Y1, C1 - Complex Numbers

Question 1

What are imaginary numbers

Answer 1

Numbers in the form bi

Question 2

What are complex numbers

Answer 2

Numbers in the form a + bi where a is a real part and b is the imaginary part

Question 3

Solve z^2 + 3z + 5 by completing the square

Answer 3

= (z + 3/2)^2 = -11/4
z + 3/2 = ±root(-11/4)
z = -3/2 ± (root(11) / 4) * i

Question 4

If z = 3 + 2i what is z* (complex conjugate)

Answer 4

z* = 3 - 2i

Question 5

What happens if you add or multiply complex conjugates (complex numbers)?

Answer 5

You are left with only real parts

Question 6

How do you ‘realise’ the denominator of a complex fraction

Answer 6

Multiply both the numerator and denominator by the conjugate of the denominator

Question 7

When ‘realising’ a denominator, what is the difference of two squares and difference of two squares ‘i’ version

Answer 7

(a + b)(a - b) = a^2 - b^2
(a + bi)(a - bi) = a^2 + b^2

Question 8

If α and β are the roots of the quadratic ax^2 + bx + c, how can the quadratic be equated to the roots?

Answer 8

ax^2 + bx + c = a(x - α)(x - β)

Question 9

If α and β are the roots of the quadratic ax^2 + bx + c, what are the equations for the sum and product of the roots?

Answer 9

Sum of roots = α + β = -b / a
Product of roots = αβ = c / a

Question 10

If z is the root of a quadratic equation and z is a complex number, what is the other root

Answer 10

z*

Question 11

If 2 - 4i is a root of the equation z^2 + pz + q= 0, find p and q

Answer 11

α + β = -b / a
αβ = c / a
α = 2 - 4i
β = 2 + 4i
Assume a to be 1
p = -4
q = 20