| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Range of parameter for intersection |
| Difficulty | Standard +0.3 Part (a) is routine completion of the square to find centre coordinates. Part (b) requires understanding that 'not cutting or touching the x-axis' means the discriminant condition when y=0, or equivalently that the distance from centre to x-axis exceeds the radius. This is a standard circle-line intersection problem with straightforward algebraic manipulation, slightly above average due to the parameter inequality 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 |
| \((x-3)^2 + (y+5)^2 \pm \ldots = \ldots\) | M1 | For sight of \((x\pm3)^2 \pm (y\pm5)^2 \pm\ldots=\ldots\) or one coordinate of centre from \((\pm3, \pm5)\) |
| Centre \((3, -5)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Deduces \(k=9\) is a critical point | B1ft | Allow from their \(("5")^2\); condone \(\frac{36}{4}\) |
| Recognises radius \(> 0\): \("9"+"25"-k > 0\) | M1 | \((x\pm3)^2+(y\pm5)^2 = ("3")^2+("5")^2 - k\) and radius² must be positive; condone \(\geq 0\) |
| \(9 < k < 34\) | A1 | Condone \(9 < k \leq 34\); allow as separate inequalities \(k>9\), \(k<34\); must not be combined as \(\{k:k>9\}\cup\{k:k<34\}\) |
## Question 6:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x-3)^2 + (y+5)^2 \pm \ldots = \ldots$ | M1 | For sight of $(x\pm3)^2 \pm (y\pm5)^2 \pm\ldots=\ldots$ or one coordinate of centre from $(\pm3, \pm5)$ |
| Centre $(3, -5)$ | A1 | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Deduces $k=9$ is a critical point | B1ft | Allow from their $("5")^2$; condone $\frac{36}{4}$ |
| Recognises radius $> 0$: $"9"+"25"-k > 0$ | M1 | $(x\pm3)^2+(y\pm5)^2 = ("3")^2+("5")^2 - k$ and radius² must be positive; condone $\geq 0$ |
| $9 < k < 34$ | A1 | Condone $9 < k \leq 34$; allow as separate inequalities $k>9$, $k<34$; must not be combined as $\{k:k>9\}\cup\{k:k<34\}$ |
---\begin{enumerate}
\item The circle $C$ has equation
\end{enumerate}
$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + k = 0$$
where $k$ is a constant.\\
(a) Find the coordinates of the centre of $C$.
Given that $C$ does not cut or touch the $x$-axis,\\
(b) find the range of possible values for $k$.
\hfill \mbox{\textit{Edexcel AS Paper 1 2023 Q6 [5]}}