| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2014 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Tangent/normal meets curve again |
| Difficulty | Challenging +1.2 This is a multi-part parametric question requiring standard techniques: finding dy/dx using the chain rule, finding a normal equation (routine calculation), then solving simultaneously with the original curve. Part (c) requires substituting the line equation back into parametric equations and solving a trigonometric equation, which involves some algebraic manipulation but follows established methods. More demanding than a basic parametric question due to the intersection problem, but still within standard C4 territory without requiring novel insight. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
7. The curve $C$ has parametric equations
$$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$
where $t$ is a parameter.
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
The point $P$ lies on $C$ where $t = \frac { 2 \pi } { 3 }$\\
The line $l$ is a normal to $C$ at $P$.
\item Show that an equation for $l$ is
$$2 x - 2 \sqrt { 3 } y - 1 = 0$$
The line $l$ intersects the curve $C$ again at the point $Q$.
\item Find the exact coordinates of $Q$.
You must show clearly how you obtained your answers.\\
\includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-23_106_63_2595_1882}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2014 Q7 [13]}}