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

It is given that

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

and

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

where $a$ is a non-zero constant.

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

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

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

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

Hence, or otherwise, find the expansion of

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

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

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

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

where $s$ is a non-zero integer.

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

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

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

Find the value of $s$.

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

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

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

(i) Expand $\text{f}\left(x\right)$ 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 $\text{g}\left(x\right)$ 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 $\left|x\right|<2$.

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

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

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

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

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

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

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

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

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

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

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

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

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

and

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

where $p$ is a non-zero constant.

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

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 $p=-8q$.

Given further that $pq=-8$, 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 $\text{f}\left(x\right)$ and $\text{g}\left(x\right)$ are given as follows.

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

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

Find the binomial expansion of

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

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

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

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

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

Express

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

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

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

and

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

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

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

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

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

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.