AS 91577 · 2025 paper · a quadratic from one root, a surd squared, an equation in √x, a cubic with two unknowns, and simultaneous equations
Question 3(e) Merit to Excellence
Practises Equations, arguments, and comparing components
Five separate parts: a quadratic built from one complex root, the square of a complex number written with surds, an equation in √x, a cubic with two unknown coefficients and one root given, and a pair of simultaneous equations in two complex unknowns.
u and v are both complex numbers, and ui + 2v = 3 and u + (1 − i)v = 4.
Solve this pair of simultaneous equations, giving u and v in the form a + bi. Show your working clearly.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 Treat it as any pair of simultaneous equations. Which unknown is easiest to get on its own?
u, from the second: u = 4 − (1 − i)v. -
2 Substitute that into the first equation and expand.
(4 − (1 − i)v)i + 2v = 3
4i − (i − i2)v + 2v = 3
4i − (1 + i)v + 2v = 3
4i + (1 − i)v = 3 -
3 Get v on its own, and divide with the conjugate.
(1 − i)v = 3 − 4i
v = (3 − 4i)/(1 − i) × (1 + i)/(1 + i) = (3 + 3i − 4i + 4)/2 = (7 − i)/2 -
4 Now u. You already know (1 − i)v.
(1 − i)v = 3 − 4i, so u = 4 − (3 − 4i) = 1 + 4i. -
5 Check both equations.
ui + 2v = (i − 4) + (7 − i) = 3 ✓
u + (1 − i)v = (1 + 4i) + (3 − 4i) = 4 ✓
What the marker wanted
- Achieved An equation in one unknown only, such as (1 − i)v = 3 − 4i. The correct answer with no working also earns Achieved.
- Merit One of u and v.
- Excellence Both u and v: E8. A full solution with one minor error: E7.
In our words, from NZQA’s 2025 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.