| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Line-circle intersection points |
| Difficulty | Standard +0.3 Part (a) is routine: find radius using distance formula, then write circle equation. Part (b) requires understanding that a horizontal chord's perpendicular from center is vertical, then using the chord-perpendicular distance relationship (Pythagoras), but this is a standard textbook exercise with straightforward geometric setup and calculation. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03f Circle properties: angles, chords, tangents |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Attempts \((8-3)^2 + (5--7)^2 = ...\) | M1 | Attempts to find the radius or radius\(^2\). Must proceed to a value. Condone sign slip if attempting \(5--7\) seen as \(5-7\). May be implied by 13 or 169. |
| Writes \((x-3)^2 + (y-5)^2 = k\) | M1 | Writes equation of circle in the form \((x-3)^2 + (y-5)^2 = k\), where \(k > 0\). e.g. \(x^2 + y^2 - 6x - 10y + c = 0\) where \(c < 0\). Invisible brackets may be implied by further work. Condone \((x-3)^2+(y-5)^2 = r^2\) where no attempt has been made to find the radius or radius\(^2\). |
| \((x-3)^2 + (y-5)^2 = 169\) | A1 | o.e. e.g. \(x^2 + y^2 - 6x - 10y - 135 = 0\) isw once correct unsimplified equation found. Condone \((x-3)^2+(y-5)^2 = 13^2\). A correct equation scores M1M1A1 but must be seen in (a). |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Attempts \(d^2 + \left(2\sqrt{22}\right)^2 = 169 \Rightarrow d = ...\) | M1 | Attempts Pythagoras' theorem with "13" as hypotenuse and \(2\sqrt{22}\) to find \(d\). Not just \(d^2\). |
| States or uses \((y =)\ 5 + d\) | dM1 | States or uses equation for \(MN\), \((y=)\ 5+d\); condone stating/using \((y=)\ 5-d\). May be implied by their value for \(y\). Dependent on previous M mark. |
| \(y = 14\) | A1 | \(y = 14\) only (must be exactly 14; e.g. \(13.99... = 14\) is A0). Note \((3+2\sqrt{22},\ 14)\) as final answer is A0. Further work leading to a different equation of a line is A0. |
## Question 3(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Attempts $(8-3)^2 + (5--7)^2 = ...$ | M1 | Attempts to find the radius or radius$^2$. Must proceed to a value. Condone sign slip if attempting $5--7$ seen as $5-7$. May be implied by 13 or 169. |
| Writes $(x-3)^2 + (y-5)^2 = k$ | M1 | Writes equation of circle in the form $(x-3)^2 + (y-5)^2 = k$, where $k > 0$. e.g. $x^2 + y^2 - 6x - 10y + c = 0$ where $c < 0$. Invisible brackets may be implied by further work. Condone $(x-3)^2+(y-5)^2 = r^2$ where no attempt has been made to find the radius or radius$^2$. |
| $(x-3)^2 + (y-5)^2 = 169$ | A1 | o.e. e.g. $x^2 + y^2 - 6x - 10y - 135 = 0$ isw once correct unsimplified equation found. Condone $(x-3)^2+(y-5)^2 = 13^2$. A correct equation scores M1M1A1 but must be seen in (a). |
## Question 3(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Attempts $d^2 + \left(2\sqrt{22}\right)^2 = 169 \Rightarrow d = ...$ | M1 | Attempts Pythagoras' theorem with "13" as hypotenuse and $2\sqrt{22}$ to find $d$. Not just $d^2$. |
| States or uses $(y =)\ 5 + d$ | dM1 | States or uses equation for $MN$, $(y=)\ 5+d$; condone stating/using $(y=)\ 5-d$. May be implied by their value for $y$. Dependent on previous M mark. |
| $y = 14$ | A1 | $y = 14$ only (must be exactly 14; e.g. $13.99... = 14$ is A0). Note $(3+2\sqrt{22},\ 14)$ as final answer is A0. Further work leading to a different equation of a line is A0. |\begin{enumerate}
\item The circle $C$
\end{enumerate}
\begin{itemize}
\item has centre $A ( 3,5 )$
\item passes through the point $B ( 8 , - 7 )$\\
(a) Find an equation for $C$.
\end{itemize}
The points $M$ and $N$ lie on $C$ such that $M N$ is a chord of $C$.\\
Given that $M N$
\begin{itemize}
\item lies above the $x$-axis
\item is parallel to the $x$-axis
\item has length $4 \sqrt { 22 }$\\
(b) find an equation for the line passing through points $M$ and $N$.
\end{itemize}
\hfill \mbox{\textit{Edexcel P2 2024 Q3 [6]}}