| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find normal equation at parameter |
| Difficulty | Standard +0.2 This is a straightforward C4 parametric differentiation question requiring standard chain rule application (dy/dx = dy/dθ ÷ dx/dθ), followed by routine normal line calculation and a standard integral using the double angle identity. All techniques are textbook exercises 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 differentiation1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
4
\begin{enumerate}[label=(\alph*)]
\item A curve is defined by the parametric equations $x = 3 \cos 2 \theta , y = 2 \cos \theta$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { k \cos \theta }$, where $k$ is an integer.
\item Find an equation of the normal to the curve at the point where $\theta = \frac { \pi } { 3 }$.
\end{enumerate}\item Find the exact value of $\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \sin ^ { 2 } x \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2011 Q4 [13]}}