| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Stationary points of curves |
| Difficulty | Standard +0.8 This question requires differentiation of trigonometric functions (routine), but then demands expressing the derivative in harmonic form R cos(2x + α) and solving a non-standard trigonometric equation. The harmonic form conversion and solving cos(2x + α) = 3/R requires multiple techniques and careful angle work, making it moderately challenging but still within standard A-level scope. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State expression of form \(k_1\cos 2x + k_2\sin 2x\) | M1 | |
| State correct \(2\cos 2x - 6\sin 2x\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(R = \sqrt{40}\) or \(6.324\ldots\) | B1 FT | Following their derivative |
| Use appropriate trigonometry to find \(\alpha\) | M1 | |
| Obtain \(1.249\ldots\) | A1 | Allow \(\alpha\) in degrees at this point |
| Equate their \(R\cos(2x+\alpha)\) to 3 and find \(\cos^{-1}(3 \div R)\) | *M1 | |
| Carry out correct process to find one value of \(x\) | M1 | Dependent on *M1, allow for \(-0.086\ldots\) |
| Obtain 1.979 | A1 | |
| Carry out correct process to find second value of \(x\) within the range | M1 | Dependent on *M1 |
| Obtain 3.055 | A1 | Allow 3.056 |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State expression of form $k_1\cos 2x + k_2\sin 2x$ | M1 | |
| State correct $2\cos 2x - 6\sin 2x$ | A1 | |
---
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $R = \sqrt{40}$ or $6.324\ldots$ | B1 FT | Following their derivative |
| Use appropriate trigonometry to find $\alpha$ | M1 | |
| Obtain $1.249\ldots$ | A1 | Allow $\alpha$ in degrees at this point |
| Equate their $R\cos(2x+\alpha)$ to 3 and find $\cos^{-1}(3 \div R)$ | *M1 | |
| Carry out correct process to find one value of $x$ | M1 | Dependent on *M1, allow for $-0.086\ldots$ |
| Obtain 1.979 | A1 | |
| Carry out correct process to find second value of $x$ within the range | M1 | Dependent on *M1 |
| Obtain 3.055 | A1 | Allow 3.056 |7\\
\includegraphics[max width=\textwidth, alt={}, center]{6bf7ba66-8362-4ac0-8e5c-3f88a3ccdf86-12_424_488_260_826}
The diagram shows the curve with equation $y = \sin 2 x + 3 \cos 2 x$ for $0 \leqslant x \leqslant \pi$. At the points $P$ and $Q$ on the curve, the gradient of the curve is 3 .\\
(i) Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) By first expressing $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in the form $R \cos ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$, find the $x$-coordinates of $P$ and $Q$, giving your answers correct to 4 significant figures.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P2 2018 Q7 [10]}}