General Binomial Expansion (OCR A Level Maths A: Pure): Exam Questions

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

               $\sqrt{1+2x}$ 

giving each term in simplest form.

State the range of values of $x$ for which the expansion in part (a) is valid.

By choosing a suitable value of $x$, use your expansion from part (a) to estimate $\sqrt{1.06}$

Give your estimate to 3 significant figures.

Find, in ascending powers of $x$, the binomial expansion of

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

up to and including the term in $x^3$.

Give each term in simplest form.

The function $\mathrm{f}\left(x\right)$ is given by

           $\mathrm{f}\left(x\right)=(1-px{)}^{-4}$

where $p$ is an integer.

Find, in terms of $p$, the coefficient of the term in $x^3$ in the binomial expansion of $\mathrm{f}\left(x\right)$.

Find, in ascending powers of $x$, the binomial expansion of

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

up to and including the term in $x^3$.

Give each term in simplest form.

Given that

         $\frac{5-x}{\left(1+x\right)\left(1-x\right)}\equiv \frac{A}{1+x}+\frac{B}{1-x}$

find the values of $A$ and $B$.

Find the first three terms, in ascending powers of $x$, of the binomial expansions of

(i)  $3(1+x{)}^{-1}$

(ii) $2(1-x{)}^{-1}$

Hence show that the first three terms, in ascending powers of $x$, in the binomial expansion of

            $\frac{5-x}{\left(1+x\right)\left(1-x\right)}$

are

               $5-x+5x^2$

Find the range of values of $x$ for which the expansion of $\frac{5-x}{\left(1+x\right)\left(1-x\right)}$ converges.

Use the binomial expansion to show that the first three terms in the expansion of  $(1+2x{)}^{-3}$ are  

$1-6x+k{x}^2$

where $k$ is a constant to be found.

Hence find the first three terms, in ascending powers of $x$, of the binomial expansion of

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

giving each term in simplest form.

State also the range of values of $x$ for which the expansion is valid.

The function $\mathrm{f}\left(x\right)$ is given by

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

(i) Expand $\mathrm{f}\left(x\right)$ in ascending powers of $x$ up to and including the term in $x^2$.

(ii) Find the range of values of $x$ for which this expansion is valid.

The function $\mathrm{g}\left(x\right)$ is given by

$\mathrm{g}\left(x\right)=(2+x{)}^{-2}$

(i) Expand $\mathrm{g}\left(x\right)$ in ascending powers of $x$ up to and including the term in $x^2$.

(ii) Find the range of values of $x$ for which this expansion is valid.

(i) Find the expansion of $\frac{\sqrt{1-\frac{1}{2}x}}{{\left(2+x\right)}^2}$ in ascending powers of $x$, up to and including the term in $x^2$.

(ii) Find the range of values of $x$ for which this expansion is valid.

The function $\mathrm{f}\left(x\right)$ is given by

            $\mathrm{f}\left(x\right)=\sqrt{4-sx}$

where $s$ is a non-zero integer.

In the binomial expansion of $\mathrm{f}\left(x\right)$, find in terms of $s$

(i) the coefficient of the term in $x$

(ii) Find the coefficient of the term in $x^2$

In the binomial expansion of $\mathrm{f}\left(x\right)$, the coefficient of the term in $x$ is equal to the coefficient of the term in $x^2$.

Find the value of $s$.

Two functions are given by

$\mathrm{f}\left(x\right)=\sqrt{1+ax}$

$\mathrm{g}\left(x\right)=\sqrt[3]{1-ax}$

where $a$ is a non-zero constant.

In their binomial expansions, the coefficient of the term in $x^2$ from $\mathrm{f}\left(x\right)$ is equal to the coefficient of the term in $x$ from $\mathrm{g}\left(x\right)$.

Find the value of $a$.

Express $\frac{2}{\left(1-x\right)\left(1+x\right)}$ in partial fractions.

Find the first three terms, in ascending powers of $x$, of the binomial expansions of

(i)  ${\left(1-x\right)}^{-1}$

(ii) ${\left(1+x\right)}^{-1}$

Hence show that 

$\frac{2}{\left(1-x\right)\left(1+x\right)}=\alpha +\beta x^2+...$

where $\alpha$ and $\beta$ are constants to be found.

Find the range of values of $x$ for which the expansion in part (c) is valid.

Find, in ascending powers of $x$, the binomial expansion of

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

up to and including the term in $x^3$.

Give each term in simplest form.

Find, in ascending powers of $x$, the binomial expansion of

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

up to and including the term in $x^3$.

Give each term in simplest form.

Use the binomial expansion to expand ${\left(1-\frac{1}{2}x\right)}^{\frac{1}{3}}$  up to and including the term in $x^2$.

Give each term in simplest form.

Hence expand  $\left(1-x\right){\left(1-\frac{1}{2}x\right)}^{\frac{1}{3}}$ up to and including the term in $x^2$.

Find, in ascending powers of $x$, the binomial expansion of

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

up to and including the term in $x^3$.

Give each term in simplest form.

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

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

to estimate the value of $\sqrt[3]{1.2}$, giving your estimate to 3 significant figures.

Explain why your estimate in part (a) is valid.

In the binomial expansion of  ${\left(1-\frac{1}{4}x\right)}^n$ where $n$ is a negative integer, the coefficient of the term in $x^2$ is $\frac{3}{8}$.

Find the value of $n$.

A function is given by

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

Given that $x$ is small, such that terms in $x^3$ and higher powers of $x$ can be ignored, show that

        $\mathrm{f}\left(x\right)\approx \frac{1}{4}+\frac{1}{3}x+k{x}^2$

where $k$ is an exact constant to be found.

Find the range of values of $x$ for which the expansion in part (a) is valid.

Find, to 3 significant figures, the percentage error when using the approximation in part (a) to estimate $\mathrm{f}\left(\frac{1}{2}\right)$.

Show clear working.

Two functions are given by

$\mathrm{f}\left(x\right)=\sqrt{9+px}$

$\mathrm{g}\left(x\right)=\sqrt[4]{16+px}$

where $p$ is a non-zero constant.

In their binomial expansions, the coefficient of the term in $x^2$ from $\mathrm{f}\left(x\right)$ is equal to the coefficient of the term in $x$ from $\mathrm{g}\left(x\right)$.

Find the value of $p$.

In the binomial expansion of $\frac{1}{{\left(3+px\right)}^3}$where $p\ne 0$, the coefficient of the term in $x^2$ is double the coefficient of the term in $x^3$. 

Find the value of  $p$.

The functions $\mathrm{f}\left(x\right)$ and  $\mathrm{g}\left(x\right)$ are given by

$\mathrm{f}\left(x\right)={\left(4+3x\right)}^{\frac{1}{2}}$

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

Find the first three terms, in ascending powers of $x$, of the binomial expansion of $\mathrm{f}\left(x\right)$.

Find the first three terms, in ascending powers of $x$, of the binomial expansion of $\mathrm{g}\left(x\right)$.

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

$\sqrt{\frac{4+3x}{9-2x}}$ 

giving each term in simplest form.

Find the range of values of $x$ for which your expansion in part (c) is valid.

In the expansion of  ${\left(1-\frac{4}{3}x\right)}^n$ where $n$ is a rational number, the coefficient of the term in $x^2$ is $-\frac{16}{81}$.

Find the possible values of $n$.

Given that $x$ is small, so that terms in $x^3$ and higher powers of $x$ can be ignored, show that

         ${(2+3x)}^{-1}{(3-2x)}^{-2}\approx \frac{1}{18}-\frac{1}{108}x+\frac{19}{216}{x}^2$

Find the range of values of $x$ for which the approximation in part (a) is valid.

Find, to 1 decimal place, the percentage error when using the approximation in part (a) to estimate the value of $\frac{1}{(2+3x){(3-2x)}^2}$ at $x=0.1$

Express $\frac{12-x}{\left(x+2\right)\left(3-x\right)}$  in partial fractions.

Hence use binomial expansions to show that

$\frac{12-x}{\left(x+2\right)\left(3-x\right)}=2-\frac{1}{2}x+m{x}^2+...$

where $m$ is a constant to be found.

Find the range of validity of $x$ for the expansion in part (b).

In the binomial expansion of  $\sqrt{4+\frac{p}{q}x}$   where $p<0<q$, the coefficient of the term in $x^2$ is equal to the coefficient of the term in $x^3$.

Show that $p=-8q$.

Given that the product of $p$ and $q$ is $-8$, find the values of $p$ and $q$.

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

         $\frac{1}{\sqrt{1-\frac{1}{2}x}}$

to estimate the value of $\frac{1}{\sqrt{0.95}}$, giving your estimate to 2 decimal places.

Explain why you would not be able to use the expansion in part (a) to estimate $\frac{1}{\sqrt{3}}$.

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

$\frac{1-{\displaystyle \frac{x}{2}}}{\sqrt{9+3x}}$

giving each term in simplest form.

In the binomial expansion of   $\frac{1}{\sqrt[3]{8+2qx}}$ where $q\ne 0$,  the coefficient of the term in $x^2$ is one-seventh of the coefficient of the term in $x^3$. 

Find the value of $q$.

Expand

$\sqrt[3]{\frac{8-x}{8+2x}}$

in ascending powers of $x$, up to and including the term in $x^2$. 

Find also the range of values of $x$ for which this expansion is valid.

Two functions are given by

 $\mathrm{f}\left(x\right)=\sqrt{4+ax}$ 

$\mathrm{g}\left(x\right)=\sqrt[4]{16+bx}$

where $a$ and $b$ are non-zero constants.

The binomial expansions of  $\mathrm{f}\left(x\right)$  and  $\mathrm{g}\left(x\right)$ have the following properties:

• The coefficient of the $x^3$ term in the expansion of $\mathrm{f}\left(x\right)$ is 72 times larger than the coefficient of the $x^2$ term in the expansion of $\mathrm{g}\left(x\right)$

• The coefficient of the $x$ term in the expansion of $\mathrm{f}\left(x\right)$ is 24 times larger than the coefficient of the $x$ term in the expansion of $\mathrm{g}\left(x\right)$

Find the values of $a$ and $b$.

Use binomial expansions to show that, in ascending powers of $x$

$15{\left(x-4\right)}^{-1}{\left(5x-2\right)}^{-1}=a+bx+c{x}^2+...$

where $a$, $b$ and $c$ are constants to be found.

Explain why the expansion found in part (a) cannot be used to estimate the value of

$\frac{15}{\left(0.6-4\right)\left(5\times 0.6-2\right)}$

Use binomial expansions to show that, in ascending powers of $x$,

$\frac{2\left(2-5x+x^2\right)}{\left(x+2\right){\left(2-x\right)}^2}=\alpha +\beta x+\gamma x^2+...$

where $\alpha$, $\beta$ and $\gamma$ are constants to be found.

Find the range of values of $x$ for which the expansion is valid.

In the expansion of ${\left(16-2x\right)}^n$ where $n$ is a rational number, the coefficient of the term in $x^2$ is

$5\times 2^{4n-11}$

Given that $\left|n\right|<1$, find the value of $n$.