| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Factorization method |
| Difficulty | Standard +0.3 This question involves standard techniques: solving a quadratic in sin x, using a double angle identity to rewrite the equation, and integrating basic trigonometric functions. While it requires multiple steps and careful algebraic manipulation, all techniques are routine A-level methods with no novel problem-solving required. The 'show that' part provides scaffolding for the integration, making it more straightforward than if students had to discover the form themselves. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve three-term quadratic equation for \(\sin x\); Obtain at least \(\sin x = -\frac{1}{2}\) and no errors seen; Obtain \(x = \frac{7}{6}\pi\) | M1, A1, A1 | [3] |
| State \(\sin^2 x = \frac{1}{4} - \frac{1}{4}\cos 2x\); Obtain given \(5 + 8\sin x - 2\cos 2x\) with necessary detail seen; Integrate to obtain expression of form \(ax + b\cos x + c\sin 2x\); Obtain correct \(5x - 8\cos x - \sin 2x\); Apply limits 0 and their \(x\)-value correctly; Obtain \(\frac{35}{6}\pi + \frac{7}{2}\sqrt{3} + 8\) or exact equivalent | B1, B1, M1, A1, M1 depM, A1 | [6] |
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve three-term quadratic equation for $\sin x$; Obtain at least $\sin x = -\frac{1}{2}$ and no errors seen; Obtain $x = \frac{7}{6}\pi$ | M1, A1, A1 | [3] |
| State $\sin^2 x = \frac{1}{4} - \frac{1}{4}\cos 2x$; Obtain given $5 + 8\sin x - 2\cos 2x$ with necessary detail seen; Integrate to obtain expression of form $ax + b\cos x + c\sin 2x$; Obtain correct $5x - 8\cos x - \sin 2x$; Apply limits 0 and their $x$-value correctly; Obtain $\frac{35}{6}\pi + \frac{7}{2}\sqrt{3} + 8$ or exact equivalent | B1, B1, M1, A1, M1 depM, A1 | [6] |
---\includegraphics{figure_6}
The diagram shows part of the curve with equation
$$y = 4\sin^2 x + 8\sin x + 3$$
and its point of intersection $P$ with the $x$-axis.
\begin{enumerate}[label=(\roman*)]
\item Find the exact $x$-coordinate of $P$. [3]
\item Show that the equation of the curve can be written
$$y = 5 + 8\sin x - 2\cos 2x,$$
and use integration to find the exact area of the shaded region enclosed by the curve and the axes. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2015 Q6 [9]}}