AHL 1.12—Complex numbers introduction - IB Questionbank | RevisionDojo

AHL 1.12—Complex numbers introduction Questionbank

Practice AHL 1.12—Complex numbers introduction with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Let $z=x+y i$ be a complex number.

Given that $w=z+16$ , write down the conjugate of $w$ .

[2]

Express $|w|^{2}$ in the form $x^{2}+y^{2}+a x+b$

[2]

Express $|z+1|^{2}$ in terms of $x$ and $y$ .

[2]

Given that $|z+16|=4|z+1|$ , find the value of $|z|$ .

[4]

Question 2

HLPaper 1

Solve the following equation for $z$ , where $z$ is a complex number.

Give your answer in the form $a+b i$ , where $a$ and $b$ are both real numbers.

$\frac{z}{3+4 i}+\frac{z-1}{5 i}=\frac{5}{3-4 i}$

[4]

Question 3

HLPaper 1

Find the complex number $z$ in each of the following equations.

$(2+5 i) z+9=3+14 i$

[3]

$(2+5 i) z+9=3 z+19 i$

[3]

$(2+5 i) z+8=3 \bar{z}+20 i$

[5]

Question 4

HLPaper 2

Given that $(a+i)(2-b i)=7-i$ ,

find the value of $a$ and of $b$ , where $a, b \in \mathbf{Z}$

[4]

Consider the equation $2(p+i q)=q-i p-2(1-i)$ , where $p$ and $q$ are both real numbers. Find $p$ and $q$ .

[3]

Question 5

HLPaper 1

Let the complex number $z$ be given by $z=\frac{2}{1-i}+1-4 i$

Express $z^{2}$ analytically in the form $a+b i$ , giving the exact values of $a, b$ .

[3]

Question 6

HLPaper 2

A researcher is studying the behavior of a quadratic function to model the growth of a certain plant species under varying environmental conditions. The function is given by ${f(z) = z^2 - 8z + 20}$

Calculate the discriminant ${\Delta}$ of the quadratic function ${f}$

${f(z) = z^2 - 8z + 20}$

[2]

Find the complex roots of the equation ${f(z) = 0}$ in the form ${z = a \pm bi}$ by using the quadratic formula

[3]

Rewrite ${f(z) = z^2 - 8z + 20}$ by completing the square, expressing it in the form ${f(z) = (z - h)^2 + k}$

[2]

Question 7

HLPaper 2

Let ${z = x + iy}$ . In a robotics application, find the values of ${x}$ and ${y}$ if ${(1 - i)z = 1 - 3i}$

Find the values of ${x}$ and ${y}$ . Let ${z = x + iy}$ .

[4]

Question 8

HLPaper 2

Consider the following complex problems

Evaluate the expression ${(1 + i)^2}$

[2]

Find ${(a + ai)^2}$ in terms of ${a}$ and ${i}$

[2]

Hence, find ${(a + ai)^4}$ in terms of ${a}$

[2]

Question 9

HLPaper 2

In a data analysis project, two variables ${a}$ and ${b}$ are related through the equation $(a - 2 )+3i$ = $7 + (b - 1) i$

Find the values of ${a}$ and ${b}$ , where ${a, b \in \mathbb{Z}}$

[2]

Given the equation ${c - 2 + (d - 1)i = 0}$ , find the values of ${c}$ and ${d}$ , where ${c, d \in \mathbb{Z}}$

[2]

Question 10

HLPaper 1

Consider the complex numbers ${z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}$ and $w = \frac{{{z_1}}}{{{z_2}}}$ .

By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write downthe modulus of $w$ ;

By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write downthe argument of $w$ .

Find the smallest positive integer value of $n$ , such that ${w^n}$ is a real number.

Paper

Level

Question 1

HLPaper 1

Let $z=x+y i$ be a complex number.

Given that $w=z+16$ , write down the conjugate of $w$ .

[2]

Express $|w|^{2}$ in the form $x^{2}+y^{2}+a x+b$

[2]

Express $|z+1|^{2}$ in terms of $x$ and $y$ .

[2]

Given that $|z+16|=4|z+1|$ , find the value of $|z|$ .

[4]

Question 2

HLPaper 1

Solve the following equation for $z$ , where $z$ is a complex number.

Give your answer in the form $a+b i$ , where $a$ and $b$ are both real numbers.

$\frac{z}{3+4 i}+\frac{z-1}{5 i}=\frac{5}{3-4 i}$

[4]

Question 3

HLPaper 1

Find the complex number $z$ in each of the following equations.

$(2+5 i) z+9=3+14 i$

[3]

$(2+5 i) z+9=3 z+19 i$

[3]

$(2+5 i) z+8=3 \bar{z}+20 i$

[5]

Question 4

HLPaper 2

Given that $(a+i)(2-b i)=7-i$ ,

find the value of $a$ and of $b$ , where $a, b \in \mathbf{Z}$

[4]

Consider the equation $2(p+i q)=q-i p-2(1-i)$ , where $p$ and $q$ are both real numbers. Find $p$ and $q$ .

[3]

Question 5

HLPaper 1

Let the complex number $z$ be given by $z=\frac{2}{1-i}+1-4 i$

Express $z^{2}$ analytically in the form $a+b i$ , giving the exact values of $a, b$ .

[3]

Question 6

HLPaper 2

A researcher is studying the behavior of a quadratic function to model the growth of a certain plant species under varying environmental conditions. The function is given by ${f(z) = z^2 - 8z + 20}$

Calculate the discriminant ${\Delta}$ of the quadratic function ${f}$

${f(z) = z^2 - 8z + 20}$

[2]

Find the complex roots of the equation ${f(z) = 0}$ in the form ${z = a \pm bi}$ by using the quadratic formula

[3]

Rewrite ${f(z) = z^2 - 8z + 20}$ by completing the square, expressing it in the form ${f(z) = (z - h)^2 + k}$

[2]

Question 7

HLPaper 2

Let ${z = x + iy}$ . In a robotics application, find the values of ${x}$ and ${y}$ if ${(1 - i)z = 1 - 3i}$

Find the values of ${x}$ and ${y}$ . Let ${z = x + iy}$ .

[4]

Question 8

HLPaper 2

Consider the following complex problems

Evaluate the expression ${(1 + i)^2}$

[2]

Find ${(a + ai)^2}$ in terms of ${a}$ and ${i}$

[2]

Hence, find ${(a + ai)^4}$ in terms of ${a}$

[2]

Question 9

HLPaper 2

In a data analysis project, two variables ${a}$ and ${b}$ are related through the equation $(a - 2 )+3i$ = $7 + (b - 1) i$

Find the values of ${a}$ and ${b}$ , where ${a, b \in \mathbb{Z}}$

[2]

Given the equation ${c - 2 + (d - 1)i = 0}$ , find the values of ${c}$ and ${d}$ , where ${c, d \in \mathbb{Z}}$

[2]

Question 10

HLPaper 1

Consider the complex numbers ${z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}$ and $w = \frac{{{z_1}}}{{{z_2}}}$ .

By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write downthe modulus of $w$ ;

By expressing ${z_1}$ and ${z_2}$ in modulus-argument form write downthe argument of $w$ .

Find the smallest positive integer value of $n$ , such that ${w^n}$ is a real number.