| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 12 |
| 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) is a standard substitution integral requiring routine application of u = cos x with du = -sin x dx, followed by expanding cos²(2x) using double angle formula and integrating a polynomial in u. Part (ii) requires differentiation using product and chain rules, then solving a trigonometric equation. Both parts are straightforward applications of A-level techniques with no novel insight required, making this slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\mathrm{d}u = -\sin x\, \mathrm{d}x\) | B1 | |
| Using correct double angle formula, express the integral in terms of \(u\) and \(\mathrm{d}u\) | M1 | |
| Obtain integrand \(\pm(2u^2 - 1)^2\) | A1 | |
| Change limits and obtain correct integral \(\int_{\frac{1}{\sqrt{2}}}^{1}(2u^2-1)^2\,\mathrm{d}u\) with no errors seen | A1 | |
| Substitute limits in an integral of the form \(au^5 + bu^3 + cu\) | M1 | |
| Obtain answer \(\frac{1}{15}(7 - 4\sqrt{2})\), or exact simplified equivalent | A1 |
| 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 and use trig formulae to obtain an equation in \(\cos x\) and \(\sin x\) | M1 | |
| Use correct methods to obtain an equation in \(\cos x\) or \(\sin x\) only | M1 | |
| Obtain \(10\cos^2 x = 9\) or \(10\sin^2 x = 1\), or equivalent | A1 | |
| Obtain answer \(0.32\) | A1 |
## Question 10(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\mathrm{d}u = -\sin x\, \mathrm{d}x$ | B1 | |
| Using correct double angle formula, express the integral in terms of $u$ and $\mathrm{d}u$ | M1 | |
| Obtain integrand $\pm(2u^2 - 1)^2$ | A1 | |
| Change limits and obtain correct integral $\int_{\frac{1}{\sqrt{2}}}^{1}(2u^2-1)^2\,\mathrm{d}u$ with no errors seen | A1 | |
| Substitute limits in an integral of the form $au^5 + bu^3 + cu$ | M1 | |
| Obtain answer $\frac{1}{15}(7 - 4\sqrt{2})$, or exact simplified equivalent | A1 | |
## Question 10(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule and chain rule at least once | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and use trig formulae to obtain an equation in $\cos x$ and $\sin x$ | M1 | |
| Use correct methods to obtain an equation in $\cos x$ or $\sin x$ only | M1 | |
| Obtain $10\cos^2 x = 9$ or $10\sin^2 x = 1$, or equivalent | A1 | |
| Obtain answer $0.32$ | A1 | |10\\
\includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-18_324_677_259_734}
The diagram shows the curve $y = \sin x \cos ^ { 2 } 2 x$ for $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$ and its maximum point $M$.\\
(i) Using the substitution $u = \cos x$, find by integration the exact area of the shaded region bounded by the curve and the $x$-axis.\\
(ii) Find the $x$-coordinate of $M$. Give your answer correct to 2 decimal places.\\
\hfill \mbox{\textit{CAIE P3 2017 Q10 [12]}}