| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find parameter value given gradient condition |
| Difficulty | Moderate -0.3 This is a straightforward parametric differentiation question requiring the chain rule (dy/dx = (dy/dt)/(dx/dt)) and solving a simple equation. The differentiation is routine (polynomial and logarithm), and finding where the gradient equals 1 involves basic algebra. Slightly easier than average due to the standard technique and minimal problem-solving required. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| State \(\frac{dx}{dt} = 3 + \frac{1}{t-1}\) or \(\frac{dy}{dt} = 2t\) | B1 | |
| Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) | M1 | |
| Obtain \(\frac{dy}{dx}\) in any correct form, e.g. \(\frac{2t(t-1)}{3t-2}\) | A1 | [3 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Equate derivative to 1 and solve for \(t\) | M1 | |
| Obtain roots 2 and \(\frac{1}{2}\) | A1 | |
| State or imply that only \(t = 2\) is admissible c.w.o. | A1 | |
| Obtain coordinates (6, 5) | A1 | [4 marks] |
**(i)**
State $\frac{dx}{dt} = 3 + \frac{1}{t-1}$ or $\frac{dy}{dt} = 2t$ | B1 | |
Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ | M1 | |
Obtain $\frac{dy}{dx}$ in any correct form, e.g. $\frac{2t(t-1)}{3t-2}$ | A1 | [3 marks] |
**(ii)**
Equate derivative to 1 and solve for $t$ | M1 | |
Obtain roots 2 and $\frac{1}{2}$ | A1 | |
State or imply that only $t = 2$ is admissible c.w.o. | A1 | |
Obtain coordinates (6, 5) | A1 | [4 marks] |3 The parametric equations of a curve are
$$x = 3 t + \ln ( t - 1 ) , \quad y = t ^ { 2 } + 1 , \quad \text { for } t > 1$$
(i) Express $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.\\
(ii) Find the coordinates of the only point on the curve at which the gradient of the curve is equal to 1 .
\hfill \mbox{\textit{CAIE P2 2007 Q3 [7]}}