| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Simultaneous with substitution elimination |
| Difficulty | Standard +0.8 This question requires multiple algebraic steps: rearranging C₁ to substitute into C₂, manipulating to reach the quartic equation, recognizing it as a quadratic in x², solving for x² values, then finding coordinates and calculating distance. The quartic substitution and exact distance calculation (likely involving surds) elevate this above routine simultaneous equations, but it follows a clear path once the substitution is made. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02q Use intersection points: of graphs to solve equations1.10f Distance between points: using position vectors |
\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
\section*{Solutions relying on calculator technology are not acceptable.}
The curve $C _ { 1 }$ has equation
$$x y = \frac { 15 } { 2 } - 5 x \quad x \neq 0$$
The curve $C _ { 2 }$ has equation
$$y = x ^ { 3 } - \frac { 7 } { 2 } x - 5$$
(a) Show that $C _ { 1 }$ and $C _ { 2 }$ meet when
$$2 x ^ { 4 } - 7 x ^ { 2 } - 15 = 0$$
Given that $C _ { 1 }$ and $C _ { 2 }$ meet at points $P$ and $Q$\\
(b) find, using algebra, the exact distance $P Q$
\hfill \mbox{\textit{Edexcel P1 2023 Q8 [7]}}