| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - trigonometric functions |
| Difficulty | Challenging +1.2 Part (i) requires product rule and chain rule to differentiate sin²(2x)cos(x), then solving the resulting trigonometric equation for stationary points—moderately challenging but standard A-level technique. Part (ii) involves a guided substitution to convert the integral, requiring double angle formula manipulation and polynomial integration—straightforward once the substitution is applied. The combination of differentiation and integration with trigonometric identities places this slightly above average difficulty. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07n Stationary points: find maxima, minima using derivatives1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use product rule | M1 | |
| Obtain correct derivative in any form, e.g. \(4\sin 2x \cos 2x \cos x - \sin^2 2x \sin x\) | A1 | |
| Equate derivative to zero and use a double angle formula | M1* | |
| Reduce equation to one in a single trig function | M1 (dep*) | |
| Obtain a correct equation in any form, e.g. \(10 \cos^3 x = 6 \cos x\), \(4 = 10 \sin^2 x\) or \(4 = 10 \sin^2 x\) | A1 | |
| Solve and obtain \(x = 0.685\) | A1 | [6 marks] |
| (ii) Using \(du = \pm \cos x \, dx\), or equivalent, express integral in terms of \(u\) and \(du\) | M1 | |
| Obtain \([4u^2(1 - u^2)] \, du\), or equivalent | 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 | [4 marks total] |
**(i)** Use product rule | M1 |
Obtain correct derivative in any form, e.g. $4\sin 2x \cos 2x \cos x - \sin^2 2x \sin x$ | A1 |
Equate derivative to zero and use a double angle formula | M1* |
Reduce equation to one in a single trig function | M1 (dep*) |
Obtain a correct equation in any form, e.g. $10 \cos^3 x = 6 \cos x$, $4 = 10 \sin^2 x$ or $4 = 10 \sin^2 x$ | A1 |
Solve and obtain $x = 0.685$ | A1 | [6 marks]
**(ii)** Using $du = \pm \cos x \, dx$, or equivalent, express integral in terms of $u$ and $du$ | M1 |
Obtain $[4u^2(1 - u^2)] \, du$, or equivalent | 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 | [4 marks total]
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\includegraphics[max width=\textwidth, alt={}, center]{436d891d-92ee-4076-8369-db756d413979-3_307_601_1553_772}
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 $M$.\\
(i) Find the $x$-coordinate of $M$.\\
(ii) Using the substitution $u = \sin x$, find by integration the area of the shaded region bounded by the curve and the $x$-axis.
\hfill \mbox{\textit{CAIE P3 2013 Q9 [10]}}