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

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

giving each term in simplest form.

Explain how you could use in the expansion to find an approximation for .

There is no need to carry out the calculation.

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

giving each term in simplest form.

State the range of values of  for which the expansion in part (a) is valid.

By choosing a suitable value of , use your expansion from part (a) to estimate

Give your estimate to 3 significant figures.

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Give each term in simplest form.

The function  is given by

where  is an integer.

Find, in terms of , the coefficient of the term in  in the binomial expansion of .

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Give each term in simplest form.

Given that

find the values of and .

Find the first three terms, in ascending powers of , of the binomial expansions of

(i)  

(ii)

Hence show that the first three terms, in ascending powers of , in the binomial expansion of

are

Find the range of values of  for which the expansion of converges.

Use the binomial expansion to show that the first three terms in the expansion of  are  

where is a constant to be found.

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

giving each term in simplest form.

State also the range of values of for which the expansion is valid.

The function is given by

(i) Expand  in ascending powers of  up to and including the term in .

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

The function is given by

(i) Expand  in ascending powers of  up to and including the term in .

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

(i) Find the expansion of  in ascending powers of , up to and including the term in .

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

The function  is given by

where  is a non-zero integer.

In the binomial expansion of , find in terms of

(i) the coefficient of the term in 

(ii) Find the coefficient of the term in

In the binomial expansion of , the coefficient of the term in  is equal to the coefficient of the term in .

Find the value of .

Two functions are given by

where  is a non-zero constant.

In their binomial expansions, the coefficient of the term in  from  is equal to the coefficient of the term in from .

Find the value of .

Express in partial fractions.

Find the first three terms, in ascending powers of , of the binomial expansions of

(i)  

(ii) 

Hence show that 

where and are constants to be found.

Find the range of values of  for which the expansion in part (c) is valid.

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Give each term in simplest form.

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Give each term in simplest form.

Use the binomial expansion to expand   up to and including the term in .

Give each term in simplest form.

Hence expand   up to and including the term in .

Find, in ascending powers of , the binomial expansion of

up to and including the term in .

Give each term in simplest form.

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 , of the binomial expansion of

writing each term in simplest form.

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

giving your answer to 4 significant figures.

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

giving each coefficient in its simplest form.

The expansion can be used to find an approximation to

Possible values of that could be substituted into this expansion are

• because

• because

• because

Without evaluating your expansion,

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

(ii) state, giving a reason, which of the three values of would lead to the most accurate approximation to

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

writing each term in simplest form.

A student uses this expansion with to find an approximation for

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

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

to estimate the value of , giving your estimate to 3 significant figures.

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

In the binomial expansion of  where  is a negative integer, the coefficient of the term in  is .

Find the value of .

A function is given by

Given that  is small, such that terms in and higher powers of  can be ignored, show that

where is an exact constant to be found.

Find the range of values of  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 .

Show clear working.

Two functions are given by

where  is a non-zero constant.

In their binomial expansions, the coefficient of the term in  from  is equal to the coefficient of the term in from .

Find the value of .

In the binomial expansion of where , the coefficient of the term in  is double the coefficient of the term in . 

Find the value of  .

The functions and   are given by

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

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

Find the first three terms, in ascending powers of , of the expansion of

giving each term in simplest form.

Find the range of values of  for which your expansion in part (c) is valid.

In the expansion of   where  is a rational number, the coefficient of the term in is .

Find the possible values of .

Given that  is small, so that terms in and higher powers of  can be ignored, show that

Find the range of values of  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 at

Express   in partial fractions.

Hence use binomial expansions to show that

where is a constant to be found.

Find the range of validity of for the expansion in part (b).

In the binomial expansion of     where , the coefficient of the term in is equal to the coefficient of the term in .

Show that .

Given that the product of and is , find the values of and .

Given that can be expression in the form

where , and are constants,

(i) find the value of and the value of ,

(ii) show that .

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

where , and are simplified fractions to be found.

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

Use binomial expansions to show that .

A student substitutes into both sides of the approximation shown in part (a) in an attempt to find an approximation to .

Give a reason why the student should not use .

Substitute into

to obtain an approximation to . Give your answer as a fraction in its simplest form.

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

to estimate the value of , giving your estimate to 2 decimal places.

Explain why you would not be able to use the expansion in part (a) to estimate .

Find the first three terms in ascending powers of of the binomial expansion of

giving each term in simplest form.

In the binomial expansion of    where ,  the coefficient of the term in  is one-seventh of the coefficient of the term in . 

Find the value of .

Expand

in ascending powers of , up to and including the term in . 

Find also the range of values of for which this expansion is valid.

Two functions are given by

where and are non-zero constants.

The binomial expansions of    and  have the following properties:

• The coefficient of the  term in the expansion of  is 72 times larger than the coefficient of the  term in the expansion of

• The coefficient of the  term in the expansion of  is 24 times larger than the coefficient of the  term in the expansion of

Find the values of  and .

Use binomial expansions to show that, in ascending powers of

where , and are constants to be found.

Explain why the expansion found in part (a) cannot be used to estimate the value of

Use binomial expansions to show that, in ascending powers of ,

where , and are constants to be found.

Find the range of values of for which the expansion is valid.

In the expansion of  where  is a rational number, the coefficient of the term in  is

Given that , find the value of .