IB Math AA HL  Questionbank
Complex Numbers
Different Forms, Roots, De Moivre’s Theorem, Argand Diagram, Geometric Applications…
Paper
Difficulty
View
Question 1
no calculator
easy
[Maximum mark: 6]
Solve the equation $z^3 = 1$, giving your answers in Cartesian form.
easy
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 2
no calculator
easy
[Maximum mark: 6]
Consider the complex number $z = \dfrac{w_1}{w_2}$ where $w_1 = \sqrt{2} + \sqrt{6}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $w_2 = 3 + \sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

Express $w_1$ and $w_2$ in modulusargument form and write down

the modulus of $z$;

the argument of $z$. [4]


Find the smallest positive integer value of $n$ such that $z^n$ is a real number. [2]
easy
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 3
no calculator
medium
[Maximum mark: 7]
The complex numbers $w$ and $z$ satisfy the equations
Find $w$ and $z$ in the form $a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{Z}$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 4
no calculator
medium
[Maximum mark: 6]

Verify that $1+2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $12{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ are the second roots of $3+4{\mathrm{\hspace{0.05em}i}\mkern 1mu}$. [2]

Hence, find two distinct roots of the equation $z^2  3z + (3{\mathrm{\hspace{0.05em}i}\mkern 1mu}) = 0$, $z \in \mathbb{C}$,
giving your answers in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$. [4]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 5
no calculator
medium
[Maximum mark: 6]
Consider the equation $\dfrac{3z}{5z^{*}}=\text{i}$, where
$z=x+\text{i}y$ and $x$, $y \in \mathbb{R}$.
Find the value of $x$ and the value of $y$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 6
no calculator
medium
[Maximum mark: 7]
Consider the complex numbers $u = 1 + 2 {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $v = 2 + {\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

Given that $\dfrac{1}{u} + \dfrac{1}{v} = \dfrac{6\sqrt{2}}{w}$, express $w$ in the form $a + b {\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a,b \in \mathbb{R}$. [4]

Find $w^\ast$ and express it in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$. [3]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 7
no calculator
medium
[Maximum mark: 9]

Find three distinct roots of the equation $z^3 + 64 = 0$, $z \in \mathbb{C}$, giving your answers in modulusargument form. [6]
The roots are represented by the vertices of a triangle in an Argand diagram.
 Show that the area of the triangle is $12\sqrt{3}$.
[3]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 8
calculator
medium
[Maximum mark: 5]
Consider $z=\cos\theta+{\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin\theta$ where $z \in \mathbb{C}$, $z\neq 0$.
Show that $\dfrac{1}{2z^2}+\left(\dfrac{1}{2z^2}\right)^\ast = \cos(2\theta)$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 9
no calculator
medium
[Maximum mark: 7]
Consider the equation $2z^4 + az^3 + bz^2 +cz + d = 0$, where $a, b, c, d \in \mathbb{R}$ and $z \in \mathbb{C}$. Two of the roots of the equation are $\log_2 10$ and ${\mathrm{\hspace{0.05em}i}\mkern 1mu}\sqrt{5}$ and the sum of all the roots is $4 + \log_25$.
Show that $15a + d + 90 = 0$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 10
calculator
medium
[Maximum mark: 7]
Consider the complex numbers
where $k\in\mathbb{Z}^{+}$.
 Express $z$ and $w$ in modulusargument form and write down

the modulus of $zw$.

the argument of $zw$ in terms of $k$.[3]

Suppose that $zw\in \mathbb{Z}$.


Find the minimum value of $k$.

For the value of $k$ found in part (i), find the value of $zw$.[4]

medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 11
no calculator
medium
[Maximum mark: 12]
Consider the complex numbers $z_1 = 3 \mathop{\mathrm{cis}}(\ang{120})$ and $z_2 = 2 + 2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

Calculate $\dfrac{z_1}{z_2}$ giving your answer both in modulusargument form and
Cartesian form. [7]

Use your results from part (a) to find the exact value of $\sin \ang{15}\cdot\,\sin \ang{45} \cdot\,\sin \ang{75}$,
giving your answer in the form $\dfrac{\sqrt{a}}{b}$ where $a, b \in \mathbb{Z}^+$. [5]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 12
no calculator
medium
[Maximum mark: 18]

Express $4 + 4\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r > 0$ and $ \pi < \theta \leq \pi$. [5]
Let the roots of the equation $z^3 = 4 + 4\sqrt{3}{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ be $z_1$, $z_2$ and $z_3$.

Find $z_1$, $z_2$ and $z_3$ expressing your answers in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r > 0$ and $\pi < \theta \leq \pi$. [5]
On an Argand diagram, $z_1$, $z_2$ and $z_3$ are represented by the points A, B and C, respectively.

Find the area of the triangle ABC. [4]

By considering the sum of the roots $z_1$, $z_2$ and $z_3$, show that
$\cos\Big(\dfrac{2\pi}{9}\Big) + \cos\Big(\dfrac{4\pi}{9}\Big) + \cos\Big(\dfrac{8\pi}{9}\Big) = 0$[4]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 13
no calculator
medium
[Maximum mark: 8]
Find the six distinct roots of the equation $z^6+(18{\mathrm{\hspace{0.05em}i}\mkern 1mu})z^38{\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0$, $z \in \mathbb{C}$, giving your answers in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 14
no calculator
medium
[Maximum mark: 7]
Consider the polynomial equation $z^4 + 10z^3 + 50z^2 + 130z + 169 = 0$ where $z \in \mathbb{C}$.
Two of the roots of this equation are $a + {\mathrm{\hspace{0.05em}i}\mkern 1mu}b$ and $b + {\mathrm{\hspace{0.05em}i}\mkern 1mu}a$ where $a,b \in \mathbb{Z}$.

Write down the other two roots in terms of $a$ and $b$. [1]

Find the possible values of $a$. [6]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 15
calculator
medium
[Maximum mark: 7]
Two distinct roots for the polynomial equation $z^4  12z^3 + 57z^2  120z + 100 =0$ are $a  1 + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $a + 1  2b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $z \in \mathbb{C}$ and $a,b \in \mathbb{Z}$.

Write down the other two roots in terms of $a$ and $b$. [1]

Find the value of $a$. [2]

Hence, or otherwise, find the four roots of the polynomial. [4]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 16
no calculator
hard
[Maximum mark: 15]
Consider $w = \dfrac{z  1}{z + {\mathrm{\hspace{0.05em}i}\mkern 1mu}}$ where $z = x + {\mathrm{\hspace{0.05em}i}\mkern 1mu}y$ and ${\mathrm{\hspace{0.05em}i}\mkern 1mu}= \sqrt{1}$.

If $z = {\mathrm{\hspace{0.05em}i}\mkern 1mu}$,

write $w$ in the form $r\mathop{\mathrm{cis}}\theta$;

find the value of $w^{14}$. [5]


Show that in general,
$w = \dfrac{(x^2  x + y^2 + y) + {\mathrm{\hspace{0.05em}i}\mkern 1mu}(y  x + 1)}{x^2 + (y + 1)^2}$[4]

Find condition under which $\mathrm{Re}(w) = 1$. [2]

State condition under which $w$ is:

real;

purely imaginary. [2]


Find the modulus of $z$ given that $\arg w = \dfrac{\pi}{4}$. [2]
hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 17
no calculator
hard
[Maximum mark: 19]


Expand $(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)^4$ by using the binomial theorem.

Hence use de Moivre's theorem to prove that
$\begin{aligned} \cos 4\theta = \cos^4 \theta  6\cos^2 \theta\sin^2 \theta + \sin^4 \theta. \\ \end{aligned}$ 
State a similar expression for $\sin 4 \theta$ in terms of $\cos \theta$ and $\sin \theta$. [6]

Let
$z = r(\cos \alpha + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha)$,
where $\alpha$ is measured in degrees, be the solution
of $z^4  {\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0$ which has the
smallest positive argument.

Find the modulus and argument of $z$. [4]

Use (a) (ii) and your answer from (b) to show that $8\cos^4\alpha  8\cos^2 \alpha + 1 = 0$. [4]

Hence express $\cos \ang{22.5}$ in the form $\dfrac{\sqrt{a + b\sqrt{c}}}{d}$ where $a,b,c,d \in \mathbb{Z}$. [5]
hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 18
no calculator
hard
[Maximum mark: 19]
Let $z = \cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta$, for $\dfrac{\pi}{4} < \theta < \dfrac{\pi}{4}$.


Find $z^3$ using the binomial theorem.

Use de Moivre's theorem to show that $\cos 3\theta = 4\cos^3\theta  3\cos \theta$ and $\sin 3\theta = 3\sin\theta4\sin^3\theta$. [8]


Hence show that $\dfrac{\sin 3\theta  \sin \theta}{\cos 3\theta + \cos \theta} = \tan \theta$. [6]

Given that $\sin \theta = \dfrac{1}{3}$, find the exact value of $\tan 3\theta$. [5]
hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 19
no calculator
hard
[Maximum mark: 22]

Solve $2\sin(x+\ang{120}) = \sqrt{3}\cos(x + \ang{60})$, for $x \in [0,\ang{180}]$. [5]

Show that $\sin \ang{75} + \cos \ang{75} = \dfrac{\sqrt{6}}{2}$. [3]

Let $z = \sin 4\theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}(1  \cos 4\theta)$, for $z \in \mathbb{C}$, $\theta \in [0,\ang{90}]$.

Find the modulus and argument of $z$ in terms of $\theta$.

Hence find the fourth roots of $z$ in modulusargument form. [14]

hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 20
no calculator
hard
[Maximum mark: 17]

Solve the equation $z^3 = 27$, $z \in \mathbb{C}$, giving your answer in the form
$z = r(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)$ and in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a,b \in \mathbb{R}$. [6]

Consider the complex numbers $z_1 = 1 + {\mathrm{\hspace{0.05em}i}\mkern 1mu} \text{ and } z_2 =\dfrac{1}{\sqrt{2}}\bigg[\mathrm{cos}\bigg(\dfrac{\pi}{3}\bigg)+\mathrm{i}\,\mathrm{sin}\bigg(\dfrac{\pi}{3}\bigg)\bigg]$ .

Write $z_1$ in the form $r(\cos \theta + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \theta)$.

Calculate $z_1z_2$ and write in the form $a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a,b \in \mathbb{R}$.

Hence find the value of $\tan\left(\dfrac{\pi}{12}\right)$ in the form $c + d\sqrt{3}$ where $c,d \in \mathbb{Z}$.

Find the smallest $p \in \mathbb{Q}^+$ such that $(z_1z_2)^p$ is a positive real number. [11]

hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 21
no calculator
hard
[Maximum mark: 16]

Find the roots of $z^{16} = 1$ which satisfy the condition $0 < \arg(z) < \dfrac{\pi}{2}$,
expressing your answer in the form $re^{{\mathrm{\hspace{0.05em}i}\mkern 1mu}\theta}$, where $r, \theta \in \mathbb{R}^+$. [5]

Let $S$ be the sum of the roots found in part (a).

Show that $\mathrm{Re}(S) = \mathrm{Im}(S)$.

By writing $\dfrac{\pi}{8}$ as $\dfrac{1}{2}\cdot\dfrac{\pi}{4}$, find the value of $\cos \Big(\dfrac{\pi}{8}\Big)$ in the form $\dfrac{\sqrt{a + \sqrt{b}}}{c}$,
where $a, b$ and $c$ are integers to be determined.

Hence, or otherwise, show that $S = \dfrac{1}{2}\big(\hspace{0.1em}\sqrt{2 + \sqrt{2}} + \sqrt{2} + \sqrt{2  \sqrt{2}}\hspace{0.1em}\big)\big(1+{\mathrm{\hspace{0.05em}i}\mkern 1mu}\big)$. [11]

hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 22
no calculator
hard
[Maximum mark: 18]


Verify that $1+{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $1{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ are the second roots of $2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$.

Find two distinct roots of the equation $z^2  2z + (12i) = 0$, $z \in \mathbb{C}$, giving your answers in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$. [5]


Find six distinct roots of the equation $z^6+(8{\mathrm{\hspace{0.05em}i}\mkern 1mu})z^38{\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0$, $z \in \mathbb{C}$, giving your answers in the form $z = a + b{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ where $a, b \in \mathbb{R}$. [8]
On an Argand diagram, $z_1 = \sqrt{3}+{\mathrm{\hspace{0.05em}i}\mkern 1mu}$, $z_2 = \sqrt{3} +{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ and $z_3 = 2{\mathrm{\hspace{0.05em}i}\mkern 1mu}$ are represented by the points A, B and C, respectively.


Verify that $z_1$, $z_2$ and $z_3$ are the roots of the equation $z^3  8{\mathrm{\hspace{0.05em}i}\mkern 1mu}= 0$, $z \in \mathbb{C}$.

Show that the area of the triangle ABC is $3\sqrt{3}$. [5]

hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 23
no calculator
hard
[Maximum mark: 20]

Solve the equation $\sin (x + \ang{90}) = 2\cos(x  \ang{60})$, $\ang{0} < x < \ang{360}$. [5]

Show that $\sin \ang{15} + \cos \ang{15} = \dfrac{\sqrt{6}}{2}$. [4]

Let $z = 1  \cos 4\theta  {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin 4\theta$, for $z \in \mathbb{C}$, $0 < \theta < \dfrac{\pi}{2}$.

Find the modulus and argument of $z$. Express each answer
in its simplest form. 
Hence find the fourth roots of $z$ in modulusargument form. [11]

hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 24
no calculator
hard
[Maximum mark: 21]

Use de Moivre's theorem to find the value of $\left[\cos\left(\dfrac{\pi}{6}\right) + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \left(\dfrac{\pi}{6}\right)\right]^{12}$. [2]

Use mathematical induction to prove that
$\hspace{3.5em} (\cos \alpha  {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha)^n = \cos (n\alpha)  {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin (n\alpha) \hspace{1em} \text{for all } n \in \mathbb{Z}^+.$[6]
Let $w = \cos \alpha + {\mathrm{\hspace{0.05em}i}\mkern 1mu}\sin \alpha$.

Find an expression in terms of $\alpha$ for $w^n  (w^\ast)^n$, $n \in \mathbb{Z}^+$, where $w^\ast$ is the complex conjugate of $w$. [2]


Show that $ww^\ast = 1$.

Write down and simplify the binomial expansion of $(w  w^\ast)^3$ in terms of $w$ and $w^\ast$.

Hence show that $\sin (3\alpha) = 3\sin \alpha  4 \sin^3 \alpha$. [5]


Hence solve $4\sin^3\alpha + (2 \cos \alpha  3) \sin \alpha = 0$ for $0 \leq \alpha \leq \pi$. [6]
hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Thank you Revision Village Members
#1 IB Math Resource
Revision Village was ranked the #1 IB Math Resources by IB Students & Teachers in 2021 & 2022.
34% Grade Increase
Revision Village students scored 34% greater than the IB Global Average in their exams (2021).
70% of IB Students
More and more IB students are using Revision Village to prepare for their IB Math Exams.
Frequently Asked Questions
What is the IB Math AA HL Questionbank?
The IB Math Analysis and Approaches (AA) HL Questionbank is a comprehensive set of IB Mathematics exam style questions, categorised into syllabus topic and concept, and sorted by difficulty of question. The bank of exam style questions are accompanied by high quality stepbystep markschemes and video tutorials, taught by experienced IB Mathematics teachers. The IB Mathematics AA HL Question bank is the perfect exam revision resource for IB students looking to practice IB Math exam style questions in a particular topic or concept in their AA Higher Level course.
Where should I start in the AA HL Questionbank?
The AA HL Questionbank is designed to help IB students practice AA HL exam style questions in a specific topic or concept. Therefore, a good place to start is by identifying a concept that you would like to practice and improve in and go to that area of the AA HL Question bank. For example, if you want to practice AA HL exam style questions that involve Complex Numbers, you can go to AA SL Topic 1 (Number & Algebra) and go to the Complex Numbers area of the question bank. On this page there is a carefully designed set of IB Math AA HL exam style questions, progressing in order of difficulty from easiest to hardest. If you’re just getting started with your revision, you could start at the top of the page with Question 1, or if you already have some confidence, you could start at the medium difficulty questions and progress down.
How should I use the AA HL Questionbank?
The AA HL Questionbank is perfect for revising a particular topic or concept, indepth. For example, if you wanted to improve your knowledge of Counting Principles (Combinations & Permutations), there is a set of full length IB Math AA HL exam style questions focused specifically on this concept. Alternatively, Revision Village also has an extensive library of AA HL Practice Exams, where students can simulate the length and difficulty of an IB exam with the Mock Exam sets, as well as AA HL Key Concepts, where students can learn and revise the underlying theory, if missed or misunderstood in class.
What if I finish the AA HL Questionbank?
With an extensive and growing library of full length IB Math Analysis and Approaches (AA) HL exam style questions in the AA HL Question bank, finishing all of the questions would be a fantastic effort, and you will be in a great position for your final exams. If you were able to complete all the questions in the AA HL Question bank, then a popular option would be to go to the AA HL Practice Exams section on Revision Village and test yourself with the Mock Exam Papers, to simulate the length and difficulty of an actual IB Mathematics AA HL exam.