General Binomial Expansion (Edexcel IAL Maths): Exam Questions

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

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

up to and including the term in $x^{2}$.

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

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

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

Show that

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

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

$\sqrt{4-4x}$

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

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

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

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

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

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

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

Find the coefficient of the term in $x^{2}$ in the binomial expansion of

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

The function $f(x)$ is given by

$f(x)=(1-px)^{-4}$

where $p$ is an integer.

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

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

$(1-\frac{1}{3}x)^{-2}\approx1+\frac{2}{3}x+\frac{1}{3}x^{2}$

Using a suitable value of $x$ in the result from part (a), find an approximation for the value of $(0.94)^{-2}$.

It is given that

$\mathrm{f} ( x ) = \sqrt{1 + a x} a n d g ( x ) = \sqrt[3]{1 - a x}$

where $a$ is a non-zero constant.

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

Find the value of $a$.

Show, as partial fractions, that

$\frac{5 - x}{( 1 + x ) ( 1 - x )} \equiv \frac{3}{1 + x} + \frac{2}{1 - x}$

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

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

are

$5-x+5x^{2}$

Write down the values of $x$ for which this expansion converges.

Find the first three terms in the binomial expansion of $( 1 - x )^{- 1}$.

Write down the values of $x$ the expansion is valid for.

The first three terms of the expansion are to be used in a computer program to estimate the value of $\frac{1}{0.95}$.

Choose an appropriate value of $x$ to use in the expansion and thus find the value the computer program will use to estimate $\frac{1}{0.95}$.

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

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

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

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

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 $f(x)$ is given by

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

where $s$ is an integer.

(i) Find the coefficient of the term in $x$ in the binomial expansion of $f(x)$, in terms of $s$.

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

In the binomial expansion of $f(x),$ 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 $f(x)$ and $g(x)$ are given as follows

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

(i) Expand $f(x)$, 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 $g(x),$ 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 $\frac{ \sqrt{1 - 1 \frac{1}{2} x} }{( 2 + x )^{2}}$ 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 $( 1 - 1 \frac{1}{4} x )^{n}$, where $n$ is a negative integer, the coefficient of the term in $x^{2}$ is $\frac{3}{8}$.

Find the value of $n$.

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

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

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

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

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

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

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

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

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

Find the value of $p$.

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

Using binomial expansions, up to and including terms in $x^{2}$ show that

$\frac{12 - x}{( x + 2 ) ( 3 - x )} \approx 2 - 1 \frac{1}{2} x + \frac{5}{12} x^{2}$

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

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

Show that the expansion is valid for open vertical bar x close vertical bar less than 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

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

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

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

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

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

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

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

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

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

In the expansion of $\frac{1}{( 3 + p x )^{3}}$ 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 $f(x)$ and $g(x)$ are given as follows

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

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

Expand $\mathrm{g} ( x )$, 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 $( 1 - \frac{4}{3} x )^{n}$ , 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 $\frac{4 + 5 x - x^{2}}{( 1 - x ) ( 1 + x )^{2}}$ in partial fractions.

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

Hence express $\frac{4 + 5 x - x^{2}}{( 1 - x ) ( 1 + x )^{2}}$ 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 $\frac{1 - 7 x}{( x + 2 ) ( 3 - x )}$ 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 $\frac{1 - 7 x}{( x + 2 ) ( 3 - x )}$, 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

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

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

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

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 $\frac{1}{( 8 + 2 q x )^{1 \frac{1}{3}}}$, 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 $\mathrm{f} ( x )$ and $\mathrm{g} ( x )$ are given as follows

$f(x)=8-xg(x)=8+2x$

Find the binomial expansion of $\sqrt[3]{ \frac{ \mathrm{f} ( x )}{ \mathrm{g} ( x )} }$, 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 $( 16 - 2 x )^{n}$, where $n$ is a real number, the coefficient of the term in $x^{2}$ is $16^{n}\times\frac{5}{2048}$.

Given that $\vert n \vert < 1$ find the value of $n$.

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

Express $\frac{2 ( 2 - 5 x + x^{2} )}{( x + 2 ) ( 2 - x )^{2}}$ 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

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

The binomial expansions of $f(x)$ and $g(x)$ have the following properties:

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

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

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

Find the binomial expansion of $\frac{15}{( x - 4 ) ( 5 x - 2 )^{'} }$, 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.