| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 10 |
| 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 guided, standard application of addition and double angle formulae followed by a straightforward integration. Part (i) is routine algebraic manipulation with formulae provided in the question stem, and part (ii) is a direct rearrangement and integration using the derived identity. The question requires multiple steps but follows a clear path with no novel insight needed, 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 |
|---|---|
| Make relevant use of the \(\sin(A + B)\) formula | B1 |
| Make relevant use of \(\sin 2A\) and \(\cos 2A\) formulae | M1 |
| Obtain a correct expression in terms of \(\sin x\) and \(\cos x\) | A1 |
| Use \(\cos^2 x = 1 - \sin^2 x\) to obtain an expression in terms of \(\sin x\) | M1 (dep*) |
| Obtain given answer correctly | A1 |
| Answer | Marks |
|---|---|
| Replace integrand by \(\frac{3}{4} \sin x - \frac{1}{4} \sin 3x\), or equivalent | B1 |
| Integrate, obtaining \(-\frac{3}{4} \cos x + \frac{1}{12} \cos 3x\), or equivalent | B1\(\checkmark\) + B1\(\checkmark\) |
| Use limits correctly | M1 |
| Obtain given answer correctly | A1 |
**(i)**
| Make relevant use of the $\sin(A + B)$ formula | B1 |
| Make relevant use of $\sin 2A$ and $\cos 2A$ formulae | M1 |
| Obtain a correct expression in terms of $\sin x$ and $\cos x$ | A1 |
| Use $\cos^2 x = 1 - \sin^2 x$ to obtain an expression in terms of $\sin x$ | M1 (dep*) |
| Obtain given answer correctly | A1 |
**Total: 5 marks**
**(ii)**
| Replace integrand by $\frac{3}{4} \sin x - \frac{1}{4} \sin 3x$, or equivalent | B1 |
| Integrate, obtaining $-\frac{3}{4} \cos x + \frac{1}{12} \cos 3x$, or equivalent | B1$\checkmark$ + B1$\checkmark$ |
| Use limits correctly | M1 |
| Obtain given answer correctly | A1 |
**Total: 5 marks**7 (i) By expanding $\sin ( 2 x + x )$ and using double-angle formulae, show that
$$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$
(ii) Hence show that
$$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x d x = \frac { 5 } { 24 }$$
\hfill \mbox{\textit{CAIE P2 2005 Q7 [10]}}