Core Pure 1 - Chapter 2: Argand diagrams

Question 1

What is an argand diagram

Answer 1

New coordinate system that can plot complex numbers

Question 2

What is the structure of an argand axis

Answer 2

Real along the bottom imaginary along the top

Question 3

What is the geometric property for the modulus of z

Answer 3

The length of the line z

Question 4

How do you find the modulus of a complex number

Answer 4

Apply pythagoras

Question 5

What is the geometric property of arg(z)

Answer 5

Anticlockwise rotation in radians from the positive real axis

Question 6

How do you find the arguement?

Answer 6

arctan(y/x)

Question 7

How do you find the argument in all 4 quadrants

Answer 7

1st: z = α
2nd: z = π - α
3rd : z = -(π - α)
4th : z = -α

Question 8

What is the modulus-argument form of a complex number?

Answer 8

r(cosθ + isinθ)

Question 9

What does r and θ represent in the modulus-argument form a complex number?

Answer 9

r = modulus
θ = arguement

Question 10

How do you convert from modulus argument form back to cartesian form?

Answer 10

Expand the brackets of the modulus argument form

Question 11

How do you multiply complex numbers in modulus argument form

Answer 11

Multiply the moduli, add the arguements

Question 12

How do you divide complex numbers in modulus argument form

Answer 12

Divide the moduli, subtract the arguments

Question 13

How do you convert r(cosθ - isinθ) back into standard form

Answer 13

Negate sin and negate the argument for both

Question 14

What is meant by a loci

Answer 14

Set of points that satisfy a restriction

Question 15

How do you convert from modulus to cartesian form

Answer 15

Let z = x+iy, then group the real and imaginary terms together

Question 16

What is the general form a circle on an argand diagram

Answer 16

┃z - z₁┃= r
Where z is a general complex number and z₁ is a fixed complex number

Question 17

What geometric property does the loci form of a circle describe

Answer 17

The distance between two complex numbers at a fixed point

Question 18

How do you find the maximum/minimum argument for a circle

Answer 18

The maximum/minimum argument occurs at the point of tangency from the circle to the origin, so draw a tangent in all possible places then use geometry to work out the value of arg(z)

Question 19

How do you find the maximum/minimum modulus for a circle

Answer 19

Occurs when z passes through the centre of the circle so you will have to use the radius and Pythagoras to maximise/minimise

Question 20

What is the general form of a perpendicular bisector on an argand diagram

Answer 20

┃z - z₁┃= ┃z - z₂┃where both can vary

Question 21

How do you find the cartesian equation of a perpendicular bisector

Answer 21

Let z = x+iy
Group terms
Apply the definition of the modulus on both sides
Rearrange for y

Question 22

How do you minimise the modulus of a perpendicular bisector

Answer 22

The minimum modulus occurs at the perpendicular distance from the origin to the bisector, so work out the equation of the perpendicular line and any useful coordinates

Question 23

What is the general formula of half lines on an argand diagram

Answer 23

arg(z - z₁) = θ

Question 24

What must you include when sketching a half line?

Answer 24

Open circle at the point of argument and always state the restriction on

Question 25

How do you find a complex number from 2 different loci

Answer 25

Convert both loci to cartesian form and solve simultaneously

Question 26

How do you find a range of values for half lines

Answer 26

Find the value of the argument at 1 solution then state a range for which the amount of solutions would change

Question 27

How do you solve region problems

Answer 27

Consider each locus individually on one diagram then shade depending on intersection or union