| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find constant from gradient condition |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation problem requiring substitution of a point, then finding dy/dx using the chain rule. The algebra is routine with exponentials, and the question guides students through two clear steps with no conceptual surprises—slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use at least one of \(e^{2x} = 9\), \(e^y = 2\) and \(e^{2y} = 4\) | B1 | |
| Obtain given result \(58 + 2k = c\) | B1 | [2] AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate left-hand side term by term, reaching \(ae^{2x} + be^y\frac{dy}{dx} + ce^{2y}\frac{dy}{dx}\) | M1 | |
| Obtain \(12e^{2x} + ke^y\frac{dy}{dx} + 2e^{2y}\frac{dy}{dx}\) | A1 | |
| Substitute \((\ln 3, \ln 2)\) in an attempt involving implicit differentiation at least once, where \(\text{RHS} = 0\) | M1 | |
| Obtain \(108 - 12k - 48 = 0\) or equivalent | A1 | |
| Obtain \(k = 5\) and \(c = 68\) | A1 | [5] |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use at least one of $e^{2x} = 9$, $e^y = 2$ and $e^{2y} = 4$ | B1 | |
| Obtain given result $58 + 2k = c$ | B1 | [2] **AG** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate left-hand side term by term, reaching $ae^{2x} + be^y\frac{dy}{dx} + ce^{2y}\frac{dy}{dx}$ | M1 | |
| Obtain $12e^{2x} + ke^y\frac{dy}{dx} + 2e^{2y}\frac{dy}{dx}$ | A1 | |
| Substitute $(\ln 3, \ln 2)$ in an attempt involving implicit differentiation at least once, where $\text{RHS} = 0$ | M1 | |
| Obtain $108 - 12k - 48 = 0$ or equivalent | A1 | |
| Obtain $k = 5$ and $c = 68$ | A1 | [5] |
---5 The curve with equation
$$6 \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { y } + \mathrm { e } ^ { 2 y } = c$$
where $k$ and $c$ are constants, passes through the point $P$ with coordinates $( \ln 3 , \ln 2 )$.\\
(i) Show that $58 + 2 k = c$.\\
(ii) Given also that the gradient of the curve at $P$ is - 6 , find the values of $k$ and $c$.
\hfill \mbox{\textit{CAIE P3 2011 Q5 [7]}}