C2 June 2011 Q4

Notes

Note: Please mark parts (a) and (b) together. Answers only in (a) and/or (b) get full marks.
Note in part (a) the marks are now M1A1 and not B1B1 as on ePEN.

M1: for \((\pm 2, \pm 1)\). Otherwise, M1 for an attempt to complete the square eg. \(\underline{(x \pm 2)^2 \pm \alpha}\), \(\alpha \ne 0\) or \(\underline{\underline{(y \pm 1)^2 \pm \beta}}\), \(\beta \ne 0\). M1A1: Correct answer of \((-2, 1)\) stated from any working gets M1A1.

Alternative: M1 in part (a): For comparing with \(x^2 + y^2 + 2gx + 2fy + c = 0\) to write down centre \((-g, -f)\) directly. Condone sign errors for this M mark.

Notes

M1: to find the radius using 11, “1” and “4”, ie. \(r = \sqrt{11 \pm \text{“}1\text{”} \pm \text{“}4\text{”}}\). By applying this method candidates will usually achieve \(\sqrt{16}, \sqrt{6}, \sqrt{8}\) or \(\sqrt{14}\) and not 16, 6, 8 or 14.

Note: \((x + 2)^2 + (y - 1)^2 = -11 - 5 = -16 \Rightarrow r = \sqrt{16} = 4\) should be awarded M0A0.

Alternative: M1 in part (b): For using \(r = \sqrt{g^2 + f^2 - c}\). Condone sign errors for this method mark.

\((x + 2)^2 + (y - 1)^2 = 16 \Rightarrow r = 8\) scores M0A0, but \(r = \sqrt{16} = 8\) scores M1A1 isw.

Notes

1^st M1: Putting \(x = 0\) in either \(x^2 + y^2 + 4x - 2y - 11 = 0\) or their circle equation usually given in part (a) or part (b).    1^st A1 for a correct equation in \(y\) in any form which can be implied by later working.

2^nd M1: See rules for using the formula. Or completing the square on a 3TQ to give \(y = a \pm \sqrt{b}\), where \(\sqrt{b}\) is a surd, \(b \ne\) their 11 and \(b \gt 0\). This mark should not be given for an attempt to factorise.

2^nd A1: Need exact pair in simplified surd form of \(\{y =\}\ 1 \pm 2\sqrt{3}\). This mark is also cso.
Do not need to see \((0, 1 + 2\sqrt{3})\) and \((0, 1 - 2\sqrt{3})\). Allow 2^nd A1 for bod \((1 + 2\sqrt{3}, 0)\) and \((1 - 2\sqrt{3}, 0)\).

Any incorrect working in (c) gets penalised the final accuracy mark. So, beware: incorrect \((x - 2)^2 + (y - 1)^2 = 16\) leading to \(y^2 - 2y - 11 = 0\) and then \(y = 1 \pm 2\sqrt{3}\) scores M1A1M1A0.

Special Case for setting \(y = 0\): Award SC: M0A0M1A0 for an attempt at applying the formula \(x = \dfrac{-4 \pm \sqrt{(-4)^2 - 4(1)(-11)}}{2(1)}\ \left\{= \dfrac{-4 \pm \sqrt{60}}{2} = -2 \pm \sqrt{15}\right\}\)
Award SC: M0A0M1A0 for completing the square to their equation in \(x\) which will usually be \(x^2 + 4x - 11 = 0\) to give \(a \pm \sqrt{b}\), where \(\sqrt{b}\) is a surd, \(b \ne\) their 11 and \(b \gt 0\).

Special Case: For a candidate not using \(\pm\) but achieving one of the correct answers then award SC: M1A1 M1A0 for one of either \(y = 1 + 2\sqrt{3}\) or \(y = 1 - 2\sqrt{3}\) or \(y = 1 + \sqrt{12}\) or \(y = 1 - \sqrt{12}\).

Alternative (c) Way 2 (Appendix)

Special Case: Award Final SC: M1A1 M1A0 if candidate achieves any one of either \(y = 1 + 2\sqrt{3}\) or \(y = 1 - 2\sqrt{3}\) or \(y = 1 + \sqrt{12}\) or \(y = 1 - \sqrt{12}\).