| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Finding maximum/minimum on curve |
| Difficulty | Standard +0.8 This is a multi-part question requiring differentiation with product and chain rules to find a maximum, then a non-trivial trigonometric substitution (x = sin θ) with careful manipulation of sin²2θ, and finally integration using double-angle formulas. While the techniques are standard for P3/C4, the execution requires careful algebraic manipulation across multiple steps, placing it moderately above average difficulty. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EITHER Use product and chain rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| OR Square and differentiate LHS by chain rule and RHS by product rule or as powers | M1 | |
| Obtain correct result in any form | A1 | |
| Set \(\dfrac{dy}{dx}\) equal to zero and make reasonable attempt to solve for \(x \neq 0\) | M1 | |
| Obtain answer \(x = \sqrt{\tfrac{2}{3}}\), or exact equivalent, correctly | A1 | [4 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \(dx = \cos\theta\, d\theta\) or \(\dfrac{dx}{d\theta} = \cos\theta\) | B1 | |
| Substitute for \(x\) and \(dx\) throughout the integral \(\int y\, dx\) | M1 | |
| Obtain the given form correctly with no errors seen | A1 | [3 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt integration and reach indefinite integral of the form \(a\theta + b\sin 4\theta\), where \(ab \neq 0\) | M1* | |
| Obtain indefinite integral \(\tfrac{1}{8}\theta - \tfrac{1}{32}\sin 4\theta\), or equivalent | A1 | |
| Substitute limits correctly | M1(dep*) | |
| Obtain exact answer \(\tfrac{1}{16}\pi\) | A1 | [4 marks] |
| [Working to carry out the change of limits is needed for the A mark in (ii) but, if omitted, can be earned retrospectively if it is seen in part (iii)] |
# Question 10:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **EITHER** Use product and chain rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| **OR** Square and differentiate LHS by chain rule and RHS by product rule or as powers | M1 | |
| Obtain correct result in any form | A1 | |
| Set $\dfrac{dy}{dx}$ equal to zero and make reasonable attempt to solve for $x \neq 0$ | M1 | |
| Obtain answer $x = \sqrt{\tfrac{2}{3}}$, or exact equivalent, correctly | A1 | **[4 marks]** |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $dx = \cos\theta\, d\theta$ or $\dfrac{dx}{d\theta} = \cos\theta$ | B1 | |
| Substitute for $x$ and $dx$ throughout the integral $\int y\, dx$ | M1 | |
| Obtain the given form correctly with no errors seen | A1 | **[3 marks]** |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt integration and reach indefinite integral of the form $a\theta + b\sin 4\theta$, where $ab \neq 0$ | M1* | |
| Obtain indefinite integral $\tfrac{1}{8}\theta - \tfrac{1}{32}\sin 4\theta$, or equivalent | A1 | |
| Substitute limits correctly | M1(dep*) | |
| Obtain exact answer $\tfrac{1}{16}\pi$ | A1 | **[4 marks]** |
| [Working to carry out the change of limits is needed for the A mark in **(ii)** but, if omitted, can be earned retrospectively if it is seen in part **(iii)**] | | |10\\
\includegraphics[max width=\textwidth, alt={}, center]{0f73e750-18a0-49ad-b4cb-fd6d14f0789e-4_424_713_262_715}
The diagram shows the curve $y = x ^ { 2 } \sqrt { } \left( 1 - x ^ { 2 } \right)$ for $x \geqslant 0$ and its maximum point $M$.\\
(i) Find the exact value of the $x$-coordinate of $M$.\\
(ii) Show, by means of the substitution $x = \sin \theta$, that the area $A$ of the shaded region between the curve and the $x$-axis is given by
$$A = \frac { 1 } { 4 } \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 2 } 2 \theta \mathrm {~d} \theta$$
(iii) Hence obtain the exact value of $A$.
\hfill \mbox{\textit{CAIE P3 2009 Q10 [11]}}