| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Standard +0.3 This is a straightforward integration by substitution question with a standard trigonometric integrand. Part (i) requires basic calculus (finding maximum via differentiation) and part (ii) follows a guided substitution u=sin x that directly simplifies the integral to a polynomial form. The working is routine for P3 level with no novel insights required, making it slightly easier than average. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07n Stationary points: find maxima, minima using derivatives1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use product rule | M1* | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and obtain an equation in a single trig function | depM1* | |
| Obtain a correct equation, e.g. \(3\tan^2 x = 2\) | A1 | |
| Obtain answer \(x = 0.685\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use the given substitution and reach \(a\int(u^2 - u^4)\,du\) | M1 | |
| Obtain correct integral with \(a = 5\) and limits \(0\) and \(1\) | A1 | |
| Use correct limits in an integral of the form \(a\!\left(\frac{1}{3}u^3 - \frac{1}{5}u^5\right)\) | M1 | |
| Obtain answer \(\frac{2}{3}\) | A1 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule | M1* | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and obtain an equation in a single trig function | depM1* | |
| Obtain a correct equation, e.g. $3\tan^2 x = 2$ | A1 | |
| Obtain answer $x = 0.685$ | A1 | |
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## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use the given substitution and reach $a\int(u^2 - u^4)\,du$ | M1 | |
| Obtain correct integral with $a = 5$ and limits $0$ and $1$ | A1 | |
| Use correct limits in an integral of the form $a\!\left(\frac{1}{3}u^3 - \frac{1}{5}u^5\right)$ | M1 | |
| Obtain answer $\frac{2}{3}$ | A1 | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{c861e691-66da-4269-9057-4a343be9835e-12_357_565_260_790}
The diagram shows the curve $y = 5 \sin ^ { 2 } x \cos ^ { 3 } x$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$, and its maximum point $M$. The shaded region $R$ is bounded by the curve and the $x$-axis.\\
(i) Find the $x$-coordinate of $M$, giving your answer correct to 3 decimal places.\\
(ii) Using the substitution $u = \sin x$ and showing all necessary working, find the exact area of $R$. [4]\\
\hfill \mbox{\textit{CAIE P3 2018 Q7 [9]}}