2019 NCEA Level 3 Complex Numbers

2019 Complex Numbers AS91577 worked answers

Guided, step-by-step worked solutions for Apply the algebra of complex numbers in solving problems.

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Paper overview

Work through the 2019 questions one step at a time

This page contains 15 independent worked question walkthroughs for NCEA Level 3 Complex Numbers (AS91577). Each walkthrough offers hints, reveals the full working in a logical sequence, and focuses on the method described below.

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Worked question index

2019 Complex Numbers questions

Choose a question part to open its guided worked solution.

Question 1

Question 1 walkthroughs

Question 1(a) worked solution Completing the square to solve a complex quadratic. Question 1(b) worked solution Applying the Remainder Theorem to find p. Question 1(c) worked solution Rationalising a complex quotient and applying its modulus. Question 1(d) worked solution Conjugate roots, cubic factorisation, and coefficient matching. Question 1(e) worked solution Equating real and imaginary parts after clearing denominators.

Question 2

Question 2 walkthroughs

Question 2(a) worked solution Using a conjugate and collecting complex-number terms. Question 2(b) worked solution Rationalising a surd denominator into a + b√5. Question 2(c) worked solution Finding fourth roots in polar form for every real p. Question 2(d) worked solution Making a rationalised complex quotient purely real. Question 2(e) worked solution Proving a unit-circle quotient is purely imaginary.

Question 3

Question 3 walkthroughs

Question 3(a) worked solution Dividing moduli and subtracting arguments in polar form. Question 3(b) worked solution Solving for real x and y by matching complex parts. Question 3(c) worked solution Solving a radical equation with real-domain checks. Question 3(d) worked solution Using an argument condition and verifying the quadrant. Question 3(e) worked solution Proving a discriminant is always positive.

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Complex-number algebra Polar form and De Moivre

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