IB Mathematics Analysis and Approaches Topic AHL 1.14—complex Roots of Polynomials, Conjugate Roots, De Moivre’s, Powers & Roots of Complex Numbers Questionbank (HL)

IB Mathematics Analysis and Approaches Topic AHL 1.14—complex Roots of Polynomials, Conjugate Roots, De Moivre’s, Powers & Roots of Complex Numbers Questionbank (HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 1.14—complex Roots of Polynomials, Conjugate Roots, De Moivre’s, Powers & Roots of Complex Numbers with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 1.14—complex Roots of Polynomials, Conjugate Roots, De Moivre’s, Powers & Roots of Complex Numbers and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

The complex number $z$ is defined by $z = \frac{4 - 4 i}{1 + i}$ .

Find an expression for $z$ in the form $r e^{i θ}$ , where $r > 0$ and $- π < θ \leq π$ .

[5]

Find the two square roots of $z$ , giving your answers in the form $r e^{i θ}$ , where $r > 0$ and $- π < θ \leq π$ .

[3]

Question 2

HLPaper 1

The polynomial $p (x) = x^{4} + a x^{3} + 7 x + b$ , where $a$ and $b$ are constants, is given. It is given that $p (x)$ is divisible by $x^{2} + x - 1$ .

Find the values of $a$ and $b$ .

[5]

When $a$ and $b$ have these values, find the real roots of the equation $p (x) = 0$ .

[2]

Question 3

HLPaper 1

Given that $z = r (cos θ + i sin θ)$ , where $r$ is a positive real number and $θ$ is a real angle (in radians).

Find and simplify an expression for $z^{n}$ , where $n$ is a positive integer.

[2]

Given a real constant $k$ , find and simplify an expression for $\frac{k}{z ^{n}}$ , where $n$ is a positive integer.

[3]

Question 4

HLPaper 1

Let $u = - 1 + i$ and $p (x) = x^{4} + x^{3} + 3 x^{2} + 4 x + 6$ .

Show your working to verify that $u$ is a root of $p (x) = 0$ , and write down a second complex root of the equation.

[4]

Find the other two roots of the equation $p (x) = 0$ .

[6]

Question 5

HLPaper 1

Let $z$ be a non-zero complex number with $∣ z ∣ = 1$ . Write $z = cos θ + i sin θ$ for some real $θ$ .

Prove that $z + \frac{1}{z} = 2 cos θ$ , and hence that $z + \frac{1}{z}$ is real.

[4]

Using De Moivre’s theorem, show that for any $n \in N$ , $z^{n} + \frac{1}{z ^{n}} = 2 cos (n θ) .$

[3]

Hence solve $z^{6} + \frac{1}{z ^{6}} = 1$ for all complex numbers $z$ with $∣ z ∣ = 1$ , giving your answers in the form $cos φ + i sin φ$ .

[6]

IB Mathematics Analysis and Approaches AHL 1.14—complex Roots of Polynomials, Conjugate Roots, De Moivre’s, Powers & Roots of Complex Numbers Questionbank

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IB Mathematics Analysis and Approaches AHL 1.14—complex Roots of Polynomials, Conjugate Roots, De Moivre’s, Powers & Roots of Complex Numbers Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 1.14—complex Roots of Polynomials, Conjugate Roots, De Moivre’s, Powers & Roots of Complex Numbers Questionbank

Unlock all 28 questions with a free account