| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: differentiate using chain rule, solve a simple trigonometric equation for stationary points, then integrate. The trigonometric manipulations are routine (using double angle formula cos 2x = 2cos²x - 1 makes the derivative factorise easily), and the integration is direct. Slightly above average difficulty due to the exact value requirement and multi-step nature, but all techniques are standard A-level fare with no novel insight required. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Differentiate to obtain \(-2\sin x + 2\sin 2x\) or equivalent | B1 | |
| Use \(\sin 2x = 2\sin x \cos x\) or equivalent | B1 | |
| Equate first derivative to zero and solve for \(x\) | M1 | |
| Obtain \(\frac{1}{3}\pi\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Integrate to obtain form \(k_1 \sin x + k_2 \sin 2x\) | M1 | |
| Obtain correct \(2\sin x - \frac{1}{2}\sin 2x\) | A1 | |
| Apply limits 0 and their answer from part (i) | M1 | |
| Obtain \(\frac{3}{4}\sqrt{3}\) or exact equivalent | A1 | [4] |
**(i)**
Differentiate to obtain $-2\sin x + 2\sin 2x$ or equivalent | B1 |
Use $\sin 2x = 2\sin x \cos x$ or equivalent | B1 |
Equate first derivative to zero and solve for $x$ | M1 |
Obtain $\frac{1}{3}\pi$ | A1 | [4]
**(ii)**
Integrate to obtain form $k_1 \sin x + k_2 \sin 2x$ | M1 |
Obtain correct $2\sin x - \frac{1}{2}\sin 2x$ | A1 |
Apply limits 0 and their answer from part (i) | M1 |
Obtain $\frac{3}{4}\sqrt{3}$ or exact equivalent | A1 | [4]5\\
\includegraphics[max width=\textwidth, alt={}, center]{c703565b-8aa8-424b-9684-6592d4effdf8-2_554_689_1354_726}
The diagram shows part of the curve
$$y = 2 \cos x - \cos 2 x$$
and its maximum point $M$. The shaded region is bounded by the curve, the axes and the line through $M$ parallel to the $y$-axis.\\
(i) Find the exact value of the $x$-coordinate of $M$.\\
(ii) Find the exact value of the area of the shaded region.
\hfill \mbox{\textit{CAIE P2 2014 Q5 [8]}}