IB Maths AA HL AA 1.6 Complex numbers questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper
Paper 1, no calculatorPaper 2, calculator
Difficulty
EasyMediumHard

Question 1

Paper 2, calculatorMedium7 marks

Let u=−8+83 iu=-8+8\sqrt{3}\,i

(a)

Express uu in the form r(cos⁡θ+isin⁡θ)r(\cos\theta+i\sin\theta) where r>0r>0 and 0≤θ<2π0\leq\theta<2\pi

[2]
(b)

Hence, or otherwise, determine both square roots of uu giving your answers in the form a+bia+bi

[3]
(c)

A polynomial with real coefficients has 2+23 i2+2\sqrt{3}\,i as a root. State the other root in its conjugate pair and hence determine the monic quadratic factor associated with this pair.

[2]

Question 2

Paper 1, no calculatorMedium7 marks

Consider the equation

z3=8(cos⁡π+isin⁡π).z^3=8(\cos\pi+i\sin\pi).
(a)

Determine all distinct solutions, giving each in the form a+bia+bi

[4]
(b)

Hence determine the monic quadratic polynomial with real coefficients whose roots are the two non-real solutions of the equation.

[3]

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