| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find dy/dx at a point |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question requiring standard technique to find dy/dx, then substitution of a point. Part (ii) adds mild algebraic manipulation to show no solutions exist. The methods are routine for P3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Obtain \(2y\frac{dy}{dx}\) as derivative of \(y^2\) | B1 | |
| Obtain \(-4y - 4x\frac{dy}{dx}\) as derivative of \(-4xy\) | B1 | |
| Substitute \(x = 2\) and \(y = -3\) and find value of \(\frac{dy}{dx}\) (dependent on at least one B1 being earned and \(\frac{d(45)}{dx} = 0\)) | M1 | |
| Obtain \(\frac{12}{7}\) or equivalent | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitute \(\frac{dy}{dx} = 1\) in an expression involving \(\frac{dy}{dx}\), \(x\) and \(y\) and obtain \(ay = bx\) | M1 | |
| Obtain \(y = x\) or equivalent | A1 | |
| Use \(y = x\) in original equation and demonstrate contradiction | A1 | [3] |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Obtain $2y\frac{dy}{dx}$ as derivative of $y^2$ | B1 | |
| Obtain $-4y - 4x\frac{dy}{dx}$ as derivative of $-4xy$ | B1 | |
| Substitute $x = 2$ and $y = -3$ and find value of $\frac{dy}{dx}$ (dependent on at least one B1 being earned and $\frac{d(45)}{dx} = 0$) | M1 | |
| Obtain $\frac{12}{7}$ or equivalent | A1 | [4] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute $\frac{dy}{dx} = 1$ in an expression involving $\frac{dy}{dx}$, $x$ and $y$ and obtain $ay = bx$ | M1 | |
| Obtain $y = x$ or equivalent | A1 | |
| Use $y = x$ in original equation and demonstrate contradiction | A1 | [3] |
---6 The equation of a curve is $3 x ^ { 2 } - 4 x y + y ^ { 2 } = 45$.\\
(i) Find the gradient of the curve at the point $( 2 , - 3 )$.\\
(ii) Show that there are no points on the curve at which the gradient is 1 .
\hfill \mbox{\textit{CAIE P3 2012 Q6 [7]}}