| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Challenging +1.2 Part (i) requires standard calculus (finding maximum via differentiation of a product/composite function) and numerical solving. Part (ii) is a straightforward substitution u = cos x that converts the integral into a polynomial form, requiring recognition that sin²x = 1 - cos²x. While it involves multiple techniques, these are standard P3/A-level Further Maths procedures with no novel insight required—slightly above average due to the algebraic manipulation needed after substitution. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use product rule and chain rule at least once | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero, use Pythagoras and obtain an equation in \(\cos x\) | M1 | |
| Obtain \(\cos^2 x + 3\cos x - 1 = 0\), or 3-term equivalent | A1 | |
| Obtain answer \(x = 1.26\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Using \(du = \pm \sin x \, dx\) express integrand in terms of \(u\) and \(du\) | M1 | |
| Obtain integrand \(e^u(u^2 - 1)\) | A1 | OE |
| Commence integration by parts and reach \(ae^u(u^2-1) + b\int ue^u \, du\) | \*M1 | |
| Obtain \(e^u(u^2-1) - 2\int ue^u \, du\) | A1 | OE |
| Complete integration, obtaining \(e^u(u^2 - 2u + 1)\) | A1 | OE |
| Substitute limits \(u = 1\) and \(u = -1\) (or \(x = 0\) and \(x = \pi\)), having integrated completely | DM1 | |
| Obtain answer \(\frac{4}{e}\), or exact equivalent | A1 |
## Question 10(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule and chain rule at least once | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero, use Pythagoras and obtain an equation in $\cos x$ | M1 | |
| Obtain $\cos^2 x + 3\cos x - 1 = 0$, or 3-term equivalent | A1 | |
| Obtain answer $x = 1.26$ | A1 | |
**Total: 5 marks**
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## Question 10(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Using $du = \pm \sin x \, dx$ express integrand in terms of $u$ and $du$ | M1 | |
| Obtain integrand $e^u(u^2 - 1)$ | A1 | OE |
| Commence integration by parts and reach $ae^u(u^2-1) + b\int ue^u \, du$ | \*M1 | |
| Obtain $e^u(u^2-1) - 2\int ue^u \, du$ | A1 | OE |
| Complete integration, obtaining $e^u(u^2 - 2u + 1)$ | A1 | OE |
| Substitute limits $u = 1$ and $u = -1$ (or $x = 0$ and $x = \pi$), having integrated completely | DM1 | |
| Obtain answer $\frac{4}{e}$, or exact equivalent | A1 | |
**Total: 7 marks**10\\
\includegraphics[max width=\textwidth, alt={}, center]{5b5ed7d1-028e-4f9a-ae9e-26071d0df678-18_449_787_262_678}
The diagram shows the graph of $y = \mathrm { e } ^ { \cos x } \sin ^ { 3 } x$ for $0 \leqslant x \leqslant \pi$, and its maximum point $M$. The shaded region $R$ is bounded by the curve and the $x$-axis.\\
(i) Find the $x$-coordinate of $M$. Show all necessary working and give your answer correct to 2 decimal places.\\
(ii) By first using the substitution $u = \cos x$, find the exact value of the area of $R$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P3 2019 Q10 [12]}}