| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find normal equation at parameter |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring the chain rule (dy/dx = dy/dt รท dx/dt), followed by finding a normal line equation. Both parts use standard techniques with no novel insight required, making it slightly easier than average for A-level. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use product rule to differentiate \(y\) | M1 | |
| Obtain \(\frac{dy}{dt} = 4e^t + 4te^t\) or equivalent | A1 | |
| Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) | M1 | |
| Obtain given answer \(\frac{dy}{dx} = \frac{2(t+1)}{e^t}\) correctly | A1 | AG |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(t=0\) to evaluate derivative and find coordinates of point | B1 | |
| Obtain \(\frac{dy}{dx} = 2\) and coordinates \((1, 0)\) | B1 | |
| Form equation of normal at their point, using negative reciprocal of their \(\frac{dy}{dx}\) | M1 | |
| State correct equation of normal \(y = -\frac{1}{2}x + \frac{1}{2}\) or equivalent | A1 | |
| Total | 4 |
# Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule to differentiate $y$ | M1 | |
| Obtain $\frac{dy}{dt} = 4e^t + 4te^t$ or equivalent | A1 | |
| Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ | M1 | |
| Obtain given answer $\frac{dy}{dx} = \frac{2(t+1)}{e^t}$ correctly | A1 | AG |
| **Total** | **4** | |
# Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $t=0$ to evaluate derivative and find coordinates of point | B1 | |
| Obtain $\frac{dy}{dx} = 2$ and coordinates $(1, 0)$ | B1 | |
| Form equation of normal at their point, using negative reciprocal of their $\frac{dy}{dx}$ | M1 | |
| State correct equation of normal $y = -\frac{1}{2}x + \frac{1}{2}$ or equivalent | A1 | |
| **Total** | **4** | |
---6 The parametric equations of a curve are
$$x = \mathrm { e } ^ { 2 t } , \quad y = 4 t \mathrm { e } ^ { t } .$$
(a) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 ( t + 1 ) } { \mathrm { e } ^ { t } }$.\\
(b) Find the equation of the normal to the curve at the point where $t = 0$.\\
\hfill \mbox{\textit{CAIE P2 2020 Q6 [8]}}