| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | December |
| Marks | 7 |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Standard +0.3 This is a straightforward calculus problem requiring standard techniques: finding a normal line (differentiate, find perpendicular gradient), solving a quadratic-linear system, and computing area between curves via integration. All steps are routine A-level methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
The diagram below shows part of a curve C with equation $y = 1 + 3x - \frac{1}{2}x^2$.
\includegraphics{figure_7}
\begin{enumerate}[label=(\roman*)]
\item The curve crosses the $y$ axis at the point A. The straight line L is normal to the curve at A and meets the curve again at B. Find the equation of L and the $x$ coordinate of the point B. [4]
\item The region R is bounded by the curve C and the line L. Find the exact area of R. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q7 [7]}}