin\theta$ | B1 | |
| Attempt correct relevant identities to express in terms of $\cos\theta$ only | M1 | M0 if moving terms from side to side |
| Confirm $4\cos^3\theta - 3\cos\theta$ with sufficient detail | A1 | AG |
| | **3** | |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use identity with $\theta = \frac{5}{18}\pi$ | M1 | |
| Obtain $\frac{1}{2}\cos\frac{5}{6}\pi$ and hence $-\frac{1}{4}\sqrt{3}$ | A1 | |
| | **2** | |
---
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express integrand in form $k_1(\cos 3x + 3\cos x) + k_2(\cos 9x + 3\cos 3x)$ | M1 | |
| Obtain correct integrand $9\cos x - \cos 9x$ | A1 | OE (allow unsimplified) |
| Integrate to obtain form $k_3\sin x + k_4\sin 9x$ | M1 | |
| Obtain correct $9\sin x - \frac{1}{9}\sin 9x$ | A1 | Now simplified; condone missing $\ldots + c$ |
| | **4** | |7
\begin{enumerate}[label=(\alph*)]
\item By first expanding $\cos ( 2 \theta + \theta )$, show that $\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$.
\item Find the exact value of $2 \cos ^ { 3 } \left( \frac { 5 } { 18 } \pi \right) - \frac { 3 } { 2 } \cos \left( \frac { 5 } { 18 } \pi \right)$.
\item Find $\int \left( 12 \cos ^ { 3 } x - 4 \cos ^ { 3 } 3 x \right) \mathrm { d } x$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q7 [9]}}