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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

The complex number $w$ is defined by $w = \frac{3\sqrt{3} + 3i}{1+i\sqrt{3}}$ .

Express $w$ in the form $re^{i\theta}$ , where $r>0$ and $0 \le \theta < 2\pi$ .

[5]

Find the square roots of $w$ , giving your answers in the form $re^{i\phi}$ , where $r>0$ and $0 \le \phi < 2\pi$ .

[4]

Without using a calculator, show that $w^6$ is a real number.

[3]

Question 8

HLPaper 1

Given that $z = r(\cos\theta + i\sin\theta)$ , where $r$ is a positive real number and $\theta$ is an angle.

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 9

HLPaper 1

The complex number $u = -1+i$ is given. The polynomial $p(x) = x^4+x^3+3x^2+4x+6$ is given.

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

[4]

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

[6]

Question 10

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]

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