IB Mathematics Analysis and Approaches Topic AHL 1.12—complex Numbers – Cartesian Form and Argand Diag Questionbank (HL)

IB Mathematics Analysis and Approaches Topic AHL 1.12—complex Numbers – Cartesian Form and Argand Diag Questionbank (HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 1.12—complex Numbers – Cartesian Form and Argand Diag with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 1.12—complex Numbers – Cartesian Form and Argand Diag and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

Given the complex number $w = 5 (cos \frac{π}{4} + i sin \frac{π}{4})$ .

Convert the complex number $w$ to its Cartesian form.

[3]

Plot the complex number $w$ on an Argand diagram.

[1]

Question 2

HLPaper 1

The complex number $w$ is defined by $w = 2 e^{i \frac{π}{4}}$ . Points $A$ , $B$ and $C$ on an Argand diagram represent the complex numbers $w$ , $w^{*}$ and $w^{2}$ respectively (where $w^{*}$ denotes the complex conjugate of $w$ ).

Draw the Argand diagram showing the points $A$ , $B$ and $C$ ,.

[3]

Calculate the area of triangle $A B C$ .

[2]

Question 3

HLPaper 1

Consider the complex equation $z^{4} = 256 i$ , where $z \in C$ . The equation has four distinct roots $z_{1}, z_{2}, z_{3}, z_{4}$ , which can be written in the form $z_{j} = r (cos θ_{j} + i sin θ_{j})$ , with $r > 0$ and $0 \leq θ_{1} < θ_{2} < θ_{3} < θ_{4} < 2 π$ .

Find the values of $z_{1}, z_{2}, z_{3}$ and $z_{4}$ , expressing them in polar form.

[6]

These roots $z_{1}, z_{2}, z_{3}$ and $z_{4}$ form a geometric sequence.

Find the common ratio of the geometric sequence, expressing your answer in Cartesian form.

[3]

On an Argand diagram, the roots are represented by points $A, B, C, D$ .

Plot the points $A, B, C$ and $D$ on an Argand diagram.

[3]

The equation $v^{4} = a + bi$ ( $a, b \in R$ ) has the complex conjugates $z_{1}^{*}, z_{2}^{*}, z_{3}^{*}, z_{4}^{*}$ as its roots.

Determine the value of $a$ and the value of $b$ .

[3]

Let $A^{'}, B^{'}, C^{'}, D^{'}$ be the midpoints of the segments $[A B], [B C], [C D], [D A]$ respectively. Consider the equation $w^{p} = 2^{q}$ ( $p, q \in Z^{+}$ ) for which the four points $A^{'}, B^{'}, C^{'}, D^{'}$ are roots.

Find the least possible value of $p$ and the corresponding value of $q$ .

[5]

Question 4

HLPaper 2

Consider a complex number $z = - 3 + 3 i$ .

Calculate the modulus of $z$ .

[1]

Find the argument of $z$ in radians, expressing it to three significant figures.

[2]

Write $z$ in polar form $r cis θ$ .

[1]

Question 5

HLPaper 1

It is given that $w = \frac{2 - i}{1 + 3 i}$ .

Express $w$ in the form $x + i y$ .

[2]

Find the argument of $w$ , giving your answer in radians.

[2]

Find the modulus of $w$ , giving your answer in the form $a b$ .

[2]

IB Mathematics Analysis and Approaches AHL 1.12—complex Numbers – Cartesian Form and Argand Diag Questionbank

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IB Mathematics Analysis and Approaches AHL 1.12—complex Numbers – Cartesian Form and Argand Diag Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 1.12—complex Numbers – Cartesian Form and Argand Diag Questionbank

Unlock all 19 questions with a free account