| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Show gradient expression then find coordinates |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring standard techniques: finding dy/dx using the chain rule, solving dy/dx = 0 for stationary points, and finding where y = 0. While it involves multiple steps and the product rule for y, all techniques are routine applications with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
A curve has parametric equations
$$x = \ln(t + 1), \quad y = t^2 \ln t.$$
\begin{enumerate}[label=(\roman*)]
\item Find an expression for $\frac{dy}{dx}$ in terms of $t$. [5]
\item Find the exact value of $t$ at the stationary point. [2]
\item Find the gradient of the curve at the point where it crosses the $x$-axis. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q6 [9]}}