| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| 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 -0.8 This is a straightforward two-part question requiring standard formula recall (cos²x = (1+cos2x)/2) followed by routine integration of trigonometric functions. The integration is direct substitution with no problem-solving insight needed, making it easier than average for A-level. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State correct expression \(\frac{1}{2} + \frac{1}{2}\cos 2x\), or equivalent | B1 | [1] |
| (ii) Integrate an expression of the form \(a + b\cos 2x\), where \(ab \neq 0\), correctly | M1 | |
| State correct integral \(\frac{1}{2}x + \frac{1}{4}\sin 2x\), or equivalent | A1 | |
| Obtain correct integral (for sin \(2x\) term) of \(-\frac{1}{2}\cos 2x\) | B1 | |
| Attempt to substitute limits, using exact values | M1 | |
| Obtain given answer correctly | A1 | [5] |
**(i)** State correct expression $\frac{1}{2} + \frac{1}{2}\cos 2x$, or equivalent | B1 | [1]
**(ii)** Integrate an expression of the form $a + b\cos 2x$, where $ab \neq 0$, correctly | M1 |
State correct integral $\frac{1}{2}x + \frac{1}{4}\sin 2x$, or equivalent | A1 |
Obtain correct integral (for sin $2x$ term) of $-\frac{1}{2}\cos 2x$ | B1 |
Attempt to substitute limits, using exact values | M1 |
Obtain given answer correctly | A1 | [5]4 (i) Express $\cos ^ { 2 } x$ in terms of $\cos 2 x$.\\
(ii) Hence show that
$$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \left( \cos ^ { 2 } x + \sin 2 x \right) \mathrm { d } x = \frac { 1 } { 8 } \sqrt { } 3 + \frac { 1 } { 12 } \pi + \frac { 1 } { 4 }$$
\hfill \mbox{\textit{CAIE P2 2011 Q4 [6]}}