| Exam Board | AQA |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Point position relative to circle |
| Difficulty | Standard +0.3 Part (a) is routine completing the square for a circle equation. Part (b)(i) is straightforward substitution. Part (b)(ii) requires geometric reasoning to identify the critical vertices and verify they lie inside the circle, involving some coordinate geometry with an equilateral triangle, but the approach is relatively standard for AS level with clear guidance from the diagram and previous part. |
| 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 |
|---|---|---|
| 11(a) | Obtains correct centre. | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| AWRT 5.6 | 1.1b | B1 |
| Subtotal | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 11(b)(i) | Shows that origin is inside circle | 2.1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 11(b)(ii) | Selects an appropriate method |
| Answer | Marks | Guidance |
|---|---|---|
| y coordinate. | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| NB Correct vales are y = 2 2 | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| inside circle. | 2.2a | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| reference to symmetry. | 2.1 | R1 |
| Subtotal | 4 | |
| Question 11 Total | 7 | |
| Q | Marking instructions | AO |
Question 11:
--- 11(a) ---
11(a) | Obtains correct centre. | 1.1b | B1 | x2 – 10x + 25 + y2 = 31
(x – 5)2 + y2 = 31
Centre is (5, 0) and radius √31
Obtains correct radius.
AWRT 5.6 | 1.1b | B1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 11(b)(i) ---
11(b)(i) | Shows that origin is inside circle | 2.1 | R1 | Distance from centre to origin is 5
and 5 < √31 so vertex at origin is
inside circle
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 11(b)(ii) ---
11(b)(ii) | Selects an appropriate method
to find the y coordinate of at
least one other vertex.
Condone one slip in the
formation of the equation for the
y coordinate. | 3.1a | M1 | y
tan 30º = so y = 8tan 30º
8
8
The other vertices are (8, ± )
3
Distance from centre to other
6 4 9 1
vertices is √( 32 + ) = √( )
3 3
< √31
Both vertices are inside circle
Complete triangle is inside circle
Uses distance formula for
distance from their centre to at
least one vertex.
or
Find at least one y value on the
circle when x = 8
NB Correct vales are y = 2 2 | 3.1a | M1
91
Compares √( ) with √31
3
AWRT 5.5 and 5.6
or
8
Compares with 2 2
3
AWRT 4.6 and 4.7
and
Deduces that one other vertex is
inside circle. | 2.2a | A1
Completes proof that triangle is
completely inside circle, either
by proof for the third vertex or by
reference to symmetry. | 2.1 | R1
Subtotal | 4
Question 11 Total | 7
Q | Marking instructions | AO | Marks | Typical solution\begin{enumerate}[label=(\alph*)]
\item A circle has equation
$$x^2 + y^2 - 10x - 6 = 0$$
Find the centre and the radius of the circle.
[2 marks]
\item An equilateral triangle has one vertex at the origin, and one side along the line $x = 8$, as shown in the diagram below.
\includegraphics{figure_11}
\begin{enumerate}[label=(\roman*)]
\item Show that the vertex at the origin lies inside the circle $x^2 + y^2 - 10x - 6 = 0$
[1 mark]
\item Prove that the triangle lies completely within the circle $x^2 + y^2 - 10x - 6 = 0$
[4 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 1 2023 Q11 [7]}}