| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then find exact value and solve/integrate |
| Difficulty | Standard +0.3 This is a structured multi-part question testing standard addition formulae and product-to-sum identities. Part (a) requires applying compound angle formulae systematically but follows a predictable path. Parts (b) and (c) are direct applications of the proven identity with straightforward substitutions and routine integration. While it requires multiple techniques, each step is standard bookwork with no novel insight needed, making it slightly easier than average. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain at least either \((\frac{1}{2}\sin\theta + \frac{1}{2}\sqrt{3}\cos\theta)\) or \((\frac{1}{2}\cos\theta + \frac{1}{2}\sqrt{3}\sin\theta)\) | B1 | Allow if implied by decimal values |
| Expand and simplify with correct use of \(\sin^2\theta + \cos^2\theta = 1\) | M1 | |
| Use \(\sin\theta\cos\theta = \frac{1}{2}\sin 2\theta\) | M1 | |
| Confirm given result \(\sqrt{3} + 2\sin 2\theta\) | A1 | AG necessary detail required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Identify value of \(\theta\) is \(\frac{3}{8}\pi\) | *B1 | OE |
| Obtain \(\sqrt{3} + 2\sin\frac{3}{4}\pi\) and conclude \(\sqrt{3} + \sqrt{2}\) | DB1 | Or exact equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Identify integrand as \(\sqrt{3} + 2\sin 4x\) | B1 | |
| Integrate to obtain form \(k_1 x + k_2\cos 4x\) | M1 | Where \(k_1 k_2 \neq 0\) |
| Obtain correct \(\sqrt{3}\,x - \frac{1}{2}\cos 4x\) | A1 | |
| Obtain \(\frac{1}{8}\pi\sqrt{3} + \frac{1}{2}\) | A1 | Or exact equivalent |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain at least either $(\frac{1}{2}\sin\theta + \frac{1}{2}\sqrt{3}\cos\theta)$ or $(\frac{1}{2}\cos\theta + \frac{1}{2}\sqrt{3}\sin\theta)$ | B1 | Allow if implied by decimal values |
| Expand and simplify with correct use of $\sin^2\theta + \cos^2\theta = 1$ | M1 | |
| Use $\sin\theta\cos\theta = \frac{1}{2}\sin 2\theta$ | M1 | |
| Confirm given result $\sqrt{3} + 2\sin 2\theta$ | A1 | AG necessary detail required |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Identify value of $\theta$ is $\frac{3}{8}\pi$ | *B1 | OE |
| Obtain $\sqrt{3} + 2\sin\frac{3}{4}\pi$ and conclude $\sqrt{3} + \sqrt{2}$ | DB1 | Or exact equivalent |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Identify integrand as $\sqrt{3} + 2\sin 4x$ | B1 | |
| Integrate to obtain form $k_1 x + k_2\cos 4x$ | M1 | Where $k_1 k_2 \neq 0$ |
| Obtain correct $\sqrt{3}\,x - \frac{1}{2}\cos 4x$ | A1 | |
| Obtain $\frac{1}{8}\pi\sqrt{3} + \frac{1}{2}$ | A1 | Or exact equivalent |
---6
\begin{enumerate}[label=(\alph*)]
\item Show that $4 \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \cos \left( \theta - \frac { 1 } { 3 } \pi \right) \equiv \sqrt { 3 } + 2 \sin 2 \theta$.
\item Find the exact value of $4 \sin \frac { 17 } { 24 } \pi \cos \frac { 1 } { 24 } \pi$.
\item Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } 4 \sin \left( 2 x + \frac { 1 } { 3 } \pi \right) \cos \left( 2 x - \frac { 1 } { 3 } \pi \right) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2023 Q6 [10]}}