| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find normal equation at parameter |
| Difficulty | Moderate -0.3 This is a straightforward parametric differentiation question requiring the chain rule (dy/dx = (dy/dt)/(dx/dt)) and finding a normal equation using point-gradient form. Both parts are standard textbook exercises with routine algebraic manipulation and no conceptual challenges beyond basic parametric calculus techniques. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(\frac{dx}{dt} = \frac{1}{t + 1}\) | B1 | |
| State \(\frac{dy}{dt} = 2e^{2t} + 2\) | B1 | |
| Attempt expression for \(\frac{dy}{dx}\) | M1 | |
| Obtain \(\frac{dy}{dx} = (2e^{2t} + 2)(t + 1)\) or equivalent | A1 | [4] |
| (ii) Substitute \(t = 0\) and attempt gradient of normal | M1 | |
| Obtain \(-\frac{1}{4}\) following their expression for \(\frac{dy}{dx}\) | A1✓ | |
| Attempt to find equation of normal through point \((0, 1)\) | M1 | |
| Obtain \(x + 4y - 4 = 0\) | A1 | [4] |
**(i)** State $\frac{dx}{dt} = \frac{1}{t + 1}$ | B1 |
State $\frac{dy}{dt} = 2e^{2t} + 2$ | B1 |
Attempt expression for $\frac{dy}{dx}$ | M1 |
Obtain $\frac{dy}{dx} = (2e^{2t} + 2)(t + 1)$ or equivalent | A1 | [4]
**(ii)** Substitute $t = 0$ and attempt gradient of normal | M1 |
Obtain $-\frac{1}{4}$ following their expression for $\frac{dy}{dx}$ | A1✓ |
Attempt to find equation of normal through point $(0, 1)$ | M1 |
Obtain $x + 4y - 4 = 0$ | A1 | [4]
---5 The parametric equations of a curve are
$$x = \ln ( t + 1 ) , \quad y = \mathrm { e } ^ { 2 t } + 2 t$$
(i) Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.\\
(ii) Find the equation of the normal to the curve at the point for which $t = 0$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
\hfill \mbox{\textit{CAIE P2 2012 Q5 [8]}}