| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with trigonometric functions |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring (a) solving 3sin(x) - 3sin(2x) = 0 using the double angle formula, and (b) integrating standard trigonometric functions over a given interval. Both parts use routine A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt to find \(x\)-value from \(3\sin x - 3\sin 2x = 0\) using identity for \(\sin 2x\) | M1 | |
| Obtain at least \(\cos x = \frac{1}{2}\) | A1 | |
| Obtain \(\frac{1}{3}\pi\) | A1 | SC B3 can be spotted from \(\sin x = \sin 2x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate to obtain form \(k_1\cos x + k_2\cos 2x\) | \*M1 | non-zero constants \(k_1\), \(k_2\); M0 for \(3\cos x \pm 6\cos 2x\) |
| Obtain correct \(-3\cos x + \frac{3}{2}\cos 2x\) | A1 | |
| Attempt value of integral using their lower limit (in radians) and \(\pi\) correctly | DM1 | Allow one sign error |
| Obtain \(\frac{27}{4}\) | A1 | OE |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to find $x$-value from $3\sin x - 3\sin 2x = 0$ using identity for $\sin 2x$ | M1 | |
| Obtain at least $\cos x = \frac{1}{2}$ | A1 | |
| Obtain $\frac{1}{3}\pi$ | A1 | **SC B3** can be spotted from $\sin x = \sin 2x$ |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain form $k_1\cos x + k_2\cos 2x$ | \*M1 | non-zero constants $k_1$, $k_2$; M0 for $3\cos x \pm 6\cos 2x$ |
| Obtain correct $-3\cos x + \frac{3}{2}\cos 2x$ | A1 | |
| Attempt value of integral using their lower limit (in radians) and $\pi$ correctly | DM1 | Allow one sign error |
| Obtain $\frac{27}{4}$ | A1 | OE |
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\includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-04_531_739_258_703}
The diagram shows the curve with equation $y = 3 \sin x - 3 \sin 2 x$ for $0 \leqslant x \leqslant \pi$. The curve meets the $x$-axis at the origin and at the points with $x$-coordinates $a$ and $\pi$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $a$.
\item Find the area of the shaded region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q3 [7]}}