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AHL 1.12—Complex numbers introduction - IB Questionbank | RevisionDojo

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 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 $f(x) = x^2 - 8x + 20$ .

Calculate the discriminant, $\Delta$ , of the quadratic function $f(x) = x^2 - 8x + 20$ .

[2]

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

[3]

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

[2]

Question 4

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 5

HLPaper 1

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

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

[3]

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

[3]

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

[5]

Question 6

HLPaper 2

Consider the following complex number expressions.

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 7

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 $a, b ∈ \mathbb{Z}$ .

[2]

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

[2]

Question 8

HLPaper 2

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

find the value of $a$ and of $b$ , where $a, b ∈ \mathbb{Z}$ .

[4]

Consider the equation $2(p+i q)=q-i p-2(1-i)$ , where $p, q \in \mathbb{R}$ . Find the value of $p$ and of $q$ .

[3]

Question 9

HLPaper 1

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

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

[3]

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 down the modulus of $w$ .

[3]

By expressing $z_1$ and $z_2$ in modulus-argument form write down the argument of $w$ .

[1]

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

[2]

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 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 $f(x) = x^2 - 8x + 20$ .

Calculate the discriminant, $\Delta$ , of the quadratic function $f(x) = x^2 - 8x + 20$ .

[2]

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

[3]

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

[2]

Question 4

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 5

HLPaper 1

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

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

[3]

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

[3]

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

[5]

Question 6

HLPaper 2

Consider the following complex number expressions.

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 7

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 $a, b ∈ \mathbb{Z}$ .

[2]

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

[2]

Question 8

HLPaper 2

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

find the value of $a$ and of $b$ , where $a, b ∈ \mathbb{Z}$ .

[4]

Consider the equation $2(p+i q)=q-i p-2(1-i)$ , where $p, q \in \mathbb{R}$ . Find the value of $p$ and of $q$ .

[3]

Question 9

HLPaper 1

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

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

[3]

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 down the modulus of $w$ .

[3]

By expressing $z_1$ and $z_2$ in modulus-argument form write down the argument of $w$ .

[1]

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

[2]

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 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 $f(x) = x^2 - 8x + 20$ .

Calculate the discriminant, $\Delta$ , of the quadratic function $f(x) = x^2 - 8x + 20$ .

[2]

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

[3]

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

[2]

Question 4

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 5

HLPaper 1

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

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

[3]

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

[3]

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

[5]

Question 6

HLPaper 2

Consider the following complex number expressions.

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 7

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 $a, b ∈ \mathbb{Z}$ .

[2]

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

[2]

Question 8

HLPaper 2

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

find the value of $a$ and of $b$ , where $a, b ∈ \mathbb{Z}$ .

[4]

Consider the equation $2(p+i q)=q-i p-2(1-i)$ , where $p, q \in \mathbb{R}$ . Find the value of $p$ and of $q$ .

[3]

Question 9

HLPaper 1

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

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

[3]

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 down the modulus of $w$ .

[3]

By expressing $z_1$ and $z_2$ in modulus-argument form write down the argument of $w$ .

[1]

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

[2]

AHL 1.12—Complex numbers introduction Questionbank