Core Pure 1-Chapter 1: Complex numbers

Question 1

Define “i”

Answer 1

i = √-1

Question 2

What is the general form of a complex number?

Answer 2

z=a+bi

Question 3

How can i be used to rewrite negative roots

Answer 3

Factor out an i and rewrite as a positive root
e.g √-36 = 6i

Question 4

What is the discriminant

Answer 4

b²-4ac

Question 5

What is needed for a quadratic to have 2 complex roots

Answer 5

The discriminant must be less than 0

Question 6

Give the first 4 powers of i

Answer 6

i =√-1
i² = -1
i³ = -i
i⁴ = 1

Question 7

What is meant by a complex conjugate

Answer 7

Inverse operation of a complex number

Question 8

What happens when you multiply a complex number by its conjugate

Answer 8

The imaginary part is removed

Question 9

How can you realise a denominator

Answer 9

Multiply the entire fraction by the complex conjugate of the denominator

Question 10

What is the possible combination of complex roots for a quadratic

Answer 10

If the solutions are non-real, they occur as a complex conjugate pair

Question 11

How can you write the general complex roots for a quadratic

Answer 11

(x-α) (x-β)

Question 12

What are the possible combinations of complex roots of a cubic

Answer 12

1 real root and 2 complex roots that occur as a conjugate pair
3 real roots

Question 13

What are the possible combinations of complex roots of a quartic

Answer 13

4 real roots, 2 real roots with 1 conjugate pair, 0 real roots and 2 conjugate pairs

Question 14

Give the general method for solving quartics

Answer 14

1. If given a complex root write down the conjugate
2. Find the quadratic from the conjugate using the general form of conjugate roots
3. Divide the quartic by the quadratic to find the other quadratic
4. Solve each