IB Math AA HL  Questionbank
Binomial Theorem
Binomial Expansion & Theorem, Pascal’s Triangle & The Binomial Coefficient nCr…
Paper
Difficulty
View
Question 1
no calculator
easy
[Maximum mark: 4]
Expand $(2x + 1)^4$ in descending powers of $x$ and simplify your answer.
easy
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 2
calculator
easy
[Maximum mark: 4]
Expand $(2x  3)^4$ in descending powers of $x$ and simplify your answer.
easy
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 3
no calculator
easy
[Maximum mark: 5]
Consider the expansion of $(x+2)^5$.

Write down the number of terms in this expansion. [1]

Find the term in $x^3$. [4]
easy
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 4
calculator
easy
[Maximum mark: 6]
Consider the expansion of $(2x1)^9$.

Write down the number of terms in this expansion. [1]

Find the coefficient of the term in $x^2$. [5]
easy
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 5
calculator
easy
[Maximum mark: 6]
Consider the expansion of $x(3x+2)^7$.

Write down the number of terms in this expansion. [1]

Find the coefficient of the term in $x^3$. [5]
easy
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 6
calculator
easy
[Maximum mark: 5]
The third term in the expansion of $(x+p)^8$ is $252x^6$. Find the possible values of $p$.
easy
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 7
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]
easy
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 8
no calculator
easy
[Maximum mark: 6]
Consider $\displaystyle \binom{6}{a} = \dfrac{6!}{a!\hspace{0.06em}\cdot\hspace{0.03em}4!}$ where $a \in \mathbb{Z}^+\hspace{0.1em}$.

Find the value of $a$. [2]

Determine the sum of the first three terms of $(13x)^6$ in ascending
powers of $x$, giving each term in its simplest form. [4]
easy
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 9
calculator
easy
[Maximum mark: 6]
Consider $\displaystyle \binom{7}{b} = \dfrac{7!}{b!\hspace{0.06em}\cdot\hspace{0.03em}4!}$ where $b \in \mathbb{Z}^+\hspace{0.1em}$.

Find the value of $b$. [2]

Determine the sum of the first four terms of $(1+2x)^7$ in ascending
powers of $x$, giving each term in its simplest form. [4]
easy
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 10
no calculator
easy
[Maximum mark: 5]
In the expansion of $(xk)^5$, where $k \in \mathbb{R}$, the coefficient of the term in $x^2$ is $270$.
Find the value of $k$.
easy
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 11
calculator
easy
[Maximum mark: 5]
Consider the expansion of $\left(\dfrac{x^2}{2} + \dfrac{a}{x}\right)^6$. The constant term is $960$.
Find the possible values of $a$.
easy
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 12
calculator
medium
[Maximum mark: 5]
Consider the expansion of $x\hspace{0.2em}\left(2x^2 + \dfrac{a}{x}\right)^7$. The constant term is $20\hspace{0.15em}412$. Find $a$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 13
calculator
medium
[Maximum mark: 5]
Find the term independent of $x$ in the expansion of $\dfrac{1}{x} \left(\dfrac{1}{2x}  \dfrac{x^3}{3} \right)^7$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 14
calculator
medium
[Maximum mark: 6]
Consider the expansion of $\bigg(x^3+\dfrac{2}{x}\bigg)^8$.

Write down the number of terms in this expansion. [1]

Find the coefficient of the term in $x^4$. [5]
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: 6]
In the expansion of $px^2(5 + px)^8$, the coefficient of the term in $x^6$ is $3402$. Find the value of $p$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 16
calculator
medium
[Maximum mark: 7]
Let $f(x) = (x^2 + a)^5$.
In the expansion of the derivative, $f'(x)$, the coefficient of the term in $x^5$ is $960$. Find the possible values of $a$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 17
calculator
medium
[Maximum mark: 7]

Find the term in $x^2$ in the expansion of $(2x + 1)^5$. [3]

Hence find the term in $x^3$ in the expansion of $(x+3)(2x+1)^5$. [4]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 18
calculator
medium
[Maximum mark: 6]
Consider the expansion of $\bigg(3x + \dfrac{p}{x}\bigg)^8$, where $p > 0$. The coefficient of the term
in $x^4$ is equal to the coefficient of the term in $x^6$. Find $p$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 19
no calculator
medium
[Maximum mark: 5]
In the expansion of $(2x + 1)^n$, the coefficient of the term in $x^2$ is $40n$, where $n \in \mathbb{Z}^+$. Find $n$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 20
no calculator
medium
[Maximum mark: 5]
In the expansion of $x(2x + 1)^n$, the coefficient of the term in $x^3$ is $20n$, where $n \in \mathbb{Z}^+$. Find $n$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 21
calculator
medium
[Maximum mark: 7]
Consider the expansion of $\bigg(2x^6+\dfrac{x^2}{q}\bigg)^{10}$, $q \neq 0$. The coefficient of the term
in $x^{40}$ is twelve times the coefficient of the term in $x^{36}$. Find $q$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 22
calculator
medium
[Maximum mark: 6]
Consider the expansion of $\left(2+x^3\right)^{n+2}$, where $n \in \mathbb{Z}^{+}$.
Given that the coefficient of $x^9$ is $1792$, find the value of $n$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 23
no calculator
medium
[Maximum mark: 6]

Write down and simplify the expansion of $(3x)^5$ in descending order of powers of $x$. [3]

Hence find the exact value of $(2.9)^5$. [3]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 24
no calculator
medium
[Maximum mark: 5]
Use the fractional binomial theorem to show that $\dfrac{x}{(1+x)^2} \approx x  2x^2 + 3x^3$, $x < 1$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 25
no calculator
medium
[Maximum mark: 8]
Consider the expression $\dfrac{1}{\sqrt{1+px}}+\sqrt{1+3x}$ where $p \in \mathbb{Z}$, $p\neq 0$.
The binomial expansion of this expression, in ascending powers of $x$, as far as the term in $x^2$ is $2+3x+qx^2$, where $q\in\mathbb{Q}$.

Find the value of $p$ and of $q$.[7]

State the set of possible values of $x$ for this expansion to be valid. [1]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 26
calculator
medium
[Maximum mark: 7]
Given that $(5+nx)^2\bigg(1+\dfrac{3}{5}x\bigg)^n\hspace{0.25em}=\hspace{0.05em}25+100x+\cdots$, find the value of $n$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 27
no calculator
medium
[Maximum mark: 8]
Use the fractional binomial theorem to show that $\sqrt{\dfrac{1+x}{1x}} \approx 1 + x + \dfrac{x^2}{2}$, $x < 1$.
medium
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 28
calculator
medium
[Maximum mark: 8]
Consider the identity $\dfrac{62x}{(13x)(1+x)} = \dfrac{A}{13x} + \dfrac{B}{1+x}$, where $A, B \in \mathbb{Z}$.

Find the value of $A$ and the value of $B$. [3]

Hence, expand $\dfrac{62x}{(13x)(1+x)}$ in ascending powers of $x$, up to and including the term in $x^3$. [5]
medium
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 29
calculator
hard
[Maximum mark: 7]

Write down the quadratic expression $3x^2 + 5x  2$ in the form $(axb)(x+c)$.[2]

Hence, or otherwise, find the coefficient of the term in $x^9$ in the expansion
of $(3x^2+5x2)^5$. [5]
hard
Formula Booklet
Mark Scheme
Video (a)
Video (b)
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 30
no calculator
hard
[Maximum mark: 7]
Given that $(1 + x)^3(1 + px)^4 = 1 + qx + 93x^2 + \dots + p^4x^7$, find the possible values of $p$ and $q$.
hard
Formula Booklet
Mark Scheme
Video
Revisit
Check with RV Newton
Formula Booklet
Mark Scheme
Solutions
Revisit
Ask Newton
Question 31
calculator
hard
[Maximum mark: 6]
Find the coefficient of the term in $\dfrac{1}{x}$ in the expansion of $\bigg(\dfrac{1}{2x} + 3x\bigg)^5(x + 1)^4$.
hard
Formula Booklet
Mark Scheme
Video
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.
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.
Practice Exams
Choose your revision tool! Contains topic quizzes for focused study, Revision Village mock exams covering the whole syllabus, and the revision ladder to precisely target your learning.
Key Concepts
Helpful refreshers summarizing exactly what you need to know about the most important concepts covered in the course.
Past Papers
Full worked solutions to all past paper questions, taught by experienced IB instructors.