| 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 C1 differentiation question requiring routine techniques: substituting to verify a point lies on a curve, finding dy/dx and evaluating at a point for the tangent equation, then solving dy/dx = m to find another point with parallel tangent. All steps are standard textbook exercises with no problem-solving insight required, making it easier than average but not trivial due to the cubic differentiation and multi-part structure. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
The curve $C$ has equation $y = \frac{1}{3}x^3 - 4x^2 + 8x + 3$.
The point $P$ has coordinates $(3, 0)$.
\begin{enumerate}[label=(\alph*)]
\item Show that $P$ lies on $C$. [1]
\item Find the equation of the tangent to $C$ at $P$, giving your answer in the form $y = mx + c$, where $m$ and $c$ are constants. [5]
\end{enumerate}
Another point $Q$ also lies on $C$. The tangent to $C$ at $Q$ is parallel to the tangent to $C$ at $P$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the coordinates of $Q$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q10 [11]}}