| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of quotient |
| Difficulty | Moderate -0.8 This is a straightforward two-part differentiation question testing standard rules (chain rule for part i, quotient rule for part ii). Both are routine applications with no problem-solving required, making it easier than average but not trivial since it requires correct application of multiple differentiation rules. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain \(\frac{k\cos 2x}{1+\sin 2x}\) for any non-zero constant \(k\) | M1 | |
| Obtain \(\frac{2\cos 2x}{1+\sin 2x}\) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct quotient or product rule | M1 | |
| Obtain \(\frac{x\sec^2 x - \tan x}{x^2}\) or equivalent | A1 | [2] |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain $\frac{k\cos 2x}{1+\sin 2x}$ for any non-zero constant $k$ | M1 | |
| Obtain $\frac{2\cos 2x}{1+\sin 2x}$ | A1 | [2] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct quotient or product rule | M1 | |
| Obtain $\frac{x\sec^2 x - \tan x}{x^2}$ or equivalent | A1 | [2] |
---2 Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in each of the following cases:\\
(i) $y = \ln ( 1 + \sin 2 x )$,\\
(ii) $y = \frac { \tan x } { x }$.
\hfill \mbox{\textit{CAIE P3 2011 Q2 [4]}}