| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Show dy/dx equals given expression |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question with standard follow-up parts. Part (a)(i) requires basic implicit differentiation of a simple equation, (a)(ii) involves writing tangent equations and finding their intersection (routine coordinate geometry), and part (b) is direct substitution to verify parametric equations. All techniques are standard C4 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
4
\begin{enumerate}[label=(\alph*)]
\item A curve is defined by the equation $x ^ { 2 } - y ^ { 2 } = 8$.
\begin{enumerate}[label=(\roman*)]
\item Show that at any point $( p , q )$ on the curve, where $q \neq 0$, the gradient of the curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { q }$.\\
(2 marks)
\item Show that the tangents at the points $( p , q )$ and $( p , - q )$ intersect on the $x$-axis.\\
(4 marks)
\end{enumerate}\item Show that $x = t + \frac { 2 } { t } , y = t - \frac { 2 } { t }$ are parametric equations of the curve $x ^ { 2 } - y ^ { 2 } = 8$.\\
(2 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2013 Q4 [8]}}