| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Moderate -0.8 This is a straightforward C1 differentiation question requiring standard techniques: differentiating powers of x (including negative powers), substituting a point to verify it lies on the curve, finding the gradient and normal equation using the negative reciprocal, and calculating distance using Pythagoras. All steps are routine textbook exercises with no problem-solving insight required, making it easier than average but not trivial due to the multi-part nature and 11 total marks. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
The curve $C$ has equation $y = 4x + 3x^{-1} - 2x^2$, $x > 0$.
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac{dy}{dx}$. [3]
\item Show that the point $P(4, 8)$ lies on $C$. [1]
\item Show that an equation of the normal to $C$ at the point $P$ is
$$3y - x + 20.$$ [4]
\end{enumerate}
The normal to $C$ at $P$ cuts the $x$-axis at the point $Q$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the length $PQ$, giving your answer in a simplified surd form. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q8 [11]}}