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

Show that

         $\sqrt{4-4x}\equiv 2(1-x{)}^{\frac{1}{2}}$

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

      $\sqrt{4-4x}$

giving each term in simplest form.

Use $x=0.02$ and your expansion from part (b) to find an approximation to  $2\sqrt{0.98}$.

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

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

giving each term in simplest form.

Explain how you could use $x=\frac{1}{32}$ in the expansion to find an approximation for $\sqrt{5}$.

There is no need to carry out the calculation.

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.

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 binomial expansion to find the first four terms in ascending powers of $x$ of

$\frac{1}{\sqrt{4-2x}}$

giving each coefficient in its simplest form.

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

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

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

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

writing each term in simplest form.

Using the answer to part (a) and using algebraic integration, estimate the value of

${\int }_{0.2}^{0.4}\frac{6x}{{\left(3+x\right)}^2}dx$

giving your answer to 4 significant figures.

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

$\frac{1}{\sqrt{4-x}}$

giving each coefficient in its simplest form.

The expansion can be used to find an approximation to $\sqrt{2}$

Possible values of $x$ that could be substituted into this expansion are

• $x=-14$ because $\frac{1}{\sqrt{4-x}}=\frac{1}{\sqrt{18}}=\frac{\sqrt{2}}{6}$

• $x=2$ because $\frac{1}{\sqrt{4-x}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

• $x=-\frac{1}{2}$ because $\frac{1}{\sqrt{4-x}}=\frac{1}{\sqrt{{\displaystyle \frac{9}{2}}}}=\frac{\sqrt{2}}{3}$

Without evaluating your expansion,

(i) state, giving a reason, which of the three values of $x$ should not be used

(ii) state, giving a reason, which of the three values of $x$ would lead to the most accurate approximation to $\sqrt{2}$

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

$\sqrt{4-9x}$

writing each term in simplest form.

A student uses this expansion with $x=\frac{1}{9}$ to find an approximation for $\sqrt{3}$

Using the answer to part (a) and without doing any calculations, state whether this approximation will be an overestimate or an underestimate of $\sqrt{3}$ giving a brief reason for your answer.

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 $q$.

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.

$\mathrm{f}\left(x\right)=\frac{50x^2+38x+9}{{\left(5x+2\right)}^2\left(1-2x\right)}x\ne -\frac{2}{5}x\ne \frac{1}{2}$

Given that $\mathrm{f}\left(x\right)$ can be expression in the form

$\frac{A}{5x+2}+\frac{B}{{\left(5x+2\right)}^2}+\frac{C}{1-2x}$

where $A$, $B$ and $C$ are constants,

(i) find the value of $B$ and the value of $C$,

(ii) show that $A=0$.

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

$\mathrm{f}\left(x\right)=p+qx+r{x}^2+...$

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

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

Use binomial expansions to show that $\sqrt{\frac{1+4x}{1-x}}\approx 1+\frac{5}{2}x-\frac{5}{8}{x}^2$.

A student substitutes $x=\frac{1}{2}$ into both sides of the approximation shown in part (a) in an attempt to find an approximation to $\sqrt{6}$.

Give a reason why the student should not use $x=\frac{1}{2}$.

Substitute $x=\frac{1}{11}$ into

$\sqrt{\frac{1+4x}{1-x}}=1+\frac{5}{2}x-\frac{5}{8}{x}^2$

to obtain an approximation to $\sqrt{6}$. Give your answer as a fraction in its simplest form.

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 $\frac{1}{\sqrt{3}}$.

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 $n$.