← Complex numbers: past papers

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(d) Merit

Practises Polynomial equations with complex roots

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.

One solution of the equation 3z3 + pz2 + qz − 8 = 0 is z = 1 + i. If p and q are both real, find the other two solutions of the equation, and the value of both p and q.

The 2025 paper, page 9 ↗ Marking schedule, page 5 ↗

Work it out

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

  1. 1 p and q are real. So what is a second root?

    The conjugate, z = 1 − i.

  2. 2 Multiply the two factors to get a real quadratic factor.

    (z − 1 − i)(z − 1 + i) = (z − 1)2 − i2 = z2 − 2z + 2

  3. 3 The cubic is that quadratic times (Az + B). Match the z3 terms and the constant terms.

    z3: A = 3. Constant: 2B = −8, so B = −4. The cubic is (3z − 4)(z2 − 2z + 2).

  4. 4 Expand it to read off p and q.

    (3z − 4)(z2 − 2z + 2) = 3z3 − 6z2 + 6z − 4z2 + 8z − 8 = 3z3 − 10z2 + 14z − 8
    So p = −10 and q = 14.

  5. 5 What is the third root?

    3z − 4 = 0, so z = 4/3.
Answer The other two solutions are z = 1 − i and z = 4/3; p = −10 and q = 14.

What the marker wanted

  • Achieved The other two solutions, or one of p and q.
  • Merit The other two solutions and the values of p and q, all found by algebra with every step shown.

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

How did it go?

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