| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Related rates of change |
| Difficulty | Standard +0.8 This question requires chain rule combined with quotient rule, algebraic manipulation to reach a specific form, then a second optimization problem involving differentiating a product of polynomial and surd terms and solving the resulting equation. The multi-step nature, need to differentiate the gradient expression, and algebraic complexity elevate this above standard chain rule exercises. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use quotient or product rule to differentiate \((1-x)/(1+x)\) | M1 | |
| Obtain correct derivative in any form | A1 | |
| Use chain rule to find \(\frac{dy}{dx}\) | M1 | |
| Obtain a correct expression in any form | A1 | |
| Obtain the gradient of the normal in the given form correctly | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain \(x = \frac{1}{2}\) | A1 | [4] |
## Question 9:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use quotient or product rule to differentiate $(1-x)/(1+x)$ | M1 | |
| Obtain correct derivative in any form | A1 | |
| Use chain rule to find $\frac{dy}{dx}$ | M1 | |
| Obtain a correct expression in any form | A1 | |
| Obtain the gradient of the normal in the given form correctly | A1 | [5] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and solve for $x$ | M1 | |
| Obtain $x = \frac{1}{2}$ | A1 | [4] |
---9\\
\includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-4_611_895_255_625}
The diagram shows the curve $y = \sqrt { } \left( \frac { 1 - x } { 1 + x } \right)$.\\
(i) By first differentiating $\frac { 1 - x } { 1 + x }$, obtain an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$. Hence show that the gradient of the normal to the curve at the point $( x , y )$ is $( 1 + x ) \sqrt { } \left( 1 - x ^ { 2 } \right)$.\\
(ii) The gradient of the normal to the curve has its maximum value at the point $P$ shown in the diagram. Find, by differentiation, the $x$-coordinate of $P$.
\hfill \mbox{\textit{CAIE P3 2010 Q9 [9]}}