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

3 hours42 questions
1a
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3 marks

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

left parenthesis 3 plus x right parenthesis to the power of 4

giving each term in simplest form.

1b
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2 marks

Use your answer to part (a) to estimate left parenthesis 3.1 right parenthesis to the power of 4.

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2
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3 marks

Expand

left parenthesis x plus 2 right parenthesis to the power of 4

giving your answer in descending powers of x.

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3
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3 marks

Expand

left parenthesis 4 minus x right parenthesis to the power of 4

giving your answer in ascending powers of x.

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4
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3 marks

Find, in simplest form, the coefficient of x squared in the expansion of

left parenthesis 2 minus x right parenthesis to the power of 5

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5
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3 marks

Without using a calculator, find the value of

(i) 4 factorial

(ii) C presuperscript 5 subscript 2

(iii) C presuperscript 6 subscript 3

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6
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3 marks

Find the coefficient of x cubed in the binomial expansion of

left parenthesis 2 minus x right parenthesis to the power of 8

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1
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5 marks

Find, in simplest form, the coefficient of x to the power of 5 in the expansion of

open parentheses 5 plus 8 x squared close parentheses open parentheses 3 minus 1 half x close parentheses to the power of 6

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2
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3 marks

In the binomial expansion of

open parentheses a plus 2 x close parentheses to the power of 7 space space space space space space space space space space space space spacewhere a is a constant

the coefficient of x to the power of 4 is 15120

Find the value of a.

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3
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3 marks

In the binomial expansion of

left parenthesis p plus x right parenthesis to the power of 12

the coefficient of x to the power of 5 is 12 976 128.

Find the value of p.

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4
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3 marks

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

left parenthesis 3 plus 2 x right parenthesis to the power of 8

giving each term in simplest form.

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5a
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3 marks

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

left parenthesis 5 plus 2 x right parenthesis to the power of 5

giving each term in simplest form.

5b
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2 marks

 Use your answer to part (a) to estimate the value of left parenthesis 5.04 right parenthesis to the power of 5.

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6
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3 marks

Expand

left parenthesis 2 x minus 3 right parenthesis to the power of 6

giving your answer in descending powers of x.

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7
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3 marks

In the binomial expansion of

left parenthesis p plus x right parenthesis to the power of 4

where p is a non-zero constant, the coefficient of x squared is twice the coefficient of x

Find the value of p, giving your answer as a simplified fraction.

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8
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2 marks

In the binomial expansion of

left parenthesis a minus x right parenthesis to the power of 4

the coefficient of x squared is 96.

Given that a greater than 0, find the value of a.

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9a
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3 marks

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

left parenthesis 9 minus 2 x right parenthesis to the power of 5

giving each term in simplest form.

9b
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2 marks

Use your answer to part (a) to estimate left parenthesis 8.9 right parenthesis to the power of 5.

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10
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4 marks

In the binomial expansion of

left parenthesis a minus 2 x right parenthesis to the power of 5

the coefficient of x squared is equal to the coefficient of x cubed

Find the non-zero value of a.

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11
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3 marks

In the binomial expansion of

left parenthesis 3 plus p x right parenthesis to the power of 6

the coefficient of x to the power of 4 is four times the coefficient of x squared.

Find the possible values of p.

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12a
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3 marks

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

left parenthesis 3 plus 2 x right parenthesis to the power of 8

giving each term in simplest form.

12b
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3 marks

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

         left parenthesis 1 plus x right parenthesis left parenthesis 3 plus 2 x right parenthesis to the power of 8 almost equal to 6561 plus 41553 x plus 116640 x squared                   

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13
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4 marks

In the binomial expansion of

left parenthesis p plus q x right parenthesis to the power of 5

where p not equal to 0 and q not equal to 0 the coefficients of x squared and the coefficient of x cubed are equal.

Find an expression for p in terms of q.

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14
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3 marks

Find the coefficient of x to the power of 4 in the binomial expansion of 

left parenthesis 3 plus 2 x right parenthesis to the power of 9

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15a
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3 marks

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

left parenthesis 5 minus 2 x right parenthesis to the power of 4

giving each term in simplest form.

15b
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2 marks

Use your answer to part (a) to estimate left parenthesis 4.5 right parenthesis to the power of 4.

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16a
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3 marks

Find the first 3 terms, in ascending powers of x, of the binomial expansion of

left parenthesis 2 minus 3 x right parenthesis to the power of 7

giving each term in simplest form.

16b
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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 2 x right parenthesis left parenthesis 2 minus 3 x right parenthesis to the power of 7 almost equal to 128 minus 1600 x plus 8736 x squared                   

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17
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3 marks

In the binomial expansion of

left parenthesis 4 minus p x right parenthesis to the power of 6

the coefficient of x to the power of 4 is 19 440.

Given that p is a positive integer, find the value of p.

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18
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4 marks

Expand

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

giving your answer in ascending powers of x.

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19
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3 marks

Expand

left parenthesis 3 minus 2 x right parenthesis to the power of 5

giving your answer in ascending powers of x.

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20
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3 marks

Find the coefficient of x to the power of 4 in the expansion of

left parenthesis 4 minus 3 x right parenthesis to the power of 7

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21
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3 marks

In the binomial expansion of

left parenthesis m minus 1 fourth x right parenthesis to the power of 5

the coefficient of x cubed is -10.

Find the possible values of m.

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1a
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4 marks

Find the first 4 terms, in ascending powers of x, of the binomial expansion of

open parentheses 3 minus fraction numerator 2 x over denominator 9 end fraction close parentheses to the power of 8

giving each term in simplest form.

1b
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2 marks

straight f open parentheses x close parentheses equals open parentheses fraction numerator x minus 1 over denominator 2 x end fraction close parentheses open parentheses 3 minus fraction numerator 2 x over denominator 9 end fraction close parentheses to the power of 8

Find the coefficient of x squared in the series expansion of straight f open parentheses x close parentheses, giving your answer as a simplified fraction.

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2a
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4 marks

straight g open parentheses x close parentheses equals open parentheses 2 plus a x close parentheses to the power of 8 space space space space space space space space space space space space spacewhere a is a constant

Given that one of the terms in the binomial expansion of straight g open parentheses x close parentheses is 3402 x to the power of 5

find the value of a.

2b
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3 marks

Using this value of a,

find the constant term in the expansion of

open parentheses 1 plus 1 over x to the power of 4 close parentheses open parentheses 2 plus a x close parentheses to the power of 8

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3
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3 marks

In the binomial expansion of

left parenthesis 3 a minus 2 x right parenthesis to the power of 6 

the coefficient of x cubed is equal to the coefficient of x to the power of 4

Find the non-zero value of a.

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4
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3 marks

In the binomial expansion of

left parenthesis p plus q x right parenthesis to the power of 8

where p not equal to 0 and q not equal to 0 the coefficient of x squared is equal to the coefficient of x to the power of 6.

Find p in terms of q.

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5a
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3 marks

In the binomial expansion of

left parenthesis a plus b x right parenthesis to the power of 4

the coefficient of x squared is equal to the coefficient of x cubed.

Find the value of a over b

5b
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2 marks

Given that a and b are integers, and that 10 less than b less than 15, find the values of a and b.

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6
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5 marks

In the binomial expansion of left parenthesis a plus b x right parenthesis to the power of 4 comma the coefficient of x cubed is 216.

In the binomial expansion of left parenthesis a plus b x right parenthesis to the power of 6 comma the coefficient of x to the power of 4 is 4860.

Find the possible values of a and b.

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7
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2 marks

Use the formula

C presuperscript n subscript r equals open parentheses table row n row r end table close parentheses equals fraction numerator n factorial over denominator r factorial open parentheses n minus r close parentheses factorial end fraction

to prove that

      C presuperscript k subscript 1 space equals k

for all positive integer values of k.

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8
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3 marks

In the binomial expansion of 

left parenthesis 3 a plus begin inline style 1 half end style x right parenthesis to the power of 6

the coefficient of x cubed is equal to the coefficient of x to the power of 5.  

Find the values of a, giving your answers in the form fraction numerator square root of m over denominator n end fraction where m and n are integers to be found.

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9
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3 marks

In the binomial expansion of

left parenthesis p plus q x right parenthesis to the power of 9

where p not equal to 0 and q not equal to 0, the coefficient of x cubed is double the coefficient of x to the power of 5

Find p in terms of q.

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10
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5 marks

In the binomial expansion of left parenthesis a plus b x right parenthesis to the power of 8 comma the coefficient of x to the power of 5 is -870 912.

In the binomial expansion of left parenthesis a plus b x right parenthesis to the power of 12 comma the coefficient of x cubed is -1 557 135 360.

Find the possible values of a and b.

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1
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3 marks

Given that

C presuperscript n subscript 3 equals 35

use algebra to show that n satisfies the cubic equation

n open parentheses n minus 1 close parentheses open parentheses n minus 2 close parentheses equals 7 cross times 6 cross times 5

and hence write down the value of n.

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2
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4 marks

In the binomial expansion of

left parenthesis 1 plus x right parenthesis to the power of n

the coefficient of x cubed is 84.

Find the value of n.

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3a
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5 marks

Use the first 3 terms, in ascending powers of x, of the expansion of 

left parenthesis 3 minus 5 x right parenthesis to the power of 4

to find an approximation for left parenthesis 2.6 right parenthesis to the power of 4.

3b
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2 marks

Find the percentage error in the approximation from part (a) to the exact value of left parenthesis 2.6 right parenthesis to the power of 4

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4
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6 marks

Given that x represents a very small value, so that x cubed (and higher powers of x) can be ignored, show that

            left parenthesis 3 minus 2 x squared right parenthesis left parenthesis 4 minus 3 x right parenthesis to the power of 9 almost equal to p plus q x plus r x squared

where p, q and r are integers to be found.

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5
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4 marks

In the binomial expansion of

left parenthesis 1 minus 3 x right parenthesis to the power of n

the coefficient of x cubed is -3240.

Find the value of n.

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