L3 Complex Numbers Past Papers
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AS91577Complex Numbers past exam papers
Every NZQA exam for AS91577 (Apply the algebra of complex numbers in solving problems) from 2013 to 2025, with its marking schedule, and from 2020 the examiners’ report, which says where students lost marks.
Papers by year
2020Archived copy
2019Archived copy
2018Archived copy
2017Archived copy
2016Archived copy
2015Archived copy
2014Archived copy
2013Archived copy
Marked student answers at each grade (NZQA exemplars), 2024: Achievement · Merit · Excellence.
What the examiners keep saying
With De Moivre’s theorem, everything depends on the starting modulus and argument: get either wrong and every root is wrong. Sketch the number on an Argand diagram first to see its quadrant, then find all of the roots.
Reports: 2021–2025Squaring or taking a square root can create answers that don’t work. Check every answer in the original equation and reject any that fail.
Reports: 2021–2025Give the answer in the form the question asks for (a + bi or r cis θ), or in terms of the letter it names.
Reports: 2021–2025Know what “purely real” and “purely imaginary” mean, and how to put z = x + iy into an equation and split it into a real equation and an imaginary one.
Reports: 2021–2025In proofs, the best answers equated the real and imaginary parts and often completed the square, with every step shown.
Reports: 2021–2025Be able to convert to polar form, find a modulus and argument, and solve a quadratic with unknown coefficients by hand. The examiners have warned about leaning on the calculator every year since 2022.
Reports: 2022–2025Take extra care with the argument when the number isn’t in the first quadrant. In 2025 many gave a positive argument where it should have been negative.
Reports: 2023–2025
How to use these papers
- Start with the most recent papers. They are the closest to this year’s exam.
- Sit a whole paper without notes, then mark it with its schedule. The schedule shows what earns Achievement, Merit and Excellence in each part, not only the answer.
- After marking, read that year’s examiners’ report to see where other students lost marks.
- Check the 2026 assessment specification for what this year’s exam covers. An older paper can include something that has since changed.
About the older papers
NZQA takes papers off its website after five years. The older ones here are NZQA’s own files as saved by the Internet Archive, a non-profit library of the web, so they can be slow to open. We could only find the examiners’ reports from 2020 on.
Practise each topic in AS91577 with worked answers.
Open the Complex Numbers question bank →