| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2026 |
| Session | November |
| Marks | 12 |
| Topic | Tangents, normals and gradients |
| Type | Find normal line equation at given point |
| Difficulty | Moderate -0.3 This is a straightforward calculus question requiring differentiation of polynomial and reciprocal terms, verification of a point on a curve, and finding a normal line equation. All techniques are standard A-level procedures with no novel problem-solving required, though the multi-step nature and algebraic manipulation (especially handling the normal's negative reciprocal gradient) elevate it slightly above pure routine recall. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations |
The curve $C$ has equation
$$y = \frac{1}{2}x^3 - 9x^2 + \frac{8}{x} + 30, \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac{dy}{dx}$.
[4]
\item Show that the point $P(4, -8)$ lies on $C$.
[2]
\item Find an equation of the normal to $C$ at the point $P$, giving your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.
[6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2026 Q3 [12]}}