← Complex numbers: past papers

AS 91577 · 2023 paper · a squared surd, no real roots, a real part of zero, a complex cubic, and a modulus equation

Question 1(a) Achieved

Practises Arithmetic & the conjugate

Five separate parts: a bracket with a surd squared, a quadratic with no real roots, a proof from a real part of zero, a cubic with one complex root given, and a constant found from a modulus.

Write (5 − 2√p)2 in the form a + bp + c√p, where a, b and c are integers.

The 2023 paper, page 2 ↗ Marking schedule, page 1 ↗

Work it out

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

  1. 1 A bracket squared: the first term squared, twice the product of the two, the second squared.

    (5 − 2√p)2 = 52 − 2(5)(2√p) + (2√p)2

  2. 2 Simplify each term. What is (2√p)2?

    4p, since (√p)2 = p.
    = 25 − 20√p + 4p

  3. 3 Match it to a + bp + c√p.

    a = 25, b = 4 and c = −20.
Answer 25 + 4p − 20√p

What the marker wanted

  • Achieved The expression in the form asked for.

In our words, from NZQA’s 2023 marking schedule.

✕

Common mistake

Squaring each term and dropping the middle one, −20√p. NZQA’s 2023 report, in its general comments on the paper, lists not multiplying surds together correctly among the things Not Achieved scripts had in common.

How did it go?

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