| 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 Part (i) requires standard differentiation using the product rule and chain rule, then solving f'(x)=0 numerically - routine calculus. Part (ii) is a straightforward substitution integral with u=sin x, leading to a polynomial integral ∫u²(1-u²) du - this is a standard textbook exercise in integration by substitution with no novel insight required. The 9 total marks reflect routine application of techniques rather than problem-solving difficulty. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| 7(i) | 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 tan2x=2 | A1 | |
| Obtain answer x = 0.685 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 7(ii) | ( ) |
| Use the given substitution and reach a∫ u2 −u4 du | M1 |
| Obtain correct integral with a = 5 and limits 0 and 1 | A1 |
| Answer | Marks |
|---|---|
| 3 5 | M1 |
| Answer | Marks |
|---|---|
| 3 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 7:
--- 7(i) ---
7(i) | 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 tan2x=2 | A1
Obtain answer x = 0.685 | A1
5
Question | Answer | Marks | Guidance
--- 7(ii) ---
7(ii) | ( )
Use the given substitution and reach a∫ u2 −u4 du | M1
Obtain correct integral with a = 5 and limits 0 and 1 | A1
1 1
Use correct limits in an integral of the form a u3 − u5
3 5 | M1
2
Obtain answer
3 | A1
4
Question | Answer | Marks | Guidance\includegraphics{figure_7}
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.
\begin{enumerate}[label=(\roman*)]
\item Find the $x$-coordinate of $M$, giving your answer correct to 3 decimal places. [5]
\item Using the substitution $u = \sin x$ and showing all necessary working, find the exact area of $R$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q7 [9]}}