Subjects

# Binomial Theorem

Binomial Expansion & Theorem, Pascal’s Triangle & The Binomial Coefficient nCr…

Paper

Paper 1
Paper 2

Difficulty

Easy
Medium
Hard

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

##### 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

##### Question 3

calculator

easy

[Maximum mark: 6]

Consider the expansion of $(2x-1)^9$.

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

2. Find the coefficient of the term in $x^2$. 

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 4

calculator

easy

[Maximum mark: 5]

Consider the expansion of $(x-3)^8$.

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

2. Find the coefficient of the term in $x^6$. 

easy

Formula Booklet

Mark Scheme

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Revisit

##### Question 5

no calculator

easy

[Maximum mark: 6]

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

2. Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. 

easy

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### 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

##### Question 7

calculator

medium

[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 $a$.

medium

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 8

calculator

medium

[Maximum mark: 6]

Consider the expansion of $\dfrac{(x+a)^7}{bx}$, where $a > 0$. The coefficient of the term in $x^5$ is $2$, and the coefficient of the term in $x^3$ is $1690$.

Find the value of $a$ and the value of $b$.

medium

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 9

calculator

medium

[Maximum mark: 6]

Consider the expansion of $\bigg(x^3+\dfrac{2}{x}\bigg)^8$.

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

2. Find the coefficient of the term in $x^4$. 

medium

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 10

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

##### Question 11

calculator

hard

[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 $p$.

hard

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 12

calculator

hard

[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 $a$.

hard

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 13

calculator

hard

[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 $q$.

hard

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 14

no calculator

hard

[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 $n$.

hard

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 15

no calculator

hard

[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 $n$.

hard

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 16

no calculator

hard

[Maximum mark: 6]

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

2. Hence find the exact value of $(2.9)^5$. 

hard

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 17

calculator

hard

[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 $n$.

hard

Formula Booklet

Mark Scheme

Video

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

##### Question 18

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

##### Question 19

calculator

hard

[Maximum mark: 7]

1. Write down the quadratic expression $3x^2 + 5x - 2$ in the form $(ax-b)(x+c)$.

2. Hence, or otherwise, find the coefficient of the term in $x^9$ in the expansion
of $(3x^2+5x-2)^5$. 

hard

Formula Booklet

Mark Scheme

Video (a)

Video (b)

Revisit

Check with RV Newton

Formula Booklet

Mark Scheme

Solutions

Revisit

Thank you Revision Village Members

## #1 IB Math Resource

Revision Village is ranked the #1 IB Math Resources by IB Students & Teachers.

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.