| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Integrate using double angle |
| Difficulty | Standard +0.3 This is a straightforward application of double angle formulae. Part (i) requires routine manipulation using cos(2θ) = 1 - 2sin²θ and cos(2θ) = 2cos²θ - 1, then combining to get cos(4θ). Part (ii) is direct integration of a constant and cos(4θ) with standard limits. While it requires knowing the formulae and careful algebra, it follows a well-practiced technique with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) Make relevant use of formula for \(\sin 2\theta\) or \(\cos 2\theta\) | M1 | |
| Make relevant use of formula for \(\cos 4\theta\) | M1 | |
| Complete proof of the given result | A1 | Total: 3 |
| (ii) Integrate and obtain \(\frac{1}{8}(\theta - \frac{1}{4}\sin 4\theta)\) or equivalent | B1 | |
| Use limits correctly with an integral of the form \(a\theta + b\sin 4\theta\), where \(ab \neq 0\) | M1 | |
| Obtain answer \(\frac{1}{8}\left(\frac{1}{3}\pi + \frac{\sqrt{3}}{8}\right)\), or exact equivalent | A1 | Total: 3 |
## Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** Make relevant use of formula for $\sin 2\theta$ or $\cos 2\theta$ | M1 | |
| Make relevant use of formula for $\cos 4\theta$ | M1 | |
| Complete proof of the given result | A1 | **Total: 3** |
| **(ii)** Integrate and obtain $\frac{1}{8}(\theta - \frac{1}{4}\sin 4\theta)$ or equivalent | B1 | |
| Use limits correctly with an integral of the form $a\theta + b\sin 4\theta$, where $ab \neq 0$ | M1 | |
| Obtain answer $\frac{1}{8}\left(\frac{1}{3}\pi + \frac{\sqrt{3}}{8}\right)$, or exact equivalent | A1 | **Total: 3** |
---5 (i) Prove the identity
$$\sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \frac { 1 } { 8 } ( 1 - \cos 4 \theta )$$
(ii) Hence find the exact value of
$$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 2 } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$$
\hfill \mbox{\textit{CAIE P3 2004 Q5 [6]}}