← Complex numbers: past papers

AS 91577 · 2024 paper · a quotient in terms of k, equal roots, a modulus from a quotient, a complex cubic, and a circle locus

Question 2(a) Achieved

Practises Arithmetic & the conjugate

Five separate parts: a quotient written as a + bi, a quadratic with equal roots, the modulus of an unknown found from a quotient, a cubic with one complex root given, and a locus that is a circle.

Write the complex number i/(2k + i) in the form a + bi, where a, b and k are real numbers, giving your answer in terms of k.

The 2024 paper, page 5 ↗ Marking schedule, page 3 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

  1. 1 To get i out of the denominator, multiply top and bottom by the conjugate of 2k + i.

    i/(2k + i) × (2k − i)/(2k − i) = (2ki − i2)/(4k2 − i2) = (1 + 2ki)/(4k2 + 1)

  2. 2 Split it into a real part and an imaginary part.

    1/(4k2 + 1) + (2k/(4k2 + 1)) i
Answer 1/(4k2 + 1) + (2k/(4k2 + 1)) i

What the marker wanted

  • Achieved The correct a + bi form.

In our words, from NZQA’s 2024 marking schedule.

How did it go?

Got it moves it further out. Review brings it back tomorrow.