| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward C2 calculus question involving standard differentiation of fractional powers, finding stationary points by setting dy/dx=0, and working with tangent lines. All techniques are routine for this level, though part (d) requires combining multiple steps. The fractional indices are standard C2 fare and no novel problem-solving is needed. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives |
The diagram shows part of a curve with a maximum point $M$.
\includegraphics{figure_5}
The equation of the curve is
$$y = 15x^{\frac{3}{2}} - x^{\frac{5}{2}}$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac{dy}{dx}$. [3]
\item Hence find the coordinates of the maximum point $M$. [4]
\item The point $P(1, 14)$ lies on the curve. Show that the equation of the tangent to the curve at $P$ is $y = 20x - 6$. [3]
\item The tangents to the curve at the points $P$ and $M$ intersect at the point $R$. Find the length of $RM$. [3]
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2009 Q5 [13]}}