| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Circle from diameter endpoints |
| Difficulty | Moderate -0.3 Part (i) involves routine completion of the square to find the centre and applying the condition r²>0 for k. Part (ii) requires knowing that if QR is a diameter, angle QPR=90°, then using perpendicular gradients or the distance formula—standard techniques but requires connecting the diameter property to the right-angle theorem, making it slightly easier than average overall. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.03f Circle properties: angles, chords, tangents |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempts \((x\pm5)^2+(y\pm6)^2\ldots=0\) | M1 | Completing the square on both \(x\) and \(y\), or states centre as \((\pm5,\pm6)\) |
| Centre \((-5, 6)\) | A1 | Allow written as separate coordinates \(x=-5,\ y=6\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sets \(k+(\pm5)^2+(\pm6)^2 > 0\) | M1 | Follow through on their \((-5,6)\); must have an inequality |
| \(k > -61\) | A1 | Allow \(k\ldots{-61}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Centre \(\left(\frac{-2+8}{2}, \frac{10+(-14)}{2}\right)\); radius \(\frac{1}{2}\sqrt{(-2-8)^2+(10-(-14))^2}\) | M1 | Attempts centre and radius (or radius squared) of \(C_2\) using correct method; diameter only scores M0 |
| Centre \((3,-2)\); radius \(= 13\) | A1 | |
| \(\Rightarrow C_2: (x-3)^2+(y+2)^2=13^2 \Rightarrow (p-3)^2+4=169\) | M1 | Uses centre and radius in correct method to find \(p\); allow if centre/radius from part (i) used |
| \(p = 3+\sqrt{165}\) only | A1 | \(3+\sqrt{165}\) ONLY |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Uses \(PQ^2+PR^2=RQ^2\) or \(\text{grad } PQ \times \text{grad } PR = -1\); e.g. \((p+2)^2+10^2+(p-8)^2+14^2=(-2-8)^2+(10+14)^2\) or \(\frac{-10}{p+2}\times\frac{14}{p-8}=-1\) | M1, A1 | Correct attempt at lengths or gradients, condone slips |
| Correct method to set up and solve 3TQ; \(p^2-6p-156=0\) | dM1 | |
| \(p = 3+\sqrt{165}\) only | A1 |
## Question 6(i):
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempts $(x\pm5)^2+(y\pm6)^2\ldots=0$ | M1 | Completing the square on both $x$ and $y$, or states centre as $(\pm5,\pm6)$ |
| Centre $(-5, 6)$ | A1 | Allow written as separate coordinates $x=-5,\ y=6$ |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sets $k+(\pm5)^2+(\pm6)^2 > 0$ | M1 | Follow through on their $(-5,6)$; must have an inequality |
| $k > -61$ | A1 | Allow $k\ldots{-61}$ |
### Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Centre $\left(\frac{-2+8}{2}, \frac{10+(-14)}{2}\right)$; radius $\frac{1}{2}\sqrt{(-2-8)^2+(10-(-14))^2}$ | M1 | Attempts centre and radius (or radius squared) of $C_2$ using correct method; diameter only scores M0 |
| Centre $(3,-2)$; radius $= 13$ | A1 | |
| $\Rightarrow C_2: (x-3)^2+(y+2)^2=13^2 \Rightarrow (p-3)^2+4=169$ | M1 | Uses centre and radius in correct method to find $p$; allow if centre/radius from part (i) used |
| $p = 3+\sqrt{165}$ only | A1 | $3+\sqrt{165}$ ONLY |
### Alt 6(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Uses $PQ^2+PR^2=RQ^2$ or $\text{grad } PQ \times \text{grad } PR = -1$; e.g. $(p+2)^2+10^2+(p-8)^2+14^2=(-2-8)^2+(10+14)^2$ or $\frac{-10}{p+2}\times\frac{14}{p-8}=-1$ | M1, A1 | Correct attempt at lengths or gradients, condone slips |
| Correct method to set up and solve 3TQ; $p^2-6p-156=0$ | dM1 | |
| $p = 3+\sqrt{165}$ only | A1 | |6. (i) The circle $C _ { 1 }$ has equation
$$x ^ { 2 } + y ^ { 2 } + 10 x - 12 y = k \quad \text { where } k \text { is a constant }$$
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the centre of $C _ { 1 }$
\item State the possible range in values for $k$.\\
(ii) The point $P ( p , 0 )$, the point $Q ( - 2,10 )$ and the point $R ( 8 , - 14 )$ lie on a different circle, $C _ { 2 }$
Given that
\begin{itemize}
\item $p$ is a positive constant
\item $Q R$ is a diameter of $C _ { 2 }$\\
find the exact value of $p$.\\
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2021 Q6 [8]}}