| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| 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 two-part question requiring expansion of a squared trigonometric expression, application of double angle formulae (standard identities), and then direct integration. The 'show that' structure guides students through the algebraic manipulation, and the integration is routine once the double angle form is established. Slightly above average due to the multi-step nature and need to recognize double angle identities, but well within standard P2/C3 expectations. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Expand to obtain 4 \(\sin^2 x + 4 \sin x \cos x + \cos^2 x\) | B1 | |
| Use 2 \(\sin x \cos x = \sin 2x\) | B1 | |
| Attempt to express \(\sin^2 x\) or \(\cos^2 x\) (or both) in terms of \(\cos 2x\) | M1 | |
| Obtain correct \(\frac{1}{2}k(1 - \cos 2x)\) for their \(k \sin^2 x\) or equivalent | A1∨ | |
| Confirm given answer \(\frac{5}{2} + 2 \sin 2x - \frac{3}{2} \cos 2x\) | A1 | [5] |
| (ii) Integrate to obtain form \(px + q \cos 2x + r \sin 2x\) | M1 | |
| Obtain \(\frac{5}{2}x - \cos 2x - \frac{3}{4} \sin 2x\) | A1 | |
| Substitute limits in integral of form \(px + q \cos 2x + r \sin 2x\) and attempt simplification | DM1 | |
| Obtain \(\frac{5}{8}\pi + \frac{1}{4}\) or exact equivalent | A1 | [4] |
**(i)** Expand to obtain 4 $\sin^2 x + 4 \sin x \cos x + \cos^2 x$ | B1 |
Use 2 $\sin x \cos x = \sin 2x$ | B1 |
Attempt to express $\sin^2 x$ or $\cos^2 x$ (or both) in terms of $\cos 2x$ | M1 |
Obtain correct $\frac{1}{2}k(1 - \cos 2x)$ for their $k \sin^2 x$ or equivalent | A1∨ |
Confirm given answer $\frac{5}{2} + 2 \sin 2x - \frac{3}{2} \cos 2x$ | A1 | [5]
**(ii)** Integrate to obtain form $px + q \cos 2x + r \sin 2x$ | M1 |
Obtain $\frac{5}{2}x - \cos 2x - \frac{3}{4} \sin 2x$ | A1 |
Substitute limits in integral of form $px + q \cos 2x + r \sin 2x$ and attempt simplification | DM1 |
Obtain $\frac{5}{8}\pi + \frac{1}{4}$ or exact equivalent | A1 | [4]7 (i) Show that $( 2 \sin x + \cos x ) ^ { 2 }$ can be written in the form $\frac { 5 } { 2 } + 2 \sin 2 x - \frac { 3 } { 2 } \cos 2 x$.\\
(ii) Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( 2 \sin x + \cos x ) ^ { 2 } \mathrm {~d} x$.
\hfill \mbox{\textit{CAIE P2 2012 Q7 [9]}}