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AS 91577 · 2025 paper · two points on an Argand diagram, a fifth power, a modulus, comparing parts, and an argument

Question 2(e) Merit to Excellence

Practises Equations, arguments, and comparing components

Five separate parts: a combination of two points read off an Argand diagram, a power of a number in polar form, an equation with a modulus in it, two unknowns found by comparing real and imaginary parts, and an unknown found from an argument.

Given that arg((d + 6i)/(1 − di)) = π/4, find the possible value(s) of d, where d is a real constant.

The 2025 paper, page 7 ↗ Marking schedule, page 4 ↗

Work it out

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

  1. 1 You cannot read an argument off a fraction. Make it a + bi first: multiply top and bottom by the conjugate of 1 − di.

    ((d + 6i)(1 + di))/((1 − di)(1 + di)) = (d + d2 i + 6i + 6di2)/(1 + d2) = (−5d + (d2 + 6)i)/(1 + d2)

  2. 2 An argument of π/4 puts the point on the 45° line in the first quadrant. What does that say about its two parts?

    They are equal, and both positive. The denominators match, so −5d = d2 + 6.

  3. 3 Solve for d.

    d2 + 5d + 6 = 0, so (d + 2)(d + 3) = 0, and d = −2 or d = −3.

  4. 4 Equal parts could also mean the third quadrant, at 5π/4. Check each d.

    The imaginary part, d2 + 6, is positive for every d, so neither is in the third quadrant.
    d = −2: the quotient is (10 + 10i)/5 = 2 + 2i ✓
    d = −3: the quotient is (15 + 15i)/10 = 1.5 + 1.5i ✓
Answer d = −2 or d = −3.

What the marker wanted

  • Achieved The quotient over a real denominator: (−5d + (d2 + 6)i)/(1 + d2).
  • Merit Its real part set equal to its imaginary part.
  • Excellence Both values of d, clearly justified: E8. Only one of them justified, or 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.