| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Rectangular hyperbola normal equation |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question on rectangular hyperbolas requiring implicit differentiation, normal equation derivation, coordinate geometry, and solving a cubic equation. While the calculus in part (a) is standard, parts (b) and (c) require careful algebraic manipulation and the final cubic equation adds complexity. It's above average difficulty but uses well-established techniques for FM students. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives |
| VIUV SIHI NI JIIIM ION OC | VI4V SIHI NI JINM IONOO | VJYV SIHI NI GLIYM LON OO |
10. The rectangular hyperbola $H$ has equation $x y = 144$. The point $P$, on $H$, has coordinates $\left( 12 p , \frac { 12 } { p } \right)$, where $p$ is a non-zero constant.
\begin{enumerate}[label=(\alph*)]
\item Show, by using calculus, that the normal to $H$ at the point $P$ has equation
$$y = p ^ { 2 } x + \frac { 12 } { p } - 12 p ^ { 3 }$$
Given that the normal through $P$ crosses the positive $x$-axis at the point $Q$ and the negative $y$-axis at the point $R$,
\item find the coordinates of $Q$ and the coordinates of $R$, giving your answers in terms of $p$.
\item Given also that the area of triangle $O Q R$ is 512 , find the possible values of $p$.\\
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VIUV SIHI NI JIIIM ION OC & VI4V SIHI NI JINM IONOO & VJYV SIHI NI GLIYM LON OO \\
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\hfill \mbox{\textit{Edexcel F1 2018 Q10 [13]}}