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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(d) Merit

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.

If (g + 2i)(3 + hi) = (10 − 4i)(3 − i), find all possible values of g and h.

The 2025 paper, page 6 ↗ Marking schedule, page 3 ↗

Work it out

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

  1. 1 Work out the right-hand side first: it has no unknowns.

    (10 − 4i)(3 − i) = 30 − 10i − 12i + 4i2 = 26 − 22i

  2. 2 Expand the left side and sort it into a real part and an imaginary part.

    (g + 2i)(3 + hi) = 3g + ghi + 6i + 2hi2 = (3g − 2h) + (gh + 6)i

  3. 3 Two complex numbers are equal when both parts match. Write the two equations.

    Real parts: 3g − 2h = 26
    Imaginary parts: gh + 6 = −22, so gh = −28

  4. 4 Substitute one into the other.

    h = (3g − 26)/2, so g × (3g − 26)/2 = −28
    3g2 − 26g + 56 = 0, which factorises as (g − 4)(3g − 14) = 0.

  5. 5 Find h for each g, and check.

    g = 4: h = (12 − 26)/2 = −7, and gh = −28 ✓
    g = 14/3: h = (14 − 26)/2 = −6, and gh = −28 ✓
Answer g = 4 and h = −7, or g = 14/3 and h = −6.

What the marker wanted

  • Achieved The two equations for g and h, from matching the real parts and the imaginary parts.
  • Merit Both pairs of values.

In our words, from NZQA’s 2025 marking schedule.

How did it go?

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