| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find stationary points coordinates |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing routine differentiation and integration techniques. Part (a) requires basic polynomial differentiation and solving a quadratic equation for stationary points. Part (b) involves standard polynomial integration and calculating a definite integral, then finding an area between a curve and line using a simple subtraction. All steps are mechanical applications of standard techniques with no problem-solving insight required, making it easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits1.08f Area between two curves: using integration |
5 The curve with equation $y = x ^ { 3 } - 10 x ^ { 2 } + 28 x$ is sketched below.\\
\includegraphics[max width=\textwidth, alt={}, center]{f2c95d73-d3fe-48f7-af07-84f12bb06727-3_483_899_402_568}
The curve crosses the $x$-axis at the origin $O$ and the point $A ( 3,21 )$ lies on the curve.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item Hence show that the curve has a stationary point when $x = 2$ and find the $x$-coordinate of the other stationary point.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find $\int \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x$.
\item Hence show that $\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x = 56 \frac { 1 } { 4 }$.
\item Hence determine the area of the shaded region bounded by the curve and the line $O A$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2006 Q5 [15]}}