| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Stationary points using trigonometry |
| Difficulty | Standard +0.8 This requires differentiating a trigonometric expression involving cos³x, setting dy/dx = 0, then solving a non-trivial trigonometric equation (likely requiring factorization or substitution), followed by a second derivative test. The algebraic manipulation of the resulting equation and finding multiple solutions in a restricted interval elevates this above routine differentiation questions, but it remains a standard P3-level problem requiring systematic application of learned techniques rather than novel insight. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| State derivative in any correct form, e.g. \(3\cos x - 12\cos^2 x \sin x\) | B1 + B1 | |
| Equate derivative to zero and solve for \(\sin 2x\), or \(\sin x\) or \(\cos x\) | M1 | |
| Obtain answer \(x = \frac{1}{12}\pi\) | A1 | |
| Obtain answer \(x = \frac{5}{12}\pi\) | A1 | |
| Obtain answer \(x = \frac{1}{2}\pi\) and no others in the given interval | A1⬇ | [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Carry out a method for determining the nature of the relevant stationary point | M1 | |
| Obtain a maximum at \(\frac{1}{12}\pi\) correctly | A1 | [2] |
**(i)**
| State derivative in any correct form, e.g. $3\cos x - 12\cos^2 x \sin x$ | B1 + B1 | |
| Equate derivative to zero and solve for $\sin 2x$, or $\sin x$ or $\cos x$ | M1 | |
| Obtain answer $x = \frac{1}{12}\pi$ | A1 | |
| Obtain answer $x = \frac{5}{12}\pi$ | A1 | |
| Obtain answer $x = \frac{1}{2}\pi$ and no others in the given interval | A1⬇ | [6] |
**Note:** [Treat answers in degrees as a misread and deduct A1 from the marks for the angles.]
**(ii)**
| Carry out a method for determining the nature of the relevant stationary point | M1 | |
| Obtain a maximum at $\frac{1}{12}\pi$ correctly | A1 | [2] |
---6 The equation of a curve is $y = 3 \sin x + 4 \cos ^ { 3 } x$.\\
(i) Find the $x$-coordinates of the stationary points of the curve in the interval $0 < x < \pi$.\\
(ii) Determine the nature of the stationary point in this interval for which $x$ is least.
\hfill \mbox{\textit{CAIE P3 2012 Q6 [8]}}