| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | March |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Standard +0.8 This question requires finding a maximum using calculus (involving product rule and double angle formula), then performing integration by substitution with trigonometric functions. Part (a) involves solving a transcendental equation numerically, and part (b) requires careful handling of the substitution including limits and simplification to exact form. The combination of optimization, trigonometric manipulation, and substitution integration elevates this above standard exercises. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct product rule or chain rule | M1 | |
| Obtain correct derivative in any form | A1 | \(\cos x \cos 2x - \sin x \cdot 2\sin 2x\) |
| Equate derivative to zero and use a correct double angle formula | \*M1 | If chain rule used then derivative set to 0 gains M1 since correct double angle formula has already been used |
| Obtain an equation in one trigonometric variable | DM1 | Allow following from coefficient errors in differentiation only |
| Obtain \(6\sin^2 x = 1\), \(6\cos^2 x = 5\) or \(5\tan^2 x = 1\) | A1 | One of these 3 expressions |
| Obtain final answer \(x = 0.421\) | A1 | Must be 3s.f. |
| 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(du = -\sin x\, dx\) | B1 | |
| Using double angle formula, express integral in terms of \(u\) and \(du\) | M1 | Use \(\cos 2x = 2\cos^2 x - 1\) |
| Integrate and obtain \(\pm\left(u - \frac{2}{3}u^3\right)\) | A1 | |
| Use limits \(u = 1\), \(u = \frac{1}{\sqrt{2}}\) in an integral of the form \(au + bu^3\), where \(ab \neq 0\) | M1 | Require both limits substituted twice in \(au + bu^3\) for M1. Do not condone decimals |
| Obtain \(\frac{1}{3}(\sqrt{2}-1)\) or \(\frac{1}{3}\sqrt{2}\cdot\frac{1}{3}\) or \(\frac{2}{3}\left(\frac{1}{\sqrt{2}}\right)\frac{1}{3}\) or simplified equivalent | A1 | ISW |
| 5 |
## Question 11(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct product rule or chain rule | **M1** | |
| Obtain correct derivative in any form | **A1** | $\cos x \cos 2x - \sin x \cdot 2\sin 2x$ |
| Equate derivative to zero and use a correct double angle formula | **\*M1** | If chain rule used then derivative set to 0 gains M1 since correct double angle formula has already been used |
| Obtain an equation in one trigonometric variable | **DM1** | Allow following from coefficient errors in differentiation only |
| Obtain $6\sin^2 x = 1$, $6\cos^2 x = 5$ or $5\tan^2 x = 1$ | **A1** | One of these 3 expressions |
| Obtain final answer $x = 0.421$ | **A1** | Must be 3s.f. |
| | **6** | |
---
## Question 11(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $du = -\sin x\, dx$ | **B1** | |
| Using double angle formula, express integral in terms of $u$ and $du$ | **M1** | Use $\cos 2x = 2\cos^2 x - 1$ |
| Integrate and obtain $\pm\left(u - \frac{2}{3}u^3\right)$ | **A1** | |
| Use limits $u = 1$, $u = \frac{1}{\sqrt{2}}$ in an integral of the form $au + bu^3$, where $ab \neq 0$ | **M1** | Require both limits substituted twice in $au + bu^3$ for M1. Do not condone decimals |
| Obtain $\frac{1}{3}(\sqrt{2}-1)$ or $\frac{1}{3}\sqrt{2}\cdot\frac{1}{3}$ or $\frac{2}{3}\left(\frac{1}{\sqrt{2}}\right)\frac{1}{3}$ or simplified equivalent | **A1** | ISW |
| | **5** | |11\\
\includegraphics[max width=\textwidth, alt={}, center]{7cdf4db7-7217-4ef1-becf-359a70cfeb62-16_556_698_274_712}
The diagram shows the curve $y = \sin x \cos 2 x$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$, and its maximum point $M$.
\begin{enumerate}[label=(\alph*)]
\item Find the $x$-coordinate of $M$, giving your answer correct to 3 significant figures.
\item Using the substitution $u = \cos x$, find the area of the shaded region enclosed by the curve and the $x$-axis in the first quadrant, giving your answer in a simplified exact form.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q11 [11]}}