(a) Sketch, on the Argand diagram below, the locus of points satisfying the equation\[|z - 2\mathrm{i}| = 2\] [2 marks]
(b) Sketch, also on the Argand diagram above, the locus of points satisfying the equation\[\arg z = \frac{\pi}{3}\] [1 mark]
(c) For the complex number \(w\) find the maximum value of \(|w|\) such that\[|w - 2\mathrm{i}| \leqslant 2 \quad \text{and} \quad 0 \leqslant \arg w \leqslant \frac{\pi}{3}\] [3 marks]
Mark scheme (a)
Scheme
Marks
AO
Draws a circle with radius 2 or centre \(2\mathrm{i}\) Condone a freehand circle if intention is clear.
M1
1.1a
Draws a circle with radius 2 and centre \(2\mathrm{i}\)
and no other curves seen.
Condone a freehand circle if intention is clear.
A1
1.1b
(2)
Typical solution
Mark scheme (b)
Scheme
Marks
AO
Draws a half-line from \(O\) into the 1st quadrant at an angle of more than \(45^\circ\) to the real axis.
and
no other straight lines seen.
B1
1.1b
(1)
Typical solution
Mark scheme (c)
Scheme
Marks
AO
Selects a method to find the maximum value of \(|w|\) eg identifies a triangle with the diameter (or radius) as a side and the intersections of their loci as two of the vertices.
eg forms a suitable equation in \(\max|w|\) or \(x_{\max|w|}\) or \(y_{\max|w|}\)
