Argand functions

Question 1

modulus of complex number

Answer 1

distance from origin
root a2 + b2

Question 2

modulus after operations

Answer 2

mod ab = mod a x mod b
mod aa* = mod a2
mod a/b = mod a / mod b

Question 3

argument

Answer 3

angle of a complex number from right axis, up to positive π on top half or negative π on bottom half

Question 4

argument after operations

Answer 4

arg ab = arg a + arg b
arg 2a = 2 arg a
arg a/b = arg a - arg b
if this leads to an argument above π then add or subtract 2π to fix

Question 5

modulus argument form

Answer 5

z = mod(cos arg + i sin arg)

Question 6

x and / in mod arg form

Answer 6

operate mod and arg separately then sub in new values

Question 7

loci for mod (z - w) ? r

Answer 7

circle with centre w and radius r
if more than r outside circle, if less then inside, if equal then circle.
cartesian equation accordingly

Question 8

loci for arg (z-w) ? r

Answer 8

start at w then draw a line with argument angle
if = then the line, if more than shade clockwise, if less then shade anticlockwise

Question 9

loci for mod z-w = mod z-x

Answer 9

perpendicular bisector of w and x

Question 10

what if you draw a dot or line to construct a loci but isnt in the actual loci

Answer 10

dotted line or small circle rather than dot

Question 11

minimising the modulus of a line

Answer 11

perpendicular bisector from origin

Question 12

min/max modulus of a circle

Answer 12

modulus of centre of circle +- radius

Question 13

min/max arg of a circle

Answer 13

draw a tangent to top/ bottom of circle
make a triangle with origin, centre, tangent line then work out angles (each triangle is congruent so one is enough)