IB Maths AA SL · Unit 1: Number and Algebra
IB Maths AA SL Binomial Theorem Questions
Exam-style IB Maths AA SL binomial theorem questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 41 questions
- Paper 1: 33
- Paper 2: 8
- 7 easy
- 16 medium
- 13 hard
- 5 very hard
- 3 worked examples
Practise Binomial Theorem questions →
AA SL formula booklet
What's examined in AA SL binomial theorem
The question bank covers these binomial theorem question types (number of questions in brackets):
- Finding Unknown Parameters (18)
- Basic Term/Coefficient (17)
- Compound & Series (6)
Key formulas
- Binomial theorem
- \((a+b)^n = \sum_{k=0}^{n}\binom{n}{k} a^{n-k} b^{k},\quad \binom{n}{k} = \dfrac{n!}{k!(n-k)!}\)
In the same notation as the IB formula booklet. All AA SL formulas →
Binomial Theorem worked examples
Worked example 1: Finding the first few terms · easy
Find the first three terms, in ascending powers of $x$, in the binomial expansion of $(3+x)^4$.
1. Identify the binomial expansion formula: $(a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots$
2. Substitute $a = 3$, $b = x$, and $n = 4$ into the first three terms: $\binom{4}{0}3^4 + \binom{4}{1}3^3(x) + \binom{4}{2}3^2(x)^2$.
3. Evaluate the binomial coefficients (from Pascal's triangle or calculator): $1$, $4$, and $6$.
4. Calculate the constant powers: $1(81) + 4(27)(x) + 6(9)(x^2)$.
5. Simplify to state the final three terms: $\mathbf{81 + 108x + 54x^2}$.
Examiner tip: A common mistake is forgetting to apply the combination coefficient $\binom{n}{r}$ entirely, simply writing the terms as $a^n + a^{n-1}b + a^{n-2}b^2$. Always check against Pascal's triangle!
Worked example 2: Finding a specific coefficient · medium
The coefficient of $x^7$ in the expansion of $x^3(ax + 3)^5$ is $1215$. Find the possible values of the real constant $a$.
1. Deduce that to get an $x^7$ term overall, we must find the $x^4$ term from the expansion of $(ax+3)^5$, since it is multiplied by $x^3$ outside the bracket.
2. Write the general term for $(ax+3)^5$: $\binom{5}{r}(ax)^{5-r}(3)^r$.
3. Set the power of $x$ to $4$, meaning $5-r = 4 \implies r = 1$.
4. Substitute $r=1$ into the general term to find the $x^4$ expression: $\binom{5}{1}(ax)^4(3)^1 = 5(a^4x^4)(3) = 15a^4x^4$.
5. Equate the coefficient to $1215$: $15a^4 = 1215$.
6. Solve for $a$: $a^4 = 81 \implies \mathbf{a = \pm 3}$.
Examiner tip: When dealing with terms like $(ax)^4$, students frequently forget to apply the power to the constant $a$, writing $ax^4$ instead of $a^4x^4$.
Worked example 3: Solving for the power $n$ · hard
In the expansion of $\left(\frac{1}{2}x + 1\right)^n$, the coefficient of the $x^2$ term is $8n$, where $n \in \mathbb{Z}^+$. Find the exact value of $n$.
1. Write the formula for the $x^2$ term using the general expansion: $\binom{n}{2}\left(\frac{1}{2}x\right)^2(1)^{n-2}$.
2. Expand the binomial coefficient algebraically: $\binom{n}{2} = \frac{n(n-1)}{2!}$.
3. Simplify the $x^2$ term expression: $\frac{n(n-1)}{2} \times \frac{1}{4}x^2 = \frac{n^2 - n}{8}x^2$.
4. Equate this coefficient to the given value of $8n$: $\frac{n^2 - n}{8} = 8n$.
5. Rearrange into a quadratic equation: $n^2 - n = 64n \implies n^2 - 65n = 0 \implies n(n - 65) = 0$.
6. Solve for $n$. Since $n$ must be a positive integer, $n \neq 0$, so $\mathbf{n = 65}$.
Examiner tip: Many students struggle with expanding $\binom{n}{2}$ algebraically. Remember it is simply $\frac{n(n-1)}{2}$. Do not divide by $n$ to solve the quadratic; always factorise to avoid losing roots, even if $n=0$ is later rejected.
Try these IB Maths AA SL binomial theorem questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 3 marks · Paper 1
Find the first three terms, in ascending powers of \(x\), in the expansion of \((9-2x)^5\).
Question 2 · medium · 5 marks · Paper 1
In the expansion of \((3+px)^6\), the coefficient of the \(x^4\) term is four times the coefficient of the \(x^2\) term. Find the possible values of \(p\).
Question 3 · hard · 6 marks · Paper 2
Consider the expansion of \(\left(2x + \frac{k}{x}\right)^9\), where \(k > 0\). The coefficient of the term in \(x^3\) is equal to the coefficient of the term in \(x^5\). Find the exact value of \(k\).
All 41 binomial theorem questions with mark schemes →
FAQ
How many IB Maths AA SL binomial theorem questions are there?
There are 41 exam-style binomial theorem questions in the AA SL question bank (Paper 1: 33 · Paper 2: 8), graded 7 easy, 16 medium, 13 hard, 5 very hard. Every question has a full IB-style mark scheme (M, A and R marks).
Is binomial theorem on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 33 · Paper 2: 8. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA SL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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