Binomial Theorem

[Maximum mark: 4]

Expand $(2x + 1)^4$ in descending powers of $x$ and simplify your answer.

[Maximum mark: 4]

Expand $(2x - 3)^4$ in descending powers of $x$ and simplify your answer.

[Maximum mark: 5]

Consider the expansion of $(x+2)^5$.

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

2. Find the term in $x^3$. [4]
   

[Maximum mark: 6]

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

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

2. Find the coefficient of the term in $x^2$. [5]
   

[Maximum mark: 6]

Consider the expansion of $x(3x+2)^7$.

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

2. Find the coefficient of the term in $x^3$. [5]
   

[Maximum mark: 5]

The third term in the expansion of $(x+p)^8$ is $252x^6$. Find the possible values of $p$.

[Maximum mark: 6]

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

2. Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]
   

[Maximum mark: 6]

Consider $\displaystyle \binom{6}{a} = \dfrac{6!}{a!\hspace{-0.06em}\cdot\hspace{-0.03em}4!}$ where $a \in \mathbb{Z}^+\hspace{-0.1em}$.

1. Find the value of $a$. [2]
   

2. Determine the sum of the first three terms of $(1-3x)^6$ in ascending
   powers of $x$, giving each term in its simplest form. [4]
   

[Maximum mark: 6]

Consider $\displaystyle \binom{7}{b} = \dfrac{7!}{b!\hspace{-0.06em}\cdot\hspace{-0.03em}4!}$ where $b \in \mathbb{Z}^+\hspace{-0.1em}$.

1. Find the value of $b$. [2]
   

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]
   

[Maximum mark: 5]

In the expansion of $(x-k)^5$, where $k \in \mathbb{R}$, the coefficient of the term in $x^2$ is $-270$.

Find the value of $k$.

[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 $a$.

[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 $a$.

[Maximum mark: 5]

Find the term independent of $x$ in the expansion of $\dfrac{1}{x} \left(\dfrac{1}{2x} - \dfrac{x^3}{3} \right)^7$.

[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. [1]
   

2. Find the coefficient of the term in $x^4$. [5]
   

[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 $p$.

[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 $a$.

[Maximum mark: 7]

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

2. Hence find the term in $x^3$ in the expansion of $(x+3)(2x+1)^5$. [4]
   

[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 $p$.

[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 $n$.

[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 $n$.

[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 $q$.

[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 $n$.

[Maximum mark: 6]

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

2. Hence find the exact value of $(2.9)^5$. [3]
   

[Maximum mark: 5]

Use the fractional binomial theorem to show that $\dfrac{x}{(1+x)^2} \approx x - 2x^2 + 3x^3$, $|x| < 1$.

[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 $q\in\mathbb{Q}$.

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

2. State the set of possible values of $x$ for this expansion to be valid. [1]
   

[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 $n$.

[Maximum mark: 8]

Use the fractional binomial theorem to show that $\sqrt{\dfrac{1+x}{1-x}} \approx 1 + x + \dfrac{x^2}{2}$, $|x| < 1$.

[Maximum mark: 8]

Consider the identity $\dfrac{6-2x}{(1-3x)(1+x)} = \dfrac{A}{1-3x} + \dfrac{B}{1+x}$, where $A, B \in \mathbb{Z}$.

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

2. Hence, expand $\dfrac{6-2x}{(1-3x)(1+x)}$ in ascending powers of $x$, up to and including the term in $x^3$. [5]
   

[Maximum mark: 7]

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

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

[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 $q$.

[Maximum mark: 6]

Find the coefficient of the term in $\dfrac{1}{x}$ in the expansion of $\bigg(\dfrac{1}{2x} + 3x\bigg)^5(x + 1)^4$.