Higher November 2022 Paper 6 Q18

18 A circle has equation \(x^2 + y^2 = 100\).

The sketch shows the circle and two points, A and B, which lie on the circumference of the circle.

(a) Write down the coordinates of point A. [1]

(b) Point B has \(x\)-coordinate 3.

Find the exact value of the \(y\)-coordinate of point B. [3]

Find the two possible pairs of coordinates for point C.
Give your answers in exact form.
You must show your working. [5]

Answer	Marks	Part marks and guidance
(10, 0)	1

Answer	Marks	Part marks and guidance
\(-\sqrt{91}\)	3	B2 for \(\sqrt{91}\) or −9.539[…] or −9.54 or B1 for 9.539[…] or 9.54 or M1 for \(9 + y^2 = 100\) oe	± answers treat as choice

Mark scheme (c)

Answer	Marks	Part marks and guidance
\((\sqrt{2},\ 7\sqrt{2})\) and \((-\sqrt{2},\ -7\sqrt{2})\) with correct working	5	B4 for \((\sqrt{2},\ 7\sqrt{2})\) with correct working or \((-\sqrt{2},\ -7\sqrt{2})\) with correct working or M2 for \(x^2 + (7x)^2 = 100\) oe A1 for [\(x = \pm\)] \(\sqrt{2}\) or M1 for 7\(x\) seen If 0 or 1 or 2 scored, instead award SC3 for answer \((\sqrt{2},\ 7\sqrt{2})\) and \((-\sqrt{2},\ -7\sqrt{2})\) with no working or insufficient working If 0 or M1 scored, instead award SC2 for answer \((\sqrt{2},\ 7\sqrt{2})\) or \((-\sqrt{2},\ -7\sqrt{2})\) or for \(\left(\frac{5\sqrt{2}}{2}, \frac{35\sqrt{2}}{2}\right)\) and \(\left(-\frac{5\sqrt{2}}{2}, -\frac{35\sqrt{2}}{2}\right)\) with no working or insufficient working If 0 scored, instead award SC1 for [\(x = \pm\)] \(\sqrt{2}\) or for \(\left(\frac{5\sqrt{2}}{2}, \frac{35\sqrt{2}}{2}\right)\) or \(\left(-\frac{5\sqrt{2}}{2}, -\frac{35\sqrt{2}}{2}\right)\) with no working	“Correct working” requires evidence of at least M2 Allow \(\sqrt{98}\) for \(7\sqrt{2}\) Condone for M1 if seen as \(x^2 + 7x^2 = 100\)