| 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 | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard harmonic form question requiring differentiation of sin/cos, converting to R cos(θ+α) form using standard formulas, then solving a trigonometric equation. All techniques are routine for P2 level with clear scaffolding provided. The 8 marks reflect multiple steps rather than conceptual difficulty. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks |
|---|---|
| 7(i) | State expression of form k cos2x+k sin2x |
| 1 2 | M1 |
| State correct 2cos2x−6sin2x | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 7(ii) | State R= 40 or 6.324… | B1 FT |
| Use appropriate trigonometry to find α | M1 | |
| Obtain 1.249… | A1 | Allow α in degrees at this point |
| Equate their Rcos(2x+α) to 3 and find cos−1(3÷R) | *M1 | |
| Carry out correct process to find one value of α | M1 | Dependent on *M1, allow for -0.086…. |
| Obtain 1.979 | A1 | |
| Carry out correct process to find second value of αwithin the range | M1 | Dependent on *M1 |
| Obtain 3.055 | A1 | Allow 3.056 |
Question 7:
--- 7(i) ---
7(i) | State expression of form k cos2x+k sin2x
1 2 | M1
State correct 2cos2x−6sin2x | A1
2
Question | Answer | Marks | Guidance
--- 7(ii) ---
7(ii) | State R= 40 or 6.324… | B1 FT | Following their derivative
Use appropriate trigonometry to find α | M1
Obtain 1.249… | A1 | Allow α in degrees at this point
Equate their Rcos(2x+α) to 3 and find cos−1(3÷R) | *M1
Carry out correct process to find one value of α | M1 | Dependent on *M1, allow for -0.086….
Obtain 1.979 | A1
Carry out correct process to find second value of αwithin the range | M1 | Dependent on *M1
Obtain 3.055 | A1 | Allow 3.056
8\includegraphics{figure_7}
The diagram shows the curve with equation $y = \sin 2x + 3\cos 2x$ for $0 \leqslant x \leqslant \pi$. At the points $P$ and $Q$ on the curve, the gradient of the curve is 3.
\begin{enumerate}[label=(\roman*)]
\item Find an expression for $\frac{dy}{dx}$. [2]
\item By first expressing $\frac{dy}{dx}$ in the form $R\cos(2x + \alpha)$, 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. [8]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2018 Q7 [10]}}