| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Substitution u = sin x or u = cos x (area/integral) |
| Difficulty | Standard +0.3 This is a straightforward two-part question: (a) finding a maximum using differentiation (product rule and chain rule with standard trig functions), and (b) a guided integration by substitution where the substitution is explicitly given. Both parts are routine A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use product rule | M1 | |
| Obtain correct derivative in any form, e.g. \(4\sin 2x \cos 2x - \sin^2 2x \sin x\) | A1 | |
| Equate derivative to zero and use a double angle formula | M1* | |
| Reduce equation to one in a single trig function | DM1 | |
| Obtain a correct equation in any form, e.g. \(10\cos^3 x = 6\cos x\), \(4 = 6\tan^2 x\), or \(4 = 10\sin^2 x\) | A1 | |
| Solve and obtain \(x = 0.685\) | A1 | Unsupported answer receives 0 marks |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Using \(du = \pm\cos x\, dx\), or equivalent, express integral in terms of \(u\) and \(du\) | M1 | |
| Obtain \(\int 4u^2(1-u^2)\,du\) | A1 | |
| Use limits \(u=0\) and \(u=1\) in an integral of the form \(au^3 + bu^5\) | M1 | |
| Obtain answer \(\frac{8}{15}\) (or \(0.533\)) | A1 | Unsupported answer receives 0 marks |
| Total | 4 |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule | M1 | |
| Obtain correct derivative in any form, e.g. $4\sin 2x \cos 2x - \sin^2 2x \sin x$ | A1 | |
| Equate derivative to zero and use a double angle formula | M1* | |
| Reduce equation to one in a single trig function | DM1 | |
| Obtain a correct equation in any form, e.g. $10\cos^3 x = 6\cos x$, $4 = 6\tan^2 x$, or $4 = 10\sin^2 x$ | A1 | |
| Solve and obtain $x = 0.685$ | A1 | Unsupported answer receives 0 marks |
| **Total** | **6** | |
## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Using $du = \pm\cos x\, dx$, or equivalent, express integral in terms of $u$ and $du$ | M1 | |
| Obtain $\int 4u^2(1-u^2)\,du$ | A1 | |
| Use limits $u=0$ and $u=1$ in an integral of the form $au^3 + bu^5$ | M1 | |
| Obtain answer $\frac{8}{15}$ (or $0.533$) | A1 | Unsupported answer receives 0 marks |
| **Total** | **4** | |9\\
\includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-16_307_593_269_735}
The diagram shows the curve $y = \sin ^ { 2 } 2 x \cos x$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$, and its maximum point $M$.\\
(a) Find the $x$-coordinate of $M$.\\
(b) Using the substitution $u = \sin x$, find the area of the shaded region bounded by the curve and the $x$-axis.\\
\hfill \mbox{\textit{CAIE P3 2020 Q9 [10]}}