| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find normal line equation at given point |
| Difficulty | Moderate -0.8 This is a straightforward C1 differentiation question requiring only routine application of power rule, substitution to find gradient, and using perpendicular gradient formula for the normal. All steps are standard textbook exercises with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
2 A curve has equation $y = x ^ { 5 } - 6 x ^ { 3 } - 3 x + 25$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item The point $P$ on the curve has coordinates $( 2,3 )$.
\begin{enumerate}[label=(\roman*)]
\item Show that the gradient of the curve at $P$ is 5 .
\item Hence find an equation of the normal to the curve at $P$, expressing your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.
\end{enumerate}\item Determine whether $y$ is increasing or decreasing when $x = 1$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2005 Q2 [10]}}