| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then evaluate integral |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring standard application of addition formulae and basic integration. Part (a) is guided algebra with sin(A±B) expansions, and part (b) involves routine integration of the simplified form with exact value evaluation. While it requires multiple steps, each step follows standard procedures with no novel insight needed, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State correct expansion of \(\sin(3x+2x)\) or \(\sin(3x-2x)\) | B1 | |
| Substitute expansions in \(\frac{1}{2}(\sin 5x + \sin x)\), or equivalent | M1 | |
| Simplify and obtain \(\frac{1}{2}(\sin 5x + \sin x) = \sin 3x \cos 2x\) | A1 | Obtain the given identity correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain integral \(-\frac{1}{10}\cos 5x - \frac{1}{2}\cos x\), or equivalent | B1 | |
| Substitute limits correctly in an expression of the form \(p\cos 5x + q\cos x\) | M1 | Correct limits and subtracted the right way around. Do not need values of trig functions for M1. Maximum one slip |
| Obtain \(\frac{1}{5}(3-\sqrt{2})\) | A1 | Substitute values and obtain the given answer following full, correct and exact working |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State correct expansion of $\sin(3x+2x)$ or $\sin(3x-2x)$ | **B1** | |
| Substitute expansions in $\frac{1}{2}(\sin 5x + \sin x)$, or equivalent | **M1** | |
| Simplify and obtain $\frac{1}{2}(\sin 5x + \sin x) = \sin 3x \cos 2x$ | **A1** | Obtain the **given identity** correctly |
**Total: 3 marks**
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain integral $-\frac{1}{10}\cos 5x - \frac{1}{2}\cos x$, or equivalent | **B1** | |
| Substitute limits correctly in an expression of the form $p\cos 5x + q\cos x$ | **M1** | Correct limits and subtracted the right way around. Do not need values of trig functions for M1. Maximum one slip |
| Obtain $\frac{1}{5}(3-\sqrt{2})$ | **A1** | Substitute values and obtain the **given answer** following full, correct and exact working |
**Total: 3 marks**6
\begin{enumerate}[label=(\alph*)]
\item Using the expansions of $\sin ( 3 x + 2 x )$ and $\sin ( 3 x - 2 x )$, show that
$$\frac { 1 } { 2 } ( \sin 5 x + \sin x ) \equiv \sin 3 x \cos 2 x$$
\item Hence show that $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin 3 x \cos 2 x \mathrm {~d} x = \frac { 1 } { 5 } ( 3 - \sqrt { 2 } )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q6 [6]}}