IB Math AA HL  Questionbank
Proofs
Proof by Mathematical Induction, Contradiction, Counterexample, Simple Deduction…
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Question 1
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easy
[Maximum mark: 4]
Consider two consecutive positive integers, $k$ and $k+1$.
Show that the difference of their squares is equal to the sum of the two integers.
easy
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Question 2
no calculator
easy
[Maximum mark: 4]
Prove that the sum of three consecutive positive integers is divisible by $3$.
easy
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Question 3
no calculator
easy
[Maximum mark: 4]
The product of three consecutive integers is increased by the middle integer.
Prove that the result is a perfect cube.
easy
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Question 4
no calculator
easy
[Maximum mark: 6]

Show that $(2n1)^3 + (2n+1)^3 = 16n^3+12n$ for $n \in \mathbb{Z}$. [3]

Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]
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Question 5
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easy
[Maximum mark: 6]
Using mathematical induction, prove that $1^2 + 2^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$ for all $n \in \mathbb{Z}^+$.
easy
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Question 6
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easy
[Maximum mark: 7]
Use the principle of mathematical induction to prove that
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Question 7
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easy
[Maximum mark: 6]
Let $r \in \mathbb{R}, r\neq 1$. Use the method of mathematical induction to prove that
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Question 8
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easy
[Maximum mark: 4]
Using the method of proof by contradiction, prove that $\sqrt{3}$ is irrational.
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Question 9
calculator
easy
[Maximum mark: 4]
Prove by contradiction that $\log_4 7$ is an irrational number.
easy
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Question 10
no calculator
easy
[Maximum mark: 7]
The Fibonacci sequence is defined as follows:
Prove by mathematical induction that $a_1^2+a_2^2+\cdots+a_n^2=a_na_{n+1}$, where $n\in\mathbb{Z}^+$.
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Question 11
no calculator
medium
[Maximum mark: 8]
Let $y = x^2 e^x$, for $x \in \mathbb{R}$.

Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$. [1]

Prove by mathematical induction that
$\hspace{4em} \dfrac{\mathrm{d}^n}{\mathrm{d}x^n}\big(x^2e^x\big) = \big(n(n1) + 2nx + x^2\big)e^x \hspace{1.5em} \text{for all $n \in \mathbb{Z}^+$, $n\geq2$.}$[7]
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Question 12
no calculator
medium
[Maximum mark: 9]
Let $f(x) = (x+1)e^{2x}$, $x \in \mathbb{R}$.

Find $f'(x)$. [2]

Prove by induction that $\dfrac{\mathrm{d}^nf}{\mathrm{d}x^n} = \big[n(2)^{n1} + (2)^n(x+1)\big]e^{2x}$ for all $n \in \mathbb{Z}^+$.[7]
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Question 13
no calculator
medium
[Maximum mark: 6]
Using the principle of mathematical induction, prove that $n(n^2+5)$ is divisible by $6$ for all integers $n \geq 1$.
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Question 14
calculator
hard
[Maximum mark: 8]

Solve the inequality $x^2 \geq 2x + 3$. [2]

Use mathematical induction to prove that $2^n > n^2  2$ for all $n \in \mathbb{Z}^+$, $n \geq 3$.[6]
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Question 15
calculator
hard
[Maximum mark: 8]

Solve the inequality $4x^2 \geq 4x + 2$. [2]

Use mathematical induction to prove that $3^n > 2n^2$ for all $n \in \mathbb{Z}^+$, $n \geq 2$.[6]
hard
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Question 16
no calculator
hard
[Maximum mark: 14]

Show that $\dfrac{1}{2\sqrt{n+1}} < \sqrt{n+1}  \sqrt{n}$, where $n \in \mathbb{Z},\hspace{0.1em} n\geq 0$. [3]

Hence show that $\dfrac{1}{\sqrt{2}} < 2\sqrt{2}  2$. [2]

Prove by mathematical induction that
$\hspace{4em} \sum_{r = 2}^n \dfrac{1}{\sqrt{r}} < 2\sqrt{n}  2 \hspace{2em} \text{for all $n \in \mathbb{Z}^+$, $n \geq 2$.}$[9]
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Question 17
no calculator
hard
[Maximum mark: 23]
Let $f(x) = (x1)e^{\frac{x}{3}}$, for $x \in \mathbb{R}$.

Find $f'(x)$. [2]

Prove by induction that $\dfrac{\mathrm{d}^nf}{\mathrm{d}x^n} = \bigg(\dfrac{3n + x  1}{3^n}\bigg)e^{\frac{x}{3}}$ for all $n \in \mathbb{Z}^+$. [7]

Find the coordinates of any local maximum and minimum points on the graph of $y = f(x)$. Justify whether such point is a maximum or a minimum. [5]

Find the coordinates of any points of inflexion on the graph of $y = f(x)$. Justify whether such point is a point of inflexion. [5]

Hence sketch the graph of $y = f(x)$, indicating clearly the points found in parts (c) and (d) and any intercepts with the axes. [4]
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Question 18
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]
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Question 19
no calculator
hard
[Maximum mark: 21]
Let $f(x) = \dfrac{1}{\sqrt{1x}}$,$x < 1$.

Show that $f''(x) = \dfrac{3}{4} (1x)^{5/2}$. [3]

Use mathematical induction to prove that[9]
$f^{(n)}(x) = \left(\dfrac{1}{4}\right)^n \dfrac{(2n)!}{n!} (1x)^{1/2n} \quad n\in \mathbb{Z},\enskip n\geq 2.$
Let $g(x)=\cos (mx)$, $m\in \mathbb{Q}$.
Consider the function $h$ defined by $h(x)=f(x) \times g(x)$ for $x<1$.
The $x^2$ term in the Maclaurin series for $h(x)$ has a coefficient of $\dfrac{3}{4}$.
 Find the possible values of $m$.[9]
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Question 20
no calculator
hard
[Maximum mark: 17]
The following diagram shows the graph of
$y=\arctan(2x3)+\dfrac{3\pi}{4}$ for $x\in \mathbb{R}$,
with asymptotes at $y=\dfrac{\pi}{4}$ and $y=\dfrac{5\pi}{4}$.

Describe a sequence of transformations that transforms the graph of
$y=\arctan x$ to the graph of $y=\arctan(2x3)+\dfrac{3\pi}{4}$ for $x\in \mathbb{R}$.[3]

Show that $\arctan p  \arctan q \equiv \arctan \left(\dfrac{pq}{1+pq}\right)$.[3]

Verify that $\arctan(x+2)\arctan(x+1) = \arctan\left( \dfrac{1}{(x+1)^2+(x+1)+1}\right)$.[3]

Using mathematical induction and the results from part (b) and (c), prove that[8]
$\sum_{r=1}^n \arctan\left(\dfrac{1}{r^2+r+1}\right) = \arctan(n+1)\dfrac{\pi}{4} \hspace{0.8em} \text{for } n\in \mathbb{Z}^{+}.$
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Question 21
no calculator
easy
[Maximum mark: 6]
The first three terms of an arithmetic sequence are $u_1, 4u_19$, and $3u_1+18$.

Show that $u_1=9$. [2]

Prove that the sum of the first $n$ terms of this arithmetic sequence is a square number. [4]
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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.
More IB Math AA HL Resources
Questionbank
All the questions you could need! Sorted by topic and arranged by difficulty, with mark schemes and video solutions for every question.
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Key Concepts
Helpful refreshers summarizing exactly what you need to know about the most important concepts covered in the course.
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Full worked solutions to all past paper questions, taught by experienced IB instructors.