De Moivre's Theorem — practice questions

Complex Numbers · Part 3 of 5

6 exam-style questions with model answers, plus 8 quick multi-choice questions — every question on this part of the standard, grouped by the 2 pages of notes they come from.

Write a full answer before you reveal the model one. That comparison is where the learning happens.

De Moivre's theorem and powers

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AchievedShow

Use De Moivre's theorem to evaluate $\left(3\,\text{cis}\,\dfrac{\pi}{9}\right)^3$ , giving your answer in polar form.

MeritShow

Evaluate $(1 - i)^8$ , giving your answer in the form $a + bi$ . Show all conversion steps.

ExcellenceShow

Use De Moivre's theorem to prove that $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ .

Roots of complex numbers

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AchievedShow

Solve $z^3 = 27\,\text{cis}\,\dfrac{\pi}{2}$ , giving all solutions in polar form.

MeritShow

Solve $z^4 = -16$ , giving all solutions in the form $a + bi$ with exact values.

ExcellenceShow

The equation $z^5 = 32$ has five solutions. Show that the sum of the five solutions is zero, and explain the result geometrically.

Back to the Complex Numbers notes →

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De Moivre's Theorem — practice questions

Complex Numbers · Part 3 of 5

6 exam-style questions with model answers, plus 8 quick multi-choice questions — every question on this part of the standard, grouped by the 2 pages of notes they come from.

Write a full answer before you reveal the model one. That comparison is where the learning happens.

De Moivre's theorem and powers

Read the notes →

AchievedShow

Use De Moivre's theorem to evaluate $\left(3\,\text{cis}\,\dfrac{\pi}{9}\right)^3$ , giving your answer in polar form.

MeritShow

Evaluate $(1 - i)^8$ , giving your answer in the form $a + bi$ . Show all conversion steps.

ExcellenceShow

Use De Moivre's theorem to prove that $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ .

Roots of complex numbers

Read the notes →

AchievedShow

Solve $z^3 = 27\,\text{cis}\,\dfrac{\pi}{2}$ , giving all solutions in polar form.

MeritShow

Solve $z^4 = -16$ , giving all solutions in the form $a + bi$ with exact values.

ExcellenceShow

The equation $z^5 = 32$ has five solutions. Show that the sum of the five solutions is zero, and explain the result geometrically.

Back to the Complex Numbers notes →