| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - trigonometric functions |
| Difficulty | Standard +0.8 This question requires product rule differentiation of a complex trigonometric expression, solving a non-trivial trigonometric equation to find stationary points, and then performing integration by substitution with trigonometric functions. While the techniques are standard A-level material, the algebraic manipulation and solving sin x = 2cos x is more demanding than typical textbook exercises, placing it moderately 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 derivatives1.08e Area between curve and x-axis: using definite integrals1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Use product and chain rule | M1 | |
| Obtain correct derivative in any form, e.g. \(15\sin^2 x \cos^3 x - 10\sin^4 x \cos x\) | A1 | |
| Equate derivative to zero and obtain a relevant equation in one trigonometric function | M1 | |
| Obtain \(2\tan^2 x = 3\), \(5\cos^2 x = 2\), or \(5\sin^2 x = 3\) | A1 | |
| Obtain answer \(x = 0.886\) radians | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(du = -\sin x\,dx\), or \(\frac{du}{dx} = -\sin x\), or equivalent | B1 | |
| Express integral in terms of \(u\) and \(du\) | M1 | |
| Obtain \(\pm\int 5(u^2 - u^4)du\), or equivalent | A1 | |
| Integrate and use limits \(u = 1\) and \(u = 0\) (or \(x = 0\) and \(x = \frac{1}{2}\pi\)) | M1 | |
| Obtain answer \(\frac{2}{3}\), or equivalent, with no errors seen | A1 | [5] |
**(i)**
Use product and chain rule | M1 |
Obtain correct derivative in any form, e.g. $15\sin^2 x \cos^3 x - 10\sin^4 x \cos x$ | A1 |
Equate derivative to zero and obtain a relevant equation in one trigonometric function | M1 |
Obtain $2\tan^2 x = 3$, $5\cos^2 x = 2$, or $5\sin^2 x = 3$ | A1 |
Obtain answer $x = 0.886$ radians | A1 | [5]
**(ii)**
State or imply $du = -\sin x\,dx$, or $\frac{du}{dx} = -\sin x$, or equivalent | B1 |
Express integral in terms of $u$ and $du$ | M1 |
Obtain $\pm\int 5(u^2 - u^4)du$, or equivalent | A1 |
Integrate and use limits $u = 1$ and $u = 0$ (or $x = 0$ and $x = \frac{1}{2}\pi$) | M1 |
Obtain answer $\frac{2}{3}$, or equivalent, with no errors seen | A1 | [5]8\\
\includegraphics[max width=\textwidth, alt={}, center]{5b219e1c-e5a0-4f75-910d-fca9761e5088-3_435_895_799_625}
The diagram shows the curve $y = 5 \sin ^ { 3 } x \cos ^ { 2 } x$ 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 $u = \cos x$, find by integration the area of the shaded region bounded by the curve and the $x$-axis.
\hfill \mbox{\textit{CAIE P3 2011 Q8 [10]}}