| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2009 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Express cos²x or sin²x in terms of cos 2x |
| Difficulty | Moderate -0.3 This is a straightforward application of the double angle formula cos²θ = (1 + cos 2θ)/2, followed by a routine integration. Part (i) requires direct substitution (θ = 2x), and part (ii) involves integrating a linear combination of constants and cosine—both standard techniques with no problem-solving insight required. Slightly easier than average due to its mechanical nature. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use double angle formulae and obtain \(a + b\cos 4x\) | M1 | |
| Obtain answer \(\frac{1}{2} + \frac{1}{2}\cos 4x\), or equivalent | A1 | [2 marks] |
| (ii) Integrate and obtain \(\frac{1}{8}x + \frac{1}{8}\sin 4x\) | A1√ + A1√ | |
| Substitute limits correctly | M1 | |
| Obtain answer \(\frac{1}{16}\pi + \frac{1}{8}\), or exact equivalent | A1 | [4 marks] |
**(i)** Use double angle formulae and obtain $a + b\cos 4x$ | M1 |
Obtain answer $\frac{1}{2} + \frac{1}{2}\cos 4x$, or equivalent | A1 | [2 marks]
**(ii)** Integrate and obtain $\frac{1}{8}x + \frac{1}{8}\sin 4x$ | A1√ + A1√ |
Substitute limits correctly | M1 |
Obtain answer $\frac{1}{16}\pi + \frac{1}{8}$, or exact equivalent | A1 | [4 marks]5 (i) Express $\cos ^ { 2 } 2 x$ in terms of $\cos 4 x$.\\
(ii) Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } \cos ^ { 2 } 2 x \mathrm {~d} x$.
\hfill \mbox{\textit{CAIE P2 2009 Q5 [6]}}