| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Point position relative to circle |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing completing the square for circles (a standard technique) and basic distance formula application. Part (c)(ii) requires minimal reasoning—just comparing two numbers. All steps are routine with no problem-solving insight required, making it easier than average but not trivial since it involves multiple coordinated steps. |
| 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 |
|---|---|---|
| \((x-4)^2 + (y+3)^2\) | B2 | B1 for one term correct |
| \((11 + 16 + 9 = 36)\) RHS \(= 6^2\) | B1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Centre \((4, -3)\) | B1√ | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Radius \(= 6\) | B1√ | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(CO^2 = (-4)^2 + 3^2\) | M1 | — |
| \(CO = 5\) | A1√ | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Considering CO and radius | M1 | — |
| \(CO < r \Rightarrow O\) is inside the circle | A1√ | 2 |
## 5(a)
$(x-4)^2 + (y+3)^2$ | B2 | B1 for one term correct
$(11 + 16 + 9 = 36)$ RHS $= 6^2$ | B1 | 3 | Condone 36
## 5(b)(i)
Centre $(4, -3)$ | B1√ | 1 | Ft their a and b from part (a)
## 5(b)(ii)
Radius $= 6$ | B1√ | 1 | Ft their r from part (a)
## 5(c)(i)
$CO^2 = (-4)^2 + 3^2$ | M1 | —
$CO = 5$ | A1√ | 2 | Full marks for answer only; Accept $\pm$ or $-$ with numbers but must add
## 5(c)(ii)
Considering CO and radius | M1 | —
$CO < r \Rightarrow O$ is **inside** the circle | A1√ | 2 | Ft outside circle when 'their $CO' > r$ or on the circle when 'their $CO' = r$; **SC B1√** if no explanation given
---5 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 8 x + 6 y = 11$.
\begin{enumerate}[label=(\alph*)]
\item By completing the square, express this equation in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
\item Write down:
\begin{enumerate}[label=(\roman*)]
\item the coordinates of $C$;
\item the radius of the circle.
\end{enumerate}\item The point $O$ has coordinates $( 0,0 )$.
\begin{enumerate}[label=(\roman*)]
\item Find the length of CO .
\item Hence determine whether the point $O$ lies inside or outside the circle, giving a reason for your answer.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2006 Q5 [9]}}