| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Tangent parallel to axis condition |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring the chain rule (dy/dx = dy/dt รท dx/dt), product rule for dy/dt, and solving dy/dx = 0. All techniques are standard P2 content with no novel insight required, making it slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use product rule to differentiate \(v\) | M1 | |
| Obtain correct derivative in any form in \(t\) for \(y\) | A1 | |
| Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) | M1 | |
| Obtain given answer correctly | A1 | [4] |
| (ii) State \(t = 0\) | M1 | |
| State that \(\frac{dy}{dx} = 0\) and make correct conclusion | A1 | [2] |
| (iii) Substitute \(t = -2\) into equation for \(x\) or \(y\) | M1 | |
| Obtain \((e^{-6}, 4e^{-3}+3)\) | A1 | [2] |
**(i)** Use product rule to differentiate $v$ | M1 |
Obtain correct derivative in any form in $t$ for $y$ | A1 |
Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ | M1 |
Obtain given answer correctly | A1 | [4]
**(ii)** State $t = 0$ | M1 |
State that $\frac{dy}{dx} = 0$ and make correct conclusion | A1 | [2]
**(iii)** Substitute $t = -2$ into equation for $x$ or $y$ | M1 |
Obtain $(e^{-6}, 4e^{-3}+3)$ | A1 | [2]7 The parametric equations of a curve are
$$x = \mathrm { e } ^ { 3 t } , \quad y = t ^ { 2 } \mathrm { e } ^ { t } + 3$$
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { t ( t + 2 ) } { 3 \mathrm { e } ^ { 2 t } }$.\\
(ii) Show that the tangent to the curve at the point $( 1,3 )$ is parallel to the $x$-axis.\\
(iii) Find the exact coordinates of the other point on the curve at which the tangent is parallel to the $x$-axis.
\hfill \mbox{\textit{CAIE P2 2011 Q7 [8]}}