| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find stationary points of parametric curve |
| Difficulty | Standard +0.3 This is a standard parametric differentiation question requiring dy/dx = (dy/dt)/(dx/dt), solving dy/dx = 0 for stationary points, and finding a normal gradient. The exponential functions add mild algebraic complexity, but the techniques are routine A-level procedures with no novel problem-solving required. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identify \(t = 0\) | B1 | |
| Substitute \(t = 0\) in expression for first derivative and find negative reciprocal | M1 | |
| Obtain \(-\frac{8}{3}\) or equivalent | A1 | |
| Total: 3 |
## Question 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Identify $t = 0$ | **B1** | |
| Substitute $t = 0$ in expression for first derivative and find negative reciprocal | **M1** | |
| Obtain $-\frac{8}{3}$ or equivalent | **A1** | |
| **Total: 3** | | |
---6 The parametric equations of a curve are
$$x = 2 \mathrm { e } ^ { 2 t } + 4 \mathrm { e } ^ { t } , \quad y = 5 t \mathrm { e } ^ { 2 t }$$
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$ and hence find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places.\\
(ii) Find the gradient of the normal to the curve at the point where the curve crosses the $x$-axis.\\
\hfill \mbox{\textit{CAIE P2 2017 Q6 [9]}}