| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | November |
| 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 is a substantial multi-part question requiring proof of a non-trivial trigonometric identity using addition and double angle formulae, then applying it to integration and equation solving. Part (a) requires systematic expansion and manipulation, part (b) needs careful integration with exact values, and part (c) involves substitution and solving a transformed equation. The combination of proof, integration, and problem-solving with exact forms places this above average difficulty but within reach of well-prepared P2 students. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o 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 |
|---|---|---|
| Answer | Mark | Guidance |
| State \(4\sin x(\sin x\cos\frac{1}{6}\pi+\cos x\sin\frac{1}{6}\pi)\) | B1 | Allow marks if completed correctly, starting with RHS |
| Attempt relevant identities to express at least one term in terms of \(\cos 2x\) and \(\sin 2x\) | M1 | |
| Confirm \(\sqrt{3}-\sqrt{3}\cos 2x+\sin 2x\) with sufficient detail | A1 | AG |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain form \(k_1 x+k_2\sin 2x+k_3\cos 2x\) | *M1 | Where \(k_1k_2k_3\neq 0\) |
| Obtain correct \(\sqrt{3}x-\frac{1}{2}\sqrt{3}\sin 2x-\frac{1}{2}\cos 2x\) | A1 | |
| Apply limits correctly and attempt simplification, retaining exactness | DM1 | Condone one sign slip |
| Obtain \(\frac{5\sqrt{3}}{6}\pi+1\) or exact equivalent | A1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Apply identity with \(x\) replaced by \(2y\) | M1 | May be implied later |
| Attempt solution of \(\tan 4y=\sqrt{3}\) | M1 | |
| Obtain \(4y=\frac{1}{3}\pi\) and hence \(y=\frac{1}{12}\pi\) | A1 | |
| Total | 3 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $4\sin x(\sin x\cos\frac{1}{6}\pi+\cos x\sin\frac{1}{6}\pi)$ | B1 | Allow marks if completed correctly, starting with RHS |
| Attempt relevant identities to express at least one term in terms of $\cos 2x$ and $\sin 2x$ | M1 | |
| Confirm $\sqrt{3}-\sqrt{3}\cos 2x+\sin 2x$ with sufficient detail | A1 | AG |
| **Total** | **3** | |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $k_1 x+k_2\sin 2x+k_3\cos 2x$ | *M1 | Where $k_1k_2k_3\neq 0$ |
| Obtain correct $\sqrt{3}x-\frac{1}{2}\sqrt{3}\sin 2x-\frac{1}{2}\cos 2x$ | A1 | |
| Apply limits correctly and attempt simplification, retaining exactness | DM1 | Condone one sign slip |
| Obtain $\frac{5\sqrt{3}}{6}\pi+1$ or exact equivalent | A1 | |
| **Total** | **4** | |
---
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Apply identity with $x$ replaced by $2y$ | M1 | May be implied later |
| Attempt solution of $\tan 4y=\sqrt{3}$ | M1 | |
| Obtain $4y=\frac{1}{3}\pi$ and hence $y=\frac{1}{12}\pi$ | A1 | |
| **Total** | **3** | |7
\begin{enumerate}[label=(\alph*)]
\item Prove that $4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \equiv \sqrt { 3 } - \sqrt { 3 } \cos 2 x + \sin 2 x$.
\item Find the exact value of $\int _ { 0 } ^ { \frac { 5 } { 6 } \pi } 4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \mathrm { d } x$.
\item Find the smallest positive value of $y$ satisfying the equation
$$4 \sin ( 2 y ) \sin \left( 2 y + \frac { 1 } { 6 } \pi \right) = \sqrt { 3 } .$$
Give your answer in an exact form.\\
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 [10]}}