← Complex numbers: past papers

AS 91577 · 2024 paper · a surd fraction, a square on the Argand diagram, a conjugate equation, fourth roots, and a cubic identity

Question 3(e) Merit to Excellence

Practises Equations, arguments, and comparing components

Five separate parts: a fraction of surds simplified, a complex number squared and plotted, an equation in a number and its conjugate, the fourth roots of a negative number, and a cube of x + 1/x.

Given that x + 1/x = p, where p is a real constant, find the value of x3 + 1/x3, giving your answers in terms of p.

The 2024 paper, page 11 ↗ Marking schedule, page 6 ↗

Work it out

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

  1. 1 You want cubes. Cube both sides of x + 1/x = p.

    (x + 1/x)3 = p3

  2. 2 Expand the cube, term by term.

    x3 + 3x2(1/x) + 3x(1/x)2 + 1/x3 = x3 + 3x + 3/x + 1/x3

  3. 3 The two middle terms hide something you already know.

    3x + 3/x = 3(x + 1/x) = 3p. So x3 + 1/x3 + 3p = p3.

  4. 4 Rearrange.

    x3 + 1/x3 = p3 − 3p. You never needed x itself.
Answer x3 + 1/x3 = p3 − 3p

What the marker wanted

  • Achieved The cube expanded correctly (step 2).
  • Merit The middle terms grouped as 3(x + 1/x) (step 3).
  • Excellence The expression, with every step said clearly: E8. The right expression without that clear working, or a full answer with one minor error: E7.

In our words, from NZQA’s 2024 marking schedule.

How did it go?

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