| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent meets curve/axis — further geometry |
| Difficulty | Moderate -0.8 This is a straightforward multi-part differentiation question requiring only standard techniques: basic polynomial differentiation, solving a quadratic equation for gradient=2, finding a tangent equation using point-slope form, and calculating distance between axis intercepts using Pythagoras. All steps are routine C1 procedures with no problem-solving insight needed, making it easier than average but not trivial due to the multi-step nature and algebraic manipulation required. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
A curve $C$ has equation $y = x^3 - 5x^2 + 5x + 2$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac{dy}{dx}$ in terms of $x$. [2]
\end{enumerate}
The points $P$ and $Q$ lie on $C$. The gradient of $C$ at both $P$ and $Q$ is 2. The $x$-coordinate of $P$ is 3.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the $x$-coordinate of $Q$. [2]
\item Find an equation for the tangent to $C$ at $P$, giving your answer in the form $y = mx + c$, where $m$ and $c$ are constants. [3]
\end{enumerate}
This tangent intersects the coordinate axes at the points $R$ and $S$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the length of $RS$, giving your answer as a surd. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q11 [11]}}