Level 3 Complex Numbers Walkthrough

2025 NCEA Level 3 Complex Numbers Question 2(e)

2025 Paper

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Question

\[ \arg\left(\frac{d+6i}{1-di}\right)=\frac{\pi}{4} \]

Find the possible value(s) of \(d\), where \(d\) is a real constant.

First walkthrough idea

Focus to try first

using an argument condition to equate real and imaginary parts after rationalising a complex fraction.

Step 1

Interpret the argument

The argument describes the direction of the complex number on the Argand diagram.

Show the first step’s working

The argument describes the direction of the complex number on the Argand diagram.

Key result

The angle the point makes with the positive real axis.

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91577 — Apply the algebra of complex numbers in solving problems.

Method: Using an argument condition to solve for a real constant.

This is Question 2(e) from the 2025 NCEA Level 3 Complex Numbers paper for AS91577 — Apply the algebra of complex numbers in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91577.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise using an argument condition to solve for a real constant. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check the complex number's quadrant and the argument range before selecting the final angle.

Continue practising

← Question 2(d) Question 3(a) →