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(c) Merit
Practises Equations, arguments, and comparing components
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.
Given that w/(w + i) = 2 − i, find |w|.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 Clear the fraction: multiply both sides by w + i.
w = (2 − i)(w + i) = 2w − wi + 2i − i2 = 2w − wi + 2i + 1 -
2 Collect the w terms on one side and factorise.
w − 2w + wi = 1 + 2i, so w(−1 + i) = 1 + 2i and w = (1 + 2i)/(−1 + i). -
3 You want |w|. Is there a short cut?
The modulus of a quotient is the quotient of the moduli:
|w| = (|1 + 2i|)/(|−1 + i|) = √5/√2 = √(5/2) = √10/2 -
4 Check by finding w itself.
w = ((1 + 2i)(−1 − i))/((−1 + i)(−1 − i)) = (−1 − i − 2i + 2)/2 = 1/2 − (3/2) i, and √(1/4 + 9/4) = √(10/4) = √10/2 ✓
What the marker wanted
- Achieved w on its own: w = (1 + 2i)/(−1 + i).
- Merit The value of |w|.
In our words, from NZQA’s 2024 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.