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AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers - IB Questionbank | RevisionDojo

Practice AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

The cubic equation $2z^3 - 14z^2 + cz - 30 = 0$ , where $c \in \mathbb{R}$ , has a solution $z = 2+i$ .

Find in any order

the other two roots of the equation.

[3]

the value of $c$ .

[3]

Question 2

HLPaper 1

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

Find an expression for $z$ in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

[5]

Find the two square roots of $z$ , giving your answers in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

[3]

Question 3

HLPaper 1

Find the modulus and argument of each root of: $z^2 - 2iz - 4 = 0$ .

[3]

Given that $w = 2\left(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}\right)$ , write $w^4$ in the form $re^{i\theta}$ , where $r>0$ and $-\pi < \theta \le \pi$ .

[3]

Question 4

HLPaper 1

The complex conjugate of $z$ is denoted by $z^*$ .

Find the two solutions of the equation $(z-i)(z^*) - 2z = -1-3i$ , $z \in \mathbb{C}$ giving the answers in the form $x+iy$ , where $x \in \mathbb{R}$ and $y \in \mathbb{R}$ .

[6]

Question 5

HLPaper 1

The polynomial $p(z)$ is defined by $p(z) = z^3 - z^2 + nz - 4$ , where $n$ is a constant. It is given that $(z-1)$ is a factor of $p(z)$ .

Find the value of $n$ .

[2]

Hence, showing all your working, find

(i) the three roots of the equation $p(z)=0$ .

[5]

(ii) the six roots of the equation $p(z^2)=0$ .

[6]

Question 6

HLPaper 1

The complex conjugate of $z$ is denoted by $z^*$ .

Solve the equation $z^* = \frac{4z+5+5i}{2}$ , giving the answer in the form $x+iy$ , where $x$ and $y$ are real numbers.

[7]

Question 7

HLPaper 1

Consider the equation $w^4 = -4$ , where $w \in \mathbb{C}$ .

Solve the equation, giving the solutions in the form $c + \text{i}d$ , where $c, d \in \mathbb{R}$ .

[5]

The solutions form the vertices of a polygon in the complex plane. Find the area of the polygon.

[2]

Question 8

HLPaper 1

An electrical engineer is analyzing an AC circuit and needs to find the roots of the equation representing the power output, given by $z^3 = 8$ .

Write the number 8 in polar form, which is useful for representing complex numbers in AC analysis.

[1]

Use De Moivre's Theorem to find all cube roots of 8, which will help the engineer determine the possible phase angles for the circuit.

[3]

Question 9

HLPaper 1

A telecommunications company is analyzing a signal represented in polar form, where $z = 2 \operatorname{cis} \frac{\pi}{6}$ represents the amplitude and phase of the signal.

Use De Moivre's Theorem to find $z^4$ in polar form.

[2]

Convert $z^4$ to rectangular form.

[2]

Question 10

HLPaper 1

An electrical engineer is analyzing an AC circuit and represents a voltage as the complex number $z = 1 + i$ .

Express $z$ in polar form.

[2]

Use De Moivre's Theorem to find $z^8$ in polar form.

[2]

Convert $z^8$ to rectangular form.

[2]

AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers Questionbank