Specialist Maths — Question 18

QCAA 2020, Paper 1 · 6 marks

Q18 · 2020 · Technology-freeComplex unfamiliar6 marks

QUESTION 18 (6 marks)

Consider the function

P(z)=2z^4+az^3+6z^2+az+b

, where

a,b\in\mathbb{Z}^+

. One of the roots of

P(z)

z=-i

. Determine the possible value/s for

a

and

b

such that all remaining roots of

P(z)

have an imaginary component.

WORKED SOLUTION

6 marks

ANSWER

b=4

and

a\in\{1,2,3,4,5\}

Worked solution

Since

P(-i)=0

2+ai-6-ai+b=0

, hence

b=4

. Because the coefficients are real,

i

is also a root, so

z^2+1

is a factor. Thus

P(z)=(z^2+1)(2z^2+az+4).

The remaining quadratic must have non-real roots, so

a^2-32<0

. With

a\in\mathbb Z^+

a=1,2,3,4,5

, and

b=4

correctly applies the factor theorem to determine

b

[1 mark]

correctly uses the conjugate root of the given root to identify another factor of

P(z)

[1 mark]

correctly identifies that

(z^2+1)

is a factor of

P(z)

[1 mark]

determines the remaining quadratic factor in terms of

a

[1 mark]

applies the complex root requirement to the remaining quadratic factor

[1 mark]

determines the possible values for

a

given

a,b\in\mathbb{Z}^+

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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