| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 21 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard C2 techniques: substitution to verify a result, index notation, differentiation of power functions, second derivative test, and integration. All parts are routine applications of learned procedures with no problem-solving insight required, making it easier than average. |
| Spec | 1.07e Second derivative: as rate of change of gradient1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08b Integrate x^n: where n != -1 and sums |
7 At the point $( x , y )$, where $x > 0$, the gradient of a curve is given by
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Verify that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 4$.\\
(1 mark)
\item Write $\frac { 16 } { x ^ { 2 } }$ in the form $16 x ^ { k }$, where $k$ is an integer.
\item Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
\item Hence determine whether the point where $x = 4$ is a maximum or a minimum, giving a reason for your answer.
\end{enumerate}\item The point $P ( 1,8 )$ lies on the curve.
\begin{enumerate}[label=(\roman*)]
\item Show that the gradient of the curve at the point $P$ is 12 .
\item Find an equation of the normal to the curve at $P$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find $\int \left( 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7 \right) \mathrm { d } x$.
\item Hence find the equation of the curve which passes through the point $P ( 1,8 )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2006 Q7 [21]}}