← 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(b) Achieved

Practises Polynomial equations with complex roots

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.

Find the value(s) of r so that the quadratic equation 2x2 + (3 + 2r)x + 3 − 2r = 0 has equal roots.

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 Equal roots means what about the discriminant?

    b2 − 4ac = 0, with a = 2, b = 3 + 2r and c = 3 − 2r.

  2. 2 Write it out and expand.

    (3 + 2r)2 − 4(2)(3 − 2r) = 0
    9 + 12r + 4r2 − 24 + 16r = 0
    4r2 + 28r − 15 = 0

  3. 3 Solve for r.

    (2r − 1)(2r + 15) = 0, so r = 1/2 or r = −15/2.
Answer r = 1/2 or r = −7.5

What the marker wanted

  • Achieved Both values of r.

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

How did it go?

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