General Binomial Expansion (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

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

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

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

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

               $\sqrt{1+2x}$ 

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

Using a suitable value of $x$, use your expansion from part (a) to estimate $\sqrt{1.06}$, giving your answer to 3 significant figures.

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

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

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

Use the binomial expansion to show that the first three terms in the expansion of  $(1+2x{)}^{-3}$ are  $1-6x+24x^2.$

Hence, or otherwise, find the expansion of  $(1+x)(1+2x{)}^{-3}$ up to and including the term in $x^2$.

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

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

where $s$ is an integer.

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

(ii) Find the coefficient of the term in $x^2$ in the binomial expansion of  $\mathrm{f}\left(x\right)$, in terms of $s$.

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 $s$.

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

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

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

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

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

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

In the 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 $n$.

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

Use the binomial expansion to find the first three terms, in ascending powers of $x$, in each of  ${\left(1-x\right)}^{-1}$ and ${\left(1+x\right)}^{-1}$

Hence show that $\frac{2}{\left(1-x\right)\left(1+x\right)}\approx 2+2x^2$

Write down the values of $x$ for which your expansion in part (c) converges.

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

        ${\left(1-\frac{1}{3}x\right)}^{-1}{\left(2-x\right)}^{-2}\approx \frac{1}{4}+\frac{1}{3}x+\frac{43}{144}{x}^2$

For which values of $x$ is the approximation in part (a) valid?

(i) Use your calculator to find the exact fraction of ${\left(1-\frac{1}{3}x\right)}^{-1}$ ${\left(2-x\right)}^{-2}$  when $x=0.5$

(ii) Use your calculator to find the fraction from the approximation $\frac{1}{4}+\frac{1}{3}x+\frac{43}{144}{x}^2$ when $x=0.5$

(iii) Find the percentage error in the approximation, giving your answer to two decimal places.

 It is given that

            $\mathrm{f}\left(x\right)=\sqrt{9+px}$      and       $g\left(x\right)=\sqrt[4]{16+px}$

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

Find the value of $p$.

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

Using binomial expansions, up to and including terms in $x^2$ show that $\frac{12-x}{\left(x+2\right)\left(3-x\right)}\approx 2-\frac{1}{2}x+\frac{5}{12}{x}^2$

Explain why the approximation in part (b) is only valid for $\left|x\right|<2.$

Find the first three terms in the binomial expansion of  $\frac{1}{1-\frac{1}{2}x.}$

Show that the expansion is valid for $\left|x\right|<2.$

The expansion is to be used in a computer program to estimate the value of  $\frac{20}{19}$.
Find the value of $x$ to be used and check it meets the validity requirement from part (b).

Hence find the value the computer program will use to estimate  $\frac{20}{19}$.

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

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

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

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

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

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

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

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

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

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

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

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

In the expansion of  $\frac{1}{{\left(3+px\right)}^3}$  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 as follows

               $\mathrm{f}\left(x\right)={\left(4+3x\right)}^{\frac{1}{2}}\mathrm{g}\left(x\right)={\left(9-2x\right)}^{-\frac{1}{2}}$

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

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

Find the expansion of $\sqrt{\frac{4+3x}{9-2x}}$  in ascending powers of $x$, up to and including the term in $x^2$.

Find the 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 real number, the coefficient of the term in $x^2$ is $-\frac{16}{81}$.

Find the possible values of $n$.

Express $\frac{4+5x-x^2}{\left(1-x\right){\left(1+x\right)}^2}$  in partial fractions.

Use the binomial expansion to find the first three terms, in ascending powers of $x$, in each of ${\left(1-x\right)}^{-1}$, ${\left(1+x\right)}^{-1}$, and ${\left(1+x\right)}^{-2}$.

Hence express  $\frac{4+5x-x^2}{\left(1-x\right){\left(1+x\right)}^2}$ as the first three terms of a binomial expansion in ascending powers of  $x$.

Write down the values of $x$ for which your expansion in part (c) converges.

Given that $x$ is small such that $x^3$ 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 percentage error between your calculator answer and the approximation in part (a) when $x=0.1$, giving your answer to one decimal place.

For which values of $x$ is the approximation in part (a) valid?

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

Show that $p=-8q$.

Given further that $pq=-8$ find the values of $p$ and $q$.

Express $\frac{1-7x}{\left(x+2\right)\left(3-x\right)}$  in the form  $\frac{A}{x+2}+\frac{B}{3-x}$, where $A$ and $B$ are integers to be found.

Hence, or otherwise, find the binomial expansion of $\frac{1-7x}{\left(x+2\right)\left(3-x\right)}$, in ascending powers of $x$, up to and including the term in $x^2$.

The expansion in part (b) is to be used to approximate the value of a fraction.

(i) If $x=0.1$, which fraction is being approximated?

(ii) Which fraction does the approximation give?

Find the first four terms in the binomial expansion of  $\frac{1}{2-3x}.$

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

The expansion is to be used in a computer program to estimate the value of $\frac{5}{7}$.
Check that the expansion is valid for this purpose and use the first four terms of the expansion to estimate the value of $\frac{5}{7}$.

Find the percentage error the computer program will introduce by using the expansion as an approximation to  $\frac{5}{7}$

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 $x^3$.

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 answer to two decimal places.

Explain why you would not be able to use your expansion to approximate $\frac{1}{\sqrt{3}}$.

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

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

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

Expand $(1-\frac{1}{2}x)(9+3x{)}^{-\frac{1}{2}}$ up to and including the term in $x^2$.

In the expansion of   $\frac{1}{{\left(8+2qx\right)}^{\frac{1}{3}}}$,  the coefficient of the term in $x^2$ is one-seventh of the coefficient of the term in $x^3$.  Find the value of $q$.

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

               $\mathrm{f}\left(x\right)=8-x\mathrm{g}\left(x\right)=8+2x$     

Find the binomial expansion of   $\sqrt[3]{\frac{\mathrm{f}\left(x\right)}{\mathrm{g}\left(x\right)}}$, in ascending powers of $x$, up to and including the term in $x^2$.  Also find the values of $x$for which your expansion is valid.

In the expansion of  ${\left(16-2x\right)}^n$, where $n$ is a real number, the coefficient of the term in $x^2$ is ${16}^n\times \frac{5}{2048}$.

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

Express  $\frac{2\left(2-5x+x^2\right)}{\left(x+2\right){\left(2-x\right)}^2}$  in partial fractions.

Express $\frac{2\left(2-5x+x^2\right)}{\left(x+2\right){\left(2-x\right)}^2}$  as the first three terms of a binomial expansion in ascending powers of $x$.

Write down the values of $x$ for which your expansion in part (b) converges.

Given that $x$ is small such that $x^3$ and higher powers of $x$ can be ignored show that   
       $(4-3x{)}^{-2}(2-x{)}^{-3}\approx \frac{1}{128}+\frac{3}{128}x+\frac{87}{2048}{x}^2$

Find the percentage error between your calculator answer and the approximation in part (a) when $x=0.2$, giving your answer to one decimal place.

For which values of $x$ is the approximation in part (a) valid?

It is given that

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

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

(i) 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 $\mathrm{g}\left(x\right)$.

(ii) 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 $b$.

Find the binomial expansion of $\frac{15}{\left(x-4\right){\left(5x-2\right)}^{\text{'}}}$, in ascending powers of $x$, up to and including the term in $x^2$.

Explain why the expansion found in part (a) cannot be used when $x=0.6$.

The binomial expansion of

      $\frac{1}{\sqrt{4-2x}}$

is to be used in a computer program to estimate the reciprocal of $\sqrt{3.8}$.

The computer program needs to be accurate to at least 5 significant figures when compared to the value produced by a scientific calculator.

Find the least number of terms from the expansion that are required for the computer program.  Justify that the expansion used is valid.