| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Standard +0.3 This is a slightly above-routine question requiring finding a normal line equation, solving a quadratic intersection, and applying coordinate geometry for an angle. The steps are standard (differentiate to find gradient, write normal equation, substitute into y=x², find tan α using coordinate geometry) but the multi-step nature and the need to work with the normal (rather than tangent) and find an angle adds modest complexity beyond basic calculus exercises. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07o Increasing/decreasing: functions using sign of dy/dx1.07p Points of inflection: using second derivative |
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_4}
\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{OCR H240/02 2023 Q4 [9]}}