| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Two circles intersection or tangency |
| Difficulty | Standard +0.3 Part (a) is routine completion of the square to find centre and radius from general circle equation. Part (b) requires understanding that two circles intersect at two distinct points when |r₁ - r₂| < d < r₁ + r₂, involving distance calculation and inequality manipulation. This is a standard A-level circle geometry question with straightforward application of known results, slightly above average due to the set notation requirement and inequality reasoning in part (b). |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Centre is \((3, -7)\) | B1 | Allow as coordinate pair, written separately, or column vector; condone missing brackets; do not allow coordinates wrong way round |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x-3)^2 + (y+7)^2 = 49 + 9 - 33 \Rightarrow r^2 = 25\) | M1 | Requires attempt at \((x\pm3)^2+(y\pm7)^2 - 3^2 - 7^2 + 33 = 0\) with at least one of \(a=3\) or \(b=7\) |
| \(r = 5\) | A1 | Do not allow \(\pm5\) or \(\sqrt{25}\); may be scored following \((x\pm3)^2+(y\pm7)^2=25\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Distance between centres \(= \sqrt{(3+6)^2+(-7+8)^2} = \sqrt{82}\) | M1 A1ft | Uses Pythagoras on centre from (a) and given centre \((-6,-8)\); condone one sign slip if intention clear; allow awrt 9.06 for \(\sqrt{82}\) |
| \(\sqrt{82} - 5\) or \(\sqrt{82} + 5\) | dM1 | Correct strategy for one limit; adds or subtracts their 5 to distance between centres |
| \(\sqrt{82}-5\) and \(\sqrt{82}+5\) | A1 | Both values |
| \(\{k : \sqrt{82}-5 < k\} \cap \{k : k < \sqrt{82}+5\}\) or \(\{k : \sqrt{82}-5 < k < \sqrt{82}+5\}\) | A1 | Correct set notation |
## Question 14:
### Part (a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Centre is $(3, -7)$ | B1 | Allow as coordinate pair, written separately, or column vector; condone missing brackets; do **not** allow coordinates wrong way round |
---
### Part (a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x-3)^2 + (y+7)^2 = 49 + 9 - 33 \Rightarrow r^2 = 25$ | M1 | Requires attempt at $(x\pm3)^2+(y\pm7)^2 - 3^2 - 7^2 + 33 = 0$ with at least one of $a=3$ or $b=7$ |
| $r = 5$ | A1 | Do **not** allow $\pm5$ or $\sqrt{25}$; may be scored following $(x\pm3)^2+(y\pm7)^2=25$ |
---
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Distance between centres $= \sqrt{(3+6)^2+(-7+8)^2} = \sqrt{82}$ | M1 A1ft | Uses Pythagoras on centre from (a) and given centre $(-6,-8)$; condone one sign slip if intention clear; allow awrt 9.06 for $\sqrt{82}$ |
| $\sqrt{82} - 5$ **or** $\sqrt{82} + 5$ | dM1 | Correct strategy for one limit; adds or subtracts their 5 to distance between centres |
| $\sqrt{82}-5$ **and** $\sqrt{82}+5$ | A1 | Both values |
| $\{k : \sqrt{82}-5 < k\} \cap \{k : k < \sqrt{82}+5\}$ or $\{k : \sqrt{82}-5 < k < \sqrt{82}+5\}$ | A1 | Correct set notation |\begin{enumerate}
\item The circle $C _ { 1 }$ has equation
\end{enumerate}
$$x ^ { 2 } + y ^ { 2 } - 6 x + 14 y + 33 = 0$$
(a) Find\\
(i) the coordinates of the centre of $C _ { 1 }$\\
(ii) the radius of $C _ { 1 }$
A different circle $C _ { 2 }$
\begin{itemize}
\item has centre with coordinates (-6, -8)
\item has radius $k$, where $k$ is a constant
\end{itemize}
Given that $C _ { 1 }$ and $C _ { 2 }$ intersect at 2 distinct points,\\
(b) find the range of values of $k$, writing your answer in set notation.
\hfill \mbox{\textit{Edexcel Paper 2 2024 Q14 [8]}}