Y1, C2 - Argand Diagrams

Question 1

What goes on the axis of an argand diagram

Answer 1

y = imaginary (Im)
x = real (Re)

Question 2

What is the modulus of a complex number

Answer 2

The distance from the origin on an argand diagrams

Question 3

What is the argument of a complex number

Answer 3

The angle the modulus makes with the positive real axis
The anti-clockwise rotation in radians

Question 4

What is the range for the principal argument

Answer 4

-pi < x < pi

Question 5

If you have an imaginary number in the negative y quadrants, how do you calculate the argument

Answer 5

pi - arctan(y / x)

Question 6

What is the modulus argument form of an imaginary number

Answer 6

z = r(cosθ + isinθ)
Where r = modulus
θ = argument

Question 7

When multiplying complex numbers, what happens to the modulus and argument of the result

Answer 7

The moduli multiply
The arguments add together

Question 8

When dividing complex numbers, what happens to the modulus and argument of the result

Answer 8

The moduli are divided
The arguments are subtracted

Question 9

What do you do if a complex number is in the form r(cosθ - isinθ)

Answer 9

Take the negative of the argument and turn the sin function to positive +
( rcos(θ) + isin(-θ) )

Question 10

What is cos-θ equal to

Answer 10

cosθ

Question 11

What is -sin-θ equal to

Answer 11

sinθ

Question 12

What does it mean if a function is even

Answer 12

f(x) = f(-x)
Symmetrical in y axis

Question 13

What does it mean if a function is odd

Answer 13

f(x) = -f(-x)
Rotational symmetry of order 2

Question 14

What should you do if your value for θ is outside of your principal value

Answer 14

Subtract or add 2pi to it until it falls within the range

Question 15

With loci, what does z denote

Answer 15

A general complex number

Question 16

What does lz - z1l = r mean?

Answer 16

Circle, radius r
Centre z1

Question 17

What does lz - z1l mean in relation to vectors

Answer 17

lz - z1l is the distance between z and z1

Question 18

What is the cartesian equation of the locus lz - 5 - 3il = 3

Answer 18

z = x + iy
l (x-5) + (y-3)i l = 3
Find modulus:
root( (x-5)^2 + (y-3)^2 ) = 3
Square both sides:
(x-5)^2 + (y-3)^2 = 9

Question 19

What is the centre and radius of a locus: l 4i + 2 - z l = 4

Answer 19

= l - (z - 2 - 4i) l = 4
= l z - (2 + 4i) l = 4
Centre = 2,4
Radius = 4

Question 20

How do you find the minimum and maximum values of arg z

Answer 20

Find the tangent
Create a triangle with the radius
Use trigonometry

Question 21

How do you find the minimum and maximum values of lzl

Answer 21

Work out the distance from 0 to centre
z max = distance + radius
z min = distance - radius

Question 22

What does lz - z1l = lz - z2l mean?

Answer 22

The complex number z must be equal distance from both z1 and z2

Question 23

How would you write y = 3 in complex number form

Answer 23

z = x + 3i ???

Question 24

After finding the perpendicular bisector between two points (loci), how do you find the least possible value of lzl

Answer 24

Find the point perpendicular from the origin on the bisector and then find the intersection of the perpendicular bisector and the perpendicular line from the origin

Question 25

How would you represent arg(z) = pi/6 on a graph

Answer 25

HALF-LINE from the origin angled at pi/6 anticlockwise direction

Question 26

How do you draw a half line

Answer 26

Open circle at the end which is not included

Question 27

Find the cartesian equation of arg(z + 3 + 2i) = 3pi / 4

Answer 27

tan(3pi / 4) = (y + 2) / (x + 3)
-1 = (y + 2) / (x + 3)
-x - 3 = y + 2
y = -x - 5
For x < -3

Question 28

How would you find the complex number z that satisfies both lz + 3 + 2il = 10 and arg(z + 3 + 2i) = 3pi / 4

Answer 28

Cartesian equation:
(x+3)^2 + (y+2)^2 = 10^2
y = -x - 5
z = -3-5root(2) + (-2 + 5root(2))i
OR
Use coordinate geometry:
Draw out diagram and use triangles

Question 29

How do you find the range from pi to -pi for values of theta where an argument and a circle have no common solutions

Answer 29

Create a kite with the distance between the centre of the circle and the half line point as well as the radius of the circle and then find theta

Question 30

When should you use dotted and full lines with inequalities

Answer 30

Dotted if < or >
Full line if <= or >=

Question 31

a) Find the cartesian equation of the locus of z if l z - 3 l = l z + i l. b) Hence find the least possible value of lzl

Answer 31

a) Using the definition of the modulus:
l x + iy - 3 l = l x + iy + i l
root((x-3)^2 + y^2) = root(x^2 + (y+1)^2)
-6x + 9 = 2y + 1
y = -3x + 4
b) (perp line) 1/3 * x = -3x + 4
x = 6/5 therefore y = 2/3
z = 6/5 + i2/5
l z lmin = 2root(2/5)