IB Mathematics AA HL  Questionbank
Proofs
Proof by Mathematical Induction, Contradiction, Counterexample, Simple Deduction…
Question Type
All
Paper
Difficulty
View
Question 1
[Maximum mark: 4]
Prove that the sum of three consecutive positive integers is divisible by $3$.
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 2
[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.
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 3
[Maximum mark: 4]
The product of three consecutive integers is increased by the middle integer.
Prove that the result is a perfect cube.
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 4
[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]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 5
[Maximum mark: 6]
Using mathematical induction, prove that $1^2 + 2^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$ for all $n \in \mathbb{Z}^+$.
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 6
[Maximum mark: 5]

Prove that $\dfrac{5}{x^2} = \dfrac{5}{x(x2)}\dfrac{10}{x^2(x2)}$. [3]

Determine the set of numbers $x$ for which the proof in part (a) is valid. [2]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 7
[Maximum mark: 7]
Use the principle of mathematical induction to prove that
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 8
[Maximum mark: 4]
Using the method of proof by contradiction, prove that $\sqrt{3}$ is irrational.
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 9
[Maximum mark: 6]
Let $r \in \mathbb{R}, r\neq 1$. Use the method of mathematical induction to prove that
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 10
[Maximum mark: 6]
Prove by contradiction that the equation $3x^37x^2+5=0$ has no integer roots.
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 11
[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 $n\in\mathbb{Z}^+$.
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 12
[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]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 13
[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]
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 14
[Maximum mark: 6]
Using the principle of mathematical induction, prove that $n(n^2+5)$ is divisible by $6$ for all integers $n \geq 1$.
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 15
[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]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 16
[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]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 17
[Maximum mark: 28]
This question asks you to explore the sequence defined by
where $\alpha$ and $\beta$ are the roots of the quadratic equation $x^24x+1=0, \, \alpha > \beta$ and $n \in \mathbb{Z}^+$.

Find the value of $\alpha$ and the value of $\beta$. Give your answers in the form $a \pm \sqrt{b}$, where $a,b \in \mathbb{Z^+}$.[3]

Hence find the values of $u_1$ and $u_2$. [4]

Show that $\alpha^2 = 4\alpha 1$ and $\beta^2 = 4\beta  1$. [1]

Hence show that $u_{n+2} = 4u_{n+1}u_n$.[4]

Suppose that $u_n$ and $u_{n+1}$ are integers. Show that $u_{n+2}$ is also an integer.[2]

Hence show that $u_n$ is an integer for all $n \in \mathbb{N}$.[2]
Now consider the sequence defined by

Find the exact values of $v_1$ and $v_2$.[4]

Express $v_{n+2}$ in terms of $v_{n+1}$ and $v_n$.[4]

Hence show that $v_n$ is a multiple of $\dfrac{\sqrt{3}}{3}$ for all $n \in \mathbb{N}$.[4]
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Video (f)
Video (g)
Video (h)
Video (i)
Revisit
Check with RV Newton
Mark Scheme
Solutions
Revisit
Ask Newton
Question 18
[Maximum mark: 24]
This question asks you to investigate some properties of hexagonal numbers.
Hexagonal numbers can be represented by dots as shown below where $h_n$ denotes the $n$th hexagonal number, $n\in \mathbb{N}$.
Note that $6$ points are required to create the regular hexagon $h_2$ with side of length $1$, while $15$ points are required to create the next hexagon $h_3$ with side of length $2$, and so on.

Write down the value of $h_5$.[1]

By examining the pattern, show that $h_{n+1} = h_{n}+4n+1$, $n\in \mathbb{N}$. [3]

By expressing $h_n$ as a series, show that $h_n = 2n^2n$, $n\in \mathbb{N}$.[3]

Hence, determine whether $2016$ is a hexagonal number.[3]

Find the least hexagonal number which is greater than $80\hspace{0.10em}000$.[5]

Consider the statement:
$45$ is the only hexagonal number which is divisible by $9$.
Show that this statement is false.[2]
Matt claims that given $h_1 = 1$ and $h_{n+1} = h_n + 4n + 1$, $n \in \mathbb{N}$, then
 Show, by mathematical induction, that Matt's claim is true
for all $n\in \mathbb{N}$.[7]
Mark Scheme
Video (a)
Video (b)
Video (c)
Video (d)
Video (e)
Video (f)
Video (g)
Revisit
Check with RV Newton
Mark Scheme
Solutions
Revisit
Ask Newton
Question 19
[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]
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 20
[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]
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 21
[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]
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Video (c)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 22
[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]
$\sum_{r=1}^n \arctan\left(\dfrac{1}{r^2+r+1}\right) = \arctan(n+1)\dfrac{\pi}{4} \hspace{0.8em} \text{for } n\in \mathbb{Z}^{+}.$
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
Thank you Revision Village Members
#1 IB Math Resource
Revision Village is ranked the #1 IB Math Resources by IB Students & Teachers.
34% Grade Increase
Revision Village students scored 34% greater than the IB Global Average in their exams (2021).
80% of IB Students
More and more IB students are using Revision Village to prepare for their IB Math Exams.
More IB Math AA HL Resources
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.