General Binomial Expansion (Cambridge International A-Level Mathematics): Exam Questions

Expand

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

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

Expand

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

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

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

Show that

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

Hence obtain the expansion of

$\sqrt{4-4x}$

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

Expand

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

in ascending powers of $x$, up to and including the term in $x^{3}$, simplifying the coefficients.

Expand

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

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

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

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

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

The function $\text{f} ( x )$ is given by

$\text{f} ( x ) = ( 1 - p x )^{-4}$

where $p$ is an integer.

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

Expand

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

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

Express

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

in partial fractions.

Expand each of the following in ascending powers of $x$, up to and including the term in $x^{2}$, simplifying the coefficients.

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

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

Hence obtain the expansion of

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

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

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

Expand

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

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

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

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 obtain the expansion of

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

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

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

It is given that

$\text{f} ( x ) = \sqrt{1+ax}$

and

\text{g}(x) = \sqrt[3]{1 - ax}

where $a$ is a non-zero constant.

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

Find the value of $a$.

Expand

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

in ascending powers of $x$, up to and including the term in $x^{3}$, simplifying the coefficients.

Expand

$\sqrt{1+2x}$

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

State the set of values of $x$ for which the 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.

Expand

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

in ascending powers of $x$, up to and including the term in $x^{3}$, simplifying the coefficients.

Expand

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

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

Hence, or otherwise, find the expansion of

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

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

The function $\text{f} ( x )$ is given by

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

where $s$ is a non-zero integer.

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

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

In the binomial expansion of $\text{f} ( x )$, the coefficient of $x$ is equal to the coefficient of $x^{2}$.

Find the value of $s$.

The functions $\text{f} ( x )$ and $\text{g} ( x )$ are given as follows.

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

$\text{g} ( x ) = ( 2 + x )^{-2}$

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

(ii) State the set of values of $x$ for which the expansion is valid.

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

(ii) State the set of values of $x$ for which the expansion is valid.

(i) Find the expansion of

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

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

(ii) State the set of values of $x$ 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 $x^{2}$ is $\frac{3}{8}$.

Find the value of $n$.

Expand

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

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

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

Express

$\frac{12-x}{(x+2)(3-x)}$

in partial fractions.

Hence obtain the expansion of

$\frac{12-x}{(x+2)(3-x)}$

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

Explain why the expansion in part (b) is only valid for |x| < 2.

Expand

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

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

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

Expand

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

in ascending powers of $x$, up to and including the term in $x^{3}$, simplifying the coefficients.

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

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

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

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

Expand

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

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

Hence, or otherwise, expand

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

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

The functions $\text{f} ( x )$ and $\text{g} ( x )$ are given as follows.

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

$\text{g} ( x ) = ( 9 - 2 x )^{-\frac{1}{2}}$

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

Expand $\text{g} ( x )$ 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 $x^{2}$.

State the set of values of $x$ for which the 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 $x^{2}$ is $- \frac{16}{81}$.

Find the possible values of $n$.

Express

$\frac{4+5x-x^{2}}{(1-x)(1+x)^{2}}$

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 obtain the expansion of

$\frac{4+5x-x^{2}}{(1-x)(1+x)^{2}}$

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

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

Expand

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

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

Find the percentage error in using your expansion from part (a) to approximate the value of $( 2 + 3 x )^{-1} ( 3 - 2 x )^{-2}$ when $x = 0 . 1$, giving your answer to 1 decimal place.

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

Express

$\frac{1-7x}{(x+2)(3-x)}$

in partial fractions.

Hence obtain the expansion of

$\frac{1-7x}{(x+2)(3-x)}$

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?

Expand

$\frac{1}{2-3x}$

in ascending powers of $x$, up to and including the term in $x^{3}$, simplifying the coefficients.

State the set of values of $x$ for which the expansion in part (a) 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 $\frac{5}{7}$.

Expand

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

in ascending powers of $x$, up to and including the term in $x^{3}$, simplifying the coefficients.

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 2 decimal places.

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

Expand

\left(1 - \frac{1}{2}x\right)(9 + 3x)^{-\frac{1}{2}}

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

Expand

$( 4 - 3 x )^{-2} ( 2 - x )^{-3}$

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

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

It is given that

$\text{f} ( x ) = \sqrt{9+px}$

and

\text{g}(x) = \sqrt[4]{16 + px}

where $p$ is a non-zero constant.

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

Find the value of $p$.

Expand

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

in ascending powers of $x$, up to and including the term in $x^{3}$, simplifying the coefficients.

In the expansion of

$\frac{1}{(3+px)^{3}}$

where $p$ is a non-zero constant, the coefficient of $x^{2}$ is double the coefficient of $x^{3}$.

Find the value of $p$.

In the binomial expansion of

$\sqrt{4+\frac{p}{q}x}$

where $p < 0 < q$, the coefficient of $x^{2}$ is equal to the coefficient of $x^{3}$.

Show that $p = - 8 q$.

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

In the expansion of

$\frac{1}{(8+2qx)^{\frac{1}{3}}}$

where $q$ is a non-zero constant, the coefficient of $x^{2}$ is one-seventh of the coefficient of $x^{3}$.

Find the value of $q$.

The functions $\text{f} ( x )$ and $\text{g} ( x )$ are given as follows.

$\text{f} ( x ) = 8 - x$

$\text{g} ( x ) = 8 + 2 x$

Find the binomial expansion of

\sqrt[3]{\frac{\text{f}(x)}{\text{g}(x)}}

in ascending powers of $x$, up to and including the term in $x^{2}$, and state the set of values of $x$ for which the expansion is valid.

In the expansion of

$( 16 - 2 x )^{n}$

where $n$ is a real number, the coefficient of $x^{2}$ is $16^{n} \times \frac{5}{2048}$.

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

Express

\frac{2\left(2 - 5x + x^{2}\right)}{(x + 2)(2 - x)^{2}}

in partial fractions.

Hence obtain the expansion of

\frac{2\left(2 - 5x + x^{2}\right)}{(x + 2)(2 - x)^{2}}

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

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

Find the binomial expansion of

$\frac{15}{(x-4)(5x-2)}$

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 $x = 0 . 6$.

Expand

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

in ascending powers of $x$, up to and including the term in $x^{3}$, simplifying the coefficients.

It is given that

$\text{f} ( x ) = \sqrt{4+ax}$

and

\text{g}(x) = \sqrt[4]{16 + bx}

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

The binomial expansions of $\text{f} ( x )$ and $\text{g} ( x )$ have the following properties.

The coefficient of $x^{3}$ in the expansion of $\text{f} ( x )$ is 72 times the coefficient of $x^{2}$ in the expansion of $\text{g} ( x )$.

The coefficient of $x$ in the expansion of $\text{f} ( x )$ is 24 times the coefficient of $x$ in the expansion of $\text{g} ( x )$.

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

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 with the value produced by a scientific calculator.

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