Pure 2: Argand Diagrams

Question 1

What is one radian?

Answer 1

The measurement of an angle subtended by an arc of length r

Question 2

How many radians in 360°?

Answer 2

2π

Question 3

How many radians in 180°?

Answer 3

π

Question 4

How many radians in 90°?

Answer 4

π/2

Question 5

How many radians in 45°?

Answer 5

π/4

Question 6

How many radians in 60°?

Answer 6

π/3

Question 7

How many radians in 30°?

Answer 7

π/6

Question 8

How many radians in 270°?

Answer 8

3π/2

Question 9

How are complex numbers represented on Argand diagrams?

Answer 9

z=x+iy = (x,y)

Question 10

What do complex conjugate pairs look like on Argand diagrams?

Answer 10

Reflections of each other in the x-axis

Question 11

What happens when two points are added together on an Argand diagram?

Answer 11

The resulting complex number forms a parallelogram with the two added complex numbers and (0,0)

Question 12

What is the modulus of a complex number?

Answer 12

The magnitude of its corresponding vector

Question 13

How is the modulus calculated?

Answer 13

|z| = ✔️(x^2 + y^2)

Question 14

What is the argument of a complex number? (2)

Answer 14

The angle (θ) that the vector makes with the positive real axis
Usually given in the range -π < θ ≤ π (principal argument)

Question 15

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

Answer 15

z = r(cos(θ) +i sin(θ)) where r is the modulus and theta is the argument

Question 16

How do you rearrange into the modulus-argument form?

Answer 16

For any θ:
sin(-θ) = -sin(θ)
cos(θ) = cos(-θ)

Question 17

How are two modulus-argument forms multiplied? (2)

Answer 17

Multiply the modulus

Add the arguments

Question 18

How are two modulus-argument forms divided? (2)

Answer 18

Divide the modulus

Subtract the arguments

Question 19

What does |z - z1| represent on an Argand diagram?

Answer 19

A circle centre (x1, y1) with radius r

Question 20

How is the modulus converted into a Cartesian equation?

Answer 20

|x+iy| = ✔️(x^2 + y^2) so |z|^2 = x^2 + y^2

Question 21

How is the maximum and minimum modulus found? (4)

Answer 21

|z| means distance from (0,0)
Find distance from (0,0) to centre (using Pythagoras)
Maximum = distance to centre + radius
Minimum = distance to centre - radius

Question 22

How is the maximum and minimum argument of r=|z - z1| found? (3)

Answer 22

Draw the circle with the locus of points
Draw a tangent from (0,0) to make the biggest and smallest angle
Use trigonometry to calculate the angle

Question 23

What does |z - z1| = |z-z2| represent on an Argand diagram?

Answer 23

The perpendicular bisected of the line segment joining z1 to z2

Question 24

How is the least possible value of |z| found given that |z - z1| = |z-z2|? (2)

Answer 24

Look for the perpendicular line that passes through (0,0)

Solve the simultaneous equations to find the complex number that is on both lines and calculate its modulus

Question 25

What does ((z - z1) = θ) represent on an Argand diagram?

Answer 25

A half line from, but not including, the fixed point z1 that makes an angle θ with a line parallel to the real axis

Question 26

How is the Cartesian equation of ((z - z1) = θ) found?

Answer 26

y - y1 = tan(θ)(x-x1)