| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation only (no integral) |
| Difficulty | Standard +0.3 This is a structured multi-part question on addition formulae and double angles. Part (a) requires expanding products using addition formulae and simplifying to a given identity—straightforward but algebraically involved. Parts (b) and (c) are direct applications of the proven identity. While requiring careful manipulation, this follows standard P2 techniques without novel insight, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \((\cos\theta\cos30 - \sin\theta\sin30)(\cos\theta\cos60 - \sin\theta\sin60)\) | B1 | |
| Expand and use correct exact values | M1 | |
| Obtain \(\frac{1}{4}\sqrt{3}(\cos^2\theta + \sin^2\theta) - \sin\theta\cos\theta\) or similarly simplified equivalent | A1 | |
| Conclude \(\frac{1}{4}\sqrt{3} - \frac{1}{2}\sin2\theta\) | A1 | AG – necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use identity to obtain value for \(\sin4\alpha\) | *M1 | |
| Obtain \(\sin4\alpha = \frac{1}{2}\sqrt{3} - \frac{2}{5}\) or \(0.466\ldots\) | A1 | |
| Show correct process to obtain one value of \(\alpha\) | DM1 | |
| Obtain \(6.9\) and \(38.1\) | A1 | Or greater accuracy; and no others between \(0°\) and \(90°\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(\theta = -10\) to obtain \(\cos 20 \cos 50 = \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin(-20)\) | B1 | |
| Substitute \(\theta = 10\) to obtain \(\cos 40 \cos 70 = \frac{\sqrt{3}}{4} - \frac{1}{2}\sin 20\) | B1 | |
| Add and confirm \(\frac{1}{2}\sqrt{3}\) with clear indication that \(\sin(-20) = -\sin 20\) | B1 | AG – necessary detail needed. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Rewrite as \(\sin 70\cos 50 + \sin 50\cos 70\) or \(\cos 20\sin 40 + \cos 40\sin 20\) or \(\sin 70\sin 40 + \cos 70\cos 40\) | B1 | |
| Obtain \(\sin 120\) or \(\sin 60\) or \(\cos 30\) | B1 | |
| Confirm \(\frac{1}{2}\sqrt{3}\) | B1 | AG – necessary detail needed. |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $(\cos\theta\cos30 - \sin\theta\sin30)(\cos\theta\cos60 - \sin\theta\sin60)$ | B1 | |
| Expand and use correct exact values | M1 | |
| Obtain $\frac{1}{4}\sqrt{3}(\cos^2\theta + \sin^2\theta) - \sin\theta\cos\theta$ or similarly simplified equivalent | A1 | |
| Conclude $\frac{1}{4}\sqrt{3} - \frac{1}{2}\sin2\theta$ | A1 | AG – necessary detail needed |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use identity to obtain value for $\sin4\alpha$ | *M1 | |
| Obtain $\sin4\alpha = \frac{1}{2}\sqrt{3} - \frac{2}{5}$ or $0.466\ldots$ | A1 | |
| Show correct process to obtain one value of $\alpha$ | DM1 | |
| Obtain $6.9$ and $38.1$ | A1 | Or greater accuracy; and no others between $0°$ and $90°$ |
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $\theta = -10$ to obtain $\cos 20 \cos 50 = \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin(-20)$ | B1 | |
| Substitute $\theta = 10$ to obtain $\cos 40 \cos 70 = \frac{\sqrt{3}}{4} - \frac{1}{2}\sin 20$ | B1 | |
| Add and confirm $\frac{1}{2}\sqrt{3}$ with clear indication that $\sin(-20) = -\sin 20$ | B1 | AG – necessary detail needed. |
**Alternative solution for Question 7(c):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Rewrite as $\sin 70\cos 50 + \sin 50\cos 70$ or $\cos 20\sin 40 + \cos 40\sin 20$ or $\sin 70\sin 40 + \cos 70\cos 40$ | B1 | |
| Obtain $\sin 120$ or $\sin 60$ or $\cos 30$ | B1 | |
| Confirm $\frac{1}{2}\sqrt{3}$ | B1 | AG – necessary detail needed. |
**Total: 3 marks**7
\begin{enumerate}[label=(\alph*)]
\item Prove that $\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta + 60 ^ { \circ } \right) \equiv \frac { 1 } { 4 } \sqrt { 3 } - \frac { 1 } { 2 } \sin 2 \theta$.
\item Solve the equation $5 \cos \left( 2 \alpha + 30 ^ { \circ } \right) \cos \left( 2 \alpha + 60 ^ { \circ } \right) = 1$ for $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
\item Show that the exact value of $\cos 20 ^ { \circ } \cos 50 ^ { \circ } + \cos 40 ^ { \circ } \cos 70 ^ { \circ }$ is $\frac { 1 } { 2 } \sqrt { 3 }$.\\
If you use the following page to complete the answer to any question, the question number must be clearly shown.\\
\includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-14_2714_38_109_2010}
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q7 [11]}}