Binomial Expansion (Edexcel A Level Maths: Pure): Exam Questions

Find, in simplest form, the coefficient of $x^5$ in the expansion of

$\left(5+8x^2\right){\left(3-\frac{1}{2}x\right)}^6$

In the binomial expansion of

${\left(a+2x\right)}^7$where $a$ is a constant

the coefficient of $x^4$ is 15120

Find the value of a.

In the binomial expansion of

$(p+x{)}^{12}$

the coefficient of $x^5$ is 12 976 128.

Find the value of $p$.

Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of

$(3+2x{)}^8$

giving each term in simplest form.

Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of

$(5+2x{)}^5$

giving each term in simplest form.

 Use your answer to part (a) to estimate the value of $(5.04{)}^5$.

Expand

$(2x-3{)}^6$

giving your answer in descending powers of $x$.

In the binomial expansion of

$(p+x{)}^4$

where $p$ is a non-zero constant, the coefficient of $x^2$ is twice the coefficient of $x$. 

Find the value of $p$, giving your answer as a simplified fraction.

In the binomial expansion of

$(a-x{)}^4$

the coefficient of $x^2$ is 96.

Given that $a>0$, find the value of $a$.

Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of

$(9-2x{)}^5$

giving each term in simplest form.

Use your answer to part (a) to estimate $(8.9{)}^5$.

In the binomial expansion of

$(a-2x{)}^5$

the coefficient of $x^2$ is equal to the coefficient of $x^3$. 

Find the non-zero value of $a$.

In the binomial expansion of

$(3+px{)}^6$

the coefficient of $x^4$ is four times the coefficient of $x^2$.

Find the possible non-zero values of $p$.

In the binomial expansion of

$(p+qx{)}^5$

where $p\ne 0$ and $q\ne 0$ the coefficients of $x^2$ and the coefficient of $x^3$ are equal.

Find an expression for $p$ in terms of $q$.

Find the first 3 terms, in ascending powers of $x$, of the binomial expansion of

$(2-3x{)}^7$

giving each term in simplest form.

Given that $x$ is small, so that $x^3$and higher powers of $x$ can be ignored, it can be shown that

$(1-2x)(2-3x{)}^7\approx 128+ax+b{x}^2$

where $a$ and $b$ are integers.

Find $a$ and $b$.

In the binomial expansion of

$(4-px{)}^6$

the coefficient of $x^4$ is 19 440.

Given that $p$ is a positive integer, find the value of $p$.

Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of ${\left(2-\frac{x}{2}\right)}^8$, giving each term in its simplest form.

Use your expansion to estimate the value of ${(1.95)}^8$, giving your answer to 3 decimal places.

In the binomial expansion of

${\left(a+2x\right)}^6$

the coefficient of $x^2$ is the same as the coefficient of $x^3$.

Find the non-zero value of $a$.

Expand

${\left(2-\frac{1}{3}x\right)}^4$

giving your answer in ascending powers of $x$.

Expand

$(3-2x{)}^5$

giving your answer in ascending powers of $x$.

Find the coefficient of $x^4$ in the expansion of

$(4-3x{)}^7$

In the binomial expansion of

${\left(m-\frac{1}{4}x\right)}^5$

the coefficient of $x^3$ is $-10$.

Find the possible values of $m$.

Find the first $4$ terms, in ascending powers of $x$, of the binomial expansion of

${\left(3-\frac{2x}{9}\right)}^8$

giving each term in simplest form.

$\mathrm{f}\left(x\right)=\left(\frac{x-1}{2x}\right){\left(3-\frac{2x}{9}\right)}^8$

Find the coefficient of $x^2$ in the series expansion of $\mathrm{f}\left(x\right)$, giving your answer as a simplified fraction.

$\mathrm{g}\left(x\right)={\left(2+ax\right)}^8$where $a$ is a constant

Given that one of the terms in the binomial expansion of $\mathrm{g}\left(x\right)$ is $3402x^5$

find the value of $a$.

Using this value of $a$,

find the constant term in the expansion of

$\left(1+\frac{1}{x^4}\right){\left(2+ax\right)}^8$

In the binomial expansion of

$(3a-2x{)}^6$ 

the coefficient of $x^3$ is equal to the coefficient of $x^4$. 

Find the non-zero value of $a$.

In the binomial expansion of

${(p+qx)}^8$

where $p\ne 0$ and $q\ne 0$ the coefficient of $x^2$ is equal to the coefficient of $x^6$.

Find the two possible expressions for $p$ in terms of $q$.

In the binomial expansion of

$(a+bx{)}^4$

the coefficient of $x^2$ is equal to the coefficient of $x^3$.

Given that $a$ and $b$ are non-zero, find the value of $\frac{a}{b}$

Given that $a$ and $b$ are integers, and that $10<b<15$, find the possible values of $a$ and $b$.

Use the formula

${}^{n}C_r=\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n!}{r!\left(n-r\right)!}$

to prove that

      ${}^{k}C_1=k$

for all positive integer values of $k$.

In the binomial expansion of 

${\left(3a+\frac{1}{2}x\right)}^6$

the coefficient of $x^3$ is equal to the coefficient of $x^5$.  

Find the non-zero values of $a$, giving your answers in the form $\frac{\sqrt{m}}{n}$ where $m$ and $n$ are integers to be found.

In the binomial expansion of $(a+bx{)}^4,$ the coefficient of $x^3$ is 216.

In the binomial expansion of $(a+bx{)}^6,$ the coefficient of $x^4$ is 4860.

Find the possible values of $a$ and $b$.

Use the first 3 terms, in ascending powers of $x$, of the expansion of 

$(3-5x{)}^4$

to find an approximation for $(2.6{)}^4$.

Find the percentage error in the approximation from part (a) to the exact value of $(2.6{)}^4$

Solutions relying on calculator technology are not acceptable.

Given that

${}^{n}C_3=35$

use algebra to show that $n$ satisfies the cubic equation

$n\left(n-1\right)\left(n-2\right)=7\times 6\times 5$

and hence write down the value of $n$.

In the binomial expansion of

$(1+x{)}^n$

where $n$ is a positive integer greater than 3, the coefficient of $x^3$ is 84.

Use algebra to show that $n$ satisfies

$\left(n-9\right)\left(n^2+pn+q\right)=0$

where $p$ and $q$ are integers to be found.

Hence, find all possible values of $n$.

Given that $x$ is a very small value, so that $x^3$ (and higher powers of $x$) can be ignored, show that

            $(3-p{x}^2)(4-3x{)}^9\approx q+rx+15400960x^2$

where $p$, $q$ and $r$ are integers to be found.

In the binomial expansion of

$(1-3x{)}^n$

the coefficient of $x^3$ is -3240.

Use algebra to find the value of $n$.

In the binomial expansion of $(a+bx{)}^8,$ the coefficient of $x^5$ is -870 912.

In the binomial expansion of $(a+bx{)}^{12},$ the coefficient of $x^3$ is -1 557 135 360.

Find the possible values of $a$ and $b$.