| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | October |
| Marks | 9 |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Standard +0.8 This question requires finding where a normal line intersects a parabola again (involving differentiation, equation of normal, and solving a cubic), then using coordinate geometry to find tan α via angle between two lines from origin. Multi-step with some algebraic manipulation, but follows standard techniques without requiring deep insight. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07m Tangents and normals: gradient and equations |
The diagram shows part of the graph of $y = x^2$. The normal to the curve at the point $A(1, 1)$ meets the curve again at $B$. Angle $AOB$ is denoted by $\alpha$.
\includegraphics{figure_7}
\begin{enumerate}[label=(\alph*)]
\item Determine the coordinates of $B$.
[6]
\item Hence determine the exact value of $\tan\alpha$.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q7 [9]}}