| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find stationary points coordinates |
| Difficulty | Standard +0.8 This question requires applying the product rule to a function involving exponential and trigonometric components, solving a transcendental equation (tan 4x = 8), and then using the periodicity of tangent to find which stationary point exceeds x=25. While the differentiation is standard, the equation-solving requires insight about the tangent function's period and systematic counting of solutions, making it moderately challenging but within reach of a well-prepared P3 student. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use product or quotient rule | M1 | |
| Obtain \(-5e^{-\frac{x}{5}}\sin 4x + 40e^{-\frac{x}{5}}\cos 4x\) | A1 | |
| Equate \(\frac{dy}{dx}\) to zero and obtain \(\tan 4z = k\) or \(\cos(4x \pm \alpha)\) | M1 | |
| Obtain \(\tan x = 8\) or \(\sqrt{65}\cos\left(4x + \tan^{-1}\frac{1}{8}\right)\) | A1 | |
| Obtain \(0.362\) or \(20.7°\) | A1 | |
| Obtain \(1.147\) or \(65.7°\) | A1 | [6] |
| (ii) State or imply that \(x\)-coordinates of \(T_n\) are increasing by \(\frac{1}{4}\pi\) or \(45°\) | B1 | |
| Attempt solution of inequality (or equation) of form \(x_{n1} + (n - 1)k\pi . 25\) | M1 | |
| Obtain \(n > \frac{4}{\pi}(25 - 0.362) + 1\), following through on their value of \(x_1\) | A1✓ | |
| \(n = 33\) | A1 | [4] |
(i) Use product or quotient rule | M1 |
Obtain $-5e^{-\frac{x}{5}}\sin 4x + 40e^{-\frac{x}{5}}\cos 4x$ | A1 |
Equate $\frac{dy}{dx}$ to zero and obtain $\tan 4z = k$ or $\cos(4x \pm \alpha)$ | M1 |
Obtain $\tan x = 8$ or $\sqrt{65}\cos\left(4x + \tan^{-1}\frac{1}{8}\right)$ | A1 |
Obtain $0.362$ or $20.7°$ | A1 |
Obtain $1.147$ or $65.7°$ | A1 | [6]
(ii) State or imply that $x$-coordinates of $T_n$ are increasing by $\frac{1}{4}\pi$ or $45°$ | B1 |
Attempt solution of inequality (or equation) of form $x_{n1} + (n - 1)k\pi . 25$ | M1 |
Obtain $n > \frac{4}{\pi}(25 - 0.362) + 1$, following through on their value of $x_1$ | A1✓ |
$n = 33$ | A1 | [4]10\\
\includegraphics[max width=\textwidth, alt={}, center]{b6bede75-3da4-4dda-9303-a5a692fc2572-3_556_1093_1596_523}
The diagram shows the curve $y = 10 e ^ { - \frac { 1 } { 2 } x } \sin 4 x$ for $x \geqslant 0$. The stationary points are labelled $T _ { 1 } , T _ { 2 }$, $T _ { 3 } , \ldots$ as shown.\\
(i) Find the $x$-coordinates of $T _ { 1 }$ and $T _ { 2 }$, giving each $x$-coordinate correct to 3 decimal places.\\
(ii) It is given that the $x$-coordinate of $T _ { n }$ is greater than 25 . Find the least possible value of $n$.
\hfill \mbox{\textit{CAIE P3 2014 Q10 [10]}}