| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Find centre and radius from equation |
| Difficulty | Moderate -0.8 This is a standard completing-the-square exercise for circle equations, requiring routine algebraic manipulation and direct reading of geometric properties. Part (d) adds a mild transformation element, but overall this is below-average difficulty—more mechanical than the typical A-level question requiring problem-solving or integration of multiple concepts. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((x - 5)^2 + (y + 7)^2\) | M1 | one term correct |
| \((x - 5)^2 + (y + 7)^2 = 49\) | A1cao | both terms correct and added; must see 49 not just \(7^2\); 3 marks total |
| condone \((x - 5)^2 + (y - (-7))^2 = 49\) | ||
| (b)(i) (Centre is) \((5, -7)\) | B1 | correct or FT their \(a\) and \(b\) |
| (ii) Radius \(= 7\) | B1 | condone \(\sqrt{49}\) but not \(\pm 7\) or \(\pm\sqrt{49}\); correct or FT their \(\sqrt{k}\) provided \(k > 0\) |
| (c)(i) | M1 | freehand circle with centre in correct quadrant or FT from their (b)(i); must have both axes shown clearly |
| A1 | correct position cutting negative \(y\)-axis twice and touching \(x\)-axis at \(x = 5\); 5 must be marked on \(x\)-axis or centre clearly marked as \((5, -7)\); must have correct centre and radius in (b) | |
| 2 marks total | ||
| (ii) \(x = 5\) | B1 | |
| \(y = -14\) | B1 | 2 marks total |
| \((5, -14)\) | ||
| (d) Translation | E1 | and no other transformation |
| through \(\begin{bmatrix} 6 \\ * \end{bmatrix}\) | M1 | |
| \(\begin{bmatrix} 6 \\ -7 \end{bmatrix}\) | A1cso | both components correct for A1; may describe in words or use a column vector; 3 marks total |
**(a)** $(x - 5)^2 + (y + 7)^2$ | M1 | one term correct
$(x - 5)^2 + (y + 7)^2 = 49$ | A1cao | both terms correct and added; must see 49 not just $7^2$; 3 marks total
| | | condone $(x - 5)^2 + (y - (-7))^2 = 49$
**(b)(i)** (Centre is) $(5, -7)$ | B1 | correct or FT their $a$ and $b$
**(ii)** Radius $= 7$ | B1 | condone $\sqrt{49}$ but not $\pm 7$ or $\pm\sqrt{49}$; correct or FT their $\sqrt{k}$ provided $k > 0$
**(c)(i)** | M1 | freehand circle with centre in correct quadrant or FT from their (b)(i); must have both axes shown clearly
| | | A1 | correct position cutting negative $y$-axis twice and touching $x$-axis at $x = 5$; 5 must be marked on $x$-axis or centre clearly marked as $(5, -7)$; must have correct centre and radius in (b)
| | | 2 marks total
**(ii)** $x = 5$ | B1 |
$y = -14$ | B1 | 2 marks total
| | | $(5, -14)$
**(d)** Translation | E1 | and no other transformation
through $\begin{bmatrix} 6 \\ * \end{bmatrix}$ | M1 |
$\begin{bmatrix} 6 \\ -7 \end{bmatrix}$ | A1cso | both components correct for A1; may describe in words or use a column vector; 3 marks total
**Total: 12 marks**
---3 A circle $C$ has equation
$$x ^ { 2 } + y ^ { 2 } - 10 x + 14 y + 25 = 0$$
\begin{enumerate}[label=(\alph*)]
\item Write the equation of $C$ in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
where $a , b$ and $k$ are integers.
\item Hence, for the circle $C$, write down:
\begin{enumerate}[label=(\roman*)]
\item the coordinates of its centre;
\item its radius.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Sketch the circle $C$.
\item Write down the coordinates of the point on $C$ that is furthest away from the $x$-axis.
\end{enumerate}\item Given that $k$ has the same value as in part (a), describe geometrically the transformation which maps the circle with equation $( x + 1 ) ^ { 2 } + y ^ { 2 } = k$ onto the circle $C$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2013 Q3 [12]}}