CIE A Level Maths: Pure 3

Topic Questions

1.5 General Binomial Expansion

1
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2 marks

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

         open parentheses 1 minus x close parentheses to the power of blank to the power of negative 1 end exponent end exponent

up to and including the term in x squared.

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2a
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2 marks

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

        left parenthesis 1 plus x right parenthesis to the power of negative 2 end exponent

2b
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1 mark

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

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3a
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2 marks

Show that

         square root of 4 minus 4 x end root equals 2 left parenthesis 1 minus x right parenthesis to the power of begin inline style 1 half end style end exponent

3b
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2 marks

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

      square root of 4 minus 4 x end root

3c
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2 marks

Using x equals 0.02, use your expansion from part (b) to find an approximation to  2 square root of 0.98 end root.

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4
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4 marks

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

        left parenthesis 1 plus 2 x right parenthesis to the power of negative begin inline style 1 half end style end exponent

up to and including the term in x cubed.

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5a
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4 marks

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

      left parenthesis 1 minus 1 half x right parenthesis to the power of begin inline style 1 third end style end exponent

5b
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1 mark

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

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6
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2 marks

Find the coefficient of the term in x squared in the binomial expansion of

        left parenthesis 1 minus 3 x right parenthesis to the power of negative 3 end exponent

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7
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2 marks

The function straight f left parenthesis x right parenthesis is given by

           straight f left parenthesis x right parenthesis equals left parenthesis 1 minus p x right parenthesis to the power of negative 4 end exponent

where p is an integer.

Find the coefficient of the term in x cubed in the binomial expansion of  straight f left parenthesis x right parenthesis, in terms of p.

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8a
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3 marks

Given that x is small such that x cubed and higher powers of x can be ignored show that

            left parenthesis 1 minus 1 third x right parenthesis to the power of negative 2 end exponent almost equal to 1 plus 2 over 3 x plus 1 third x squared

8b
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2 marks

Using a suitable value of x in the result from part (a), find an approximation for the value of left parenthesis 0.94 right parenthesis to the power of negative 2 end exponent.

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9
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5 marks

It is given that

               straight f left parenthesis x right parenthesis equals square root of 1 plus a x end root space space space a n d space space space g left parenthesis x right parenthesis equals cube root of 1 minus a x end root

where a is a non-zero constant.

In their binomial expansions, the coefficient of the x squared term for straight f left parenthesis x right parenthesis is equal to the coefficient of the x term for g left parenthesis x right parenthesis.

Find the value of a.

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10a
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3 marks

Show, as partial fractions, that

         fraction numerator 5 minus x over denominator open parentheses 1 plus x close parentheses open parentheses 1 minus x close parentheses end fraction identical to fraction numerator 3 over denominator 1 plus x end fraction plus fraction numerator 2 over denominator 1 minus x end fraction

10b
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4 marks

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

(i)
 3 left parenthesis 1 plus x right parenthesis to the power of negative 1 end exponent comma
(ii)
2 left parenthesis 1 minus x right parenthesis to the power of negative 1 end exponent
10c
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2 marks

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

            fraction numerator 5 minus x over denominator open parentheses 1 plus x close parentheses open parentheses 1 minus x close parentheses end fraction

are

               5 minus x plus 5 x squared

10d
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1 mark

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

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11a
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3 marks

Find the first three terms in the binomial expansion of  open parentheses 1 minus x close parentheses to the power of negative 1 end exponent.

11b
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1 mark

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

11c
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2 marks

The first three terms of the expansion are to be used in a computer program to estimate the value of  fraction numerator 1 over denominator 0.95 end fraction.
Choose an appropriate value of  x to use in the expansion and thus find the value the computer program will use to estimate  fraction numerator 1 over denominator 0.95 end fraction.

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1
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2 marks

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

            1 over open parentheses 1 minus 2 x close parentheses cubed

up to and including the term in x cubed.

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2a
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3 marks

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

              open parentheses 1 plus 4 x close parentheses to the power of begin inline style 1 third end style end exponent
               

to estimate the value of cube root of 1.2 end root, giving your answer to three significant figures.

2b
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1 mark

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

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3
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4 marks

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

         1 over open parentheses 4 plus x close parentheses cubed

up to and including the term in x cubed.

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4a
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2 marks

Use the binomial expansion to expand open parentheses 1 minus begin inline style 1 half end style x close parentheses to the power of begin inline style 1 third end style end exponent  up to and including the term in x to the power of 2. end exponent

4b
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2 marks

Hence, or otherwise, expand  open parentheses 1 minus x close parentheses open parentheses 1 minus begin inline style 1 half end style x close parentheses to the power of begin inline style 1 third end style end exponent up to and including the term in x squared.

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5
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3 marks

In the expansion of  1 over open parentheses 3 plus p x close parentheses cubed  the coefficient of the term in x squared is double the coefficient of the term in x cubed.  Find the value of  p.

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6a
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2 marks

The functions straight f left parenthesis x right parenthesis and  straight g left parenthesis x right parenthesis are given as follows

               straight f left parenthesis x right parenthesis equals open parentheses 4 plus 3 x close parentheses to the power of 1 half end exponent space space space space space space space space space space space space space space space space space straight g left parenthesis x right parenthesis equals open parentheses 9 minus 2 x close parentheses to the power of negative 1 half end exponent

Expand straight f left parenthesis x right parenthesis, in ascending powers of x up to and including the term in x squared.

6b
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2 marks

Expand straight g open parentheses x close parentheses, in ascending powers of x up to and including the term in x squared.

6c
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2 marks

Find the expansion of square root of fraction numerator 4 plus 3 x over denominator 9 minus 2 x end fraction end root  in ascending powers of x, up to and including the term in x squared.

6d
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2 marks

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

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7
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3 marks

In the expansion of  open parentheses 1 minus begin inline style 4 over 3 end style x close parentheses to the power of n , where n is a real number, the coefficient of the term in x squared is begin inline style negative 16 over 81 end style.

Find the possible values of n.

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8a
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3 marks

Express fraction numerator 4 plus 5 x minus x squared over denominator open parentheses 1 minus x close parentheses open parentheses 1 plus x close parentheses squared end fraction  in partial fractions.

8b
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3 marks

Use the binomial expansion to find the first three terms, in ascending powers of x, in each of open parentheses 1 minus x close parentheses to the power of negative 1 end exponent, open parentheses 1 plus x close parentheses to the power of negative 1 end exponent, and open parentheses 1 plus x close parentheses to the power of negative 2 end exponent.

8c
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2 marks

Hence express  fraction numerator 4 plus 5 x minus x squared over denominator open parentheses 1 minus x close parentheses open parentheses 1 plus x close parentheses squared end fraction as the first three terms of a binomial expansion in ascending powers of  x.

8d
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1 mark

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

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9a
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3 marks

Given that x is small such that x cubed and higher powers of x can be ignored show that

         left parenthesis 2 plus 3 x right parenthesis to the power of negative 1 end exponent left parenthesis 3 minus 2 x right parenthesis to the power of negative 2 end exponent almost equal to 1 over 18 minus 1 over 108 x plus 19 over 216 x squared

9b
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3 marks

Find the percentage error between your calculator answer and the approximation in part (a) when x equals 0.1, giving your answer to one decimal place.

9c
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2 marks

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

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10a
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3 marks

In the binomial expansion of  square root of 4 plus begin inline style p over q end style x end root   where p less than 0 less than q, the coefficient of the x squared term is equal to the coefficient of the x cubed term.

Show that space p equals negative 8 q.

10b
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2 marks

Given further that p q equals negative 8 space find the values of p and q.

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11a
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3 marks

Express fraction numerator 1 minus 7 x over denominator open parentheses x plus 2 close parentheses open parentheses 3 minus x close parentheses end fraction  in the form  fraction numerator A over denominator x plus 2 end fraction plus fraction numerator B over denominator 3 minus x end fraction, where A and B are integers to be found.

11b
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3 marks

Hence, or otherwise, find the binomial expansion of fraction numerator 1 minus 7 x over denominator open parentheses x plus 2 close parentheses open parentheses 3 minus x close parentheses end fraction, in ascending powers of x, up to and including the term in x squared.

11c
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2 marks

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

(i)
If x equals 0.1, which fraction is being approximated?
(ii)
Which fraction does the approximation give?

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12a
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3 marks

Find the first four terms in the binomial expansion of  fraction numerator 1 over denominator 2 minus 3 x end fraction.

12b
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2 marks

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

12c
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2 marks

The expansion is to be used in a computer program to estimate the value of begin inline style 5 over 7 end style.
Check that the expansion is valid for this purpose and use the first four terms of the expansion to estimate the value of 5 over 7.

12d
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2 marks

Find the percentage error the computer program will introduce by using the expansion as an approximation to  5 over 7

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1
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2 marks

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

         1 over open parentheses 1 minus x close parentheses squared

up to and including the term in x cubed.

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2a
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2 marks

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

               square root of 1 plus 2 x end root 

2b
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1 mark

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

2c
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1 mark

Using a suitable value of x, use your expansion from part (a) to estimate square root of 1.06 end root, giving your answer to 3 significant figures.

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3
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3 marks

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

            1 over open parentheses 4 plus 8 x close parentheses squared

up to and including the term in x cubed.

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4a
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3 marks

Use the binomial expansion to show that the first three terms in the expansion of  left parenthesis 1 plus 2 x right parenthesis to the power of negative 3 end exponent are  1 minus 6 x plus 24 x squared.

4b
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2 marks

Hence, or otherwise, find the expansion of  left parenthesis 1 plus x right parenthesis left parenthesis 1 plus 2 x right parenthesis to the power of negative 3 end exponent up to and including the term in x squared.

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5a
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2 marks

The function straight f left parenthesis x right parenthesis is given by

            f left parenthesis x right parenthesis equals square root of 4 minus s x end root

where s is an integer.

(i)
Find the coefficient of the term in x in the binomial expansion of  straight f left parenthesis x right parenthesis, in terms of s.
(ii)
Find the coefficient of the term in x squared in the binomial expansion of  straight f left parenthesis x right parenthesis, in terms of s.
5b
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2 marks

In the binomial expansion of straight f left parenthesis x right parenthesis comma space the coefficient of the term in x is equal to the coefficient of the term in x squared.
Find the value of s.

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6a
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3 marks

The functions straight f left parenthesis x right parenthesis and g left parenthesis x right parenthesis are given as follows

         f left parenthesis x right parenthesis equals open parentheses 1 minus 1 half x close parentheses to the power of 1 half end exponent space space space space space space space space g left parenthesis x right parenthesis equals left parenthesis 2 plus x right parenthesis to the power of negative 2 end exponent

(i)
Expand straight f left parenthesis x right parenthesis, in ascending powers of x up to and including the term in x squared.
(ii)
Find the values for x for which the expansion is valid.
6b
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3 marks
(i)
Expand g left parenthesis x right parenthesis comma in ascending powers of x up to and including the term in x squared.
(ii)
Find the values for x for which the expansion is valid.
6c
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2 marks
(i)
Find the expansion of fraction numerator square root of 1 minus 1 half x end root over denominator open parentheses 2 plus x close parentheses squared end fraction in ascending powers of x, up to and including the term in x squared.
(ii)
Find the values for x for which the expansion is valid.

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7
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3 marks

In the expansion of  open parentheses 1 minus 1 fourth x close parentheses to the power of n, where n is a negative integer, the coefficient of the term in x squared is begin inline style 3 over 8 end style.

Find the value of n.

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8a
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3 marks

Express fraction numerator 2 over denominator open parentheses 1 minus x close parentheses open parentheses 1 plus x close parentheses end fraction  in partial fractions.

8b
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2 marks

Use the binomial expansion to find the first three terms, in ascending powers of x, in each of  open parentheses 1 minus x close parentheses to the power of negative 1 end exponent and open parentheses 1 plus x close parentheses to the power of negative 1 end exponent

8c
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2 marks

Hence show that fraction numerator 2 over denominator open parentheses 1 minus x close parentheses open parentheses 1 plus x close parentheses end fraction almost equal to 2 plus 2 x squared

8d
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1 mark

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

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9a
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3 marks

Given that x is small such that x cubed and higher powers of x can be ignored show that

        open parentheses 1 minus 1 third x close parentheses to the power of negative 1 end exponent space open parentheses 2 minus x close parentheses to the power of negative 2 end exponent space almost equal to 1 fourth plus 1 third x plus 43 over 144 x squared

9b
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2 marks

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

9c
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3 marks
(i)
Use your calculator to find the exact fraction of open parentheses 1 minus begin inline style 1 third end style x close parentheses to the power of negative 1 end exponent open parentheses 2 minus x close parentheses to the power of negative 2 end exponent  when x equals 0.5
(ii)
Use your calculator to find the fraction from the approximation 1 fourth plus 1 third x plus 43 over 144 x squared when x equals 0.5
(iii)
Find the percentage error in the approximation, giving your answer to two decimal places.

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10
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3 marks

 It is given that

            straight f left parenthesis x right parenthesis equals square root of 9 plus p x end root      and       g left parenthesis x right parenthesis equals fourth root of 16 plus p x end root

                                  

In their binomial expansions, the coefficient of the x squared term for straight f open parentheses x close parentheses is equal to the coefficient of the x term for g left parenthesis x right parenthesis.

Find the value of p.

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11a
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2 marks

Express fraction numerator 12 minus x over denominator open parentheses x plus 2 close parentheses open parentheses 3 minus x close parentheses end fraction  in partial fractions.

11b
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3 marks

Using binomial expansions, up to and including terms in x squared show that
fraction numerator 12 minus x over denominator open parentheses x plus 2 close parentheses open parentheses 3 minus x close parentheses end fraction almost equal to 2 minus 1 half x plus 5 over 12 x squared

11c
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2 marks

Explain why the approximation in part (b) is only valid for space vertical line x vertical line less than 2.

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12a
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3 marks

Find the first three terms in the binomial expansion of  fraction numerator 1 over denominator 1 minus 1 half x. end fraction

12b
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1 mark

Show that the expansion is valid for open vertical bar x close vertical bar less than 2.

12c
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2 marks

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

12d
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2 marks

Hence find the value the computer program will use to estimate  20 over 19.

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1
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2 marks

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

         1 over open parentheses 1 minus 1 third x close parentheses to the power of 4

up to and including the term in x cubed.

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2a
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3 marks

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

         fraction numerator 1 over denominator square root of 1 minus 1 half x end root end fraction

to estimate the value of fraction numerator 1 over denominator square root of 0.95 end root end fraction, giving your answer to two decimal places.

2b
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1 mark

Explain why you would not be able to use your expansion to approximate fraction numerator 1 over denominator square root of 3 end fraction.

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3
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3 marks

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

         1 over open parentheses 3 minus 2 x close parentheses to the power of 4

up to and including the term in x cubed.

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4
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4 marks

Expand left parenthesis 1 minus begin inline style 1 half end style x right parenthesis left parenthesis 9 plus 3 x right parenthesis to the power of negative begin inline style 1 half end style end exponent up to and including the term in x squared.

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5
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4 marks

In the expansion of   1 over open parentheses 8 plus 2 q x close parentheses to the power of 1 third end exponent,  the coefficient of the term in x squared is one-seventh of the coefficient of the term in x cubed.  Find the value of q.

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6
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5 marks

The functions straight f open parentheses x close parentheses and straight g open parentheses x close parentheses are given as follows

               straight f left parenthesis x right parenthesis equals 8 minus x space space space space space space space space space space space space space space straight g left parenthesis x right parenthesis equals 8 plus 2 x     

Find the binomial expansion of   cube root of fraction numerator straight f open parentheses x close parentheses over denominator straight g open parentheses x close parentheses end fraction end root, in ascending powers of x, up to and including the term in x squared.  Also find the values of xfor which your expansion is valid.

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7
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4 marks

In the expansion of  open parentheses 16 minus 2 x close parentheses to the power of n, where n is a real number, the coefficient of the term in x squared is 16 to the power of n cross times 5 over 2048.

Given that vertical line n vertical line less than 1 find the value of n.

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8a
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3 marks

Express  fraction numerator 2 open parentheses 2 minus 5 x plus x squared close parentheses over denominator open parentheses x plus 2 close parentheses open parentheses 2 minus x close parentheses squared end fraction  in partial fractions.

8b
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5 marks

Express fraction numerator 2 open parentheses 2 minus 5 x plus x squared close parentheses over denominator open parentheses x plus 2 close parentheses open parentheses 2 minus x close parentheses squared end fraction  as the first three terms of a binomial expansion in ascending powers of x.

8c
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1 mark

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

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9a
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4 marks

Given that x is small such that x cubed and higher powers of x can be ignored show that

         left parenthesis 4 minus 3 x right parenthesis to the power of negative 2 end exponent left parenthesis 2 minus x right parenthesis to the power of negative 3 end exponent almost equal to 1 over 128 plus 3 over 128 x plus 87 over 2048 x squared

9b
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3 marks

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

9c
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2 marks

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

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10
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4 marks

It is given that

         straight f left parenthesis x right parenthesis equals square root of 4 plus a x end root        and              straight g left parenthesis x right parenthesis equals fourth root of 16 plus b x end root

The binomial expansions of  straight f left parenthesis x right parenthesis  and  straight g left parenthesis x right parenthesis have the following properties:

(i)
The coefficient of the x cubed term in the expansion of straight f open parentheses x close parentheses is 72 times larger than the coefficient of the x squared term in the expansion of straight g open parentheses x close parentheses.

(ii)

The coefficient of the x term in the expansion of straight f open parentheses x close parentheses is 24 times larger than the coefficient of the x term in the expansion of straight g open parentheses x close parentheses.

 

Find the values of a and b.

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11a
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5 marks

Find the binomial expansion of fraction numerator 15 over denominator open parentheses x minus 4 close parentheses open parentheses 5 x minus 2 close parentheses to the power of apostrophe end fraction, in ascending powers of x, up to and including the term in x squared.

11b
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2 marks

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

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12
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6 marks

The binomial expansion of

      fraction numerator 1 over denominator square root of 4 minus 2 x end root end fraction

is to be used in a computer program to estimate the reciprocal of square root of space 3.8 end root.

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.

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