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KS5 "Full Coverage": Binomial Expansion (Year 2)
This worksheet is designed to cover one question of each type seen in past papers, for each A
Level topic. This worksheet was automatically generated by the DrFrostMaths Homework
Platform: students can practice this set of questions interactively by going to
www.drfrostmaths.com, logging on, Practise β Past Papers (or Library β Past Papers for
teachers), and using the βRevisionβ tab.
Question 1 Categorisation: Determine the binomial expansion of (π + ππ)π for negative or
fractional π.
[OCR C4 June 2014 Q3i] Find the first three terms in the expansion of (1 β 2π₯)β1
2 in
ascending powers of π₯ , where |π₯| <1
2 .
..........................
Question 2 Categorisation: Determine the binomial expansion of (π + ππ)π
[Edexcel C4 Jan 2011 Q5a] Use the binomial theorem to expand
(2 β 3π₯)β2 , |π₯| <2
3 ,
in ascending powers of π₯ , up to and including the term in π₯3 . Give each coefficient as a
simplified fraction.
..........................
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Question 3 Categorisation: As above, but for fractional π.
[Edexcel A2 Specimen Papers P1 Q2a] Show that the binomial expansion of (4 + 5π₯)1
2 in
ascending powers of π₯ , up to and including the term in π₯2 is
2 +5
4π₯ + ππ₯2
giving the value of the constant π as a simplified fraction.
..........................
Question 4 Categorisation: Use a Binomial expansion to determine an approximation for a
square root.
[Edexcel A2 Specimen Papers P1 Q2bi Edited]
It can be shown that the binomial expansion of (4 + 5π₯)1
2 in ascending powers of π₯ , up to and
including the term in π₯2 is
2 +5
4π₯ β
25
64π₯2
Use this expansion with π₯ =1
10 , to find an approximate value for β2
Give your answer in the form π
π where π and π are integers.
..........................
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Question 5 Categorisation: Understand that Binomial expansions are only valid for particular
ranges of values for π.
[Edexcel A2 Specimen Papers P1 Q2bii Edited] (Continued from above)
Explain why substituting π₯ =1
10 into this binomial expansion leads to a valid approximation.
Question 6
Categorisation: Understand that ββ¦ can be written as (β¦ )ππ in order to a Binomial
expansion.
[Edexcel C4 June 2018 Q1a] Find the binomial series expansion of
β4 β 9π₯ , |π₯| <4
9
in ascending powers of π₯ , up to and including the term in π₯2 Give each coefficient in its
simplest form.
..........................
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Question 7 Categorisation: Use substitution to determine square/cube roots, but more difficult
ones where a direct comparison would lead to an invalid value of π.
[Edexcel C4 June 2013(R) Q4b Edited] The binomial expansion of β8 β 9π₯3
up to and
including the term in π₯3 is written below.
β8 β 9π₯3
= 2 β3
4π₯ β
9
32π₯2 β
45
256π₯3 , |π₯| <
8
9
Use this expansion to estimate an approximate value for β71003
, giving your answer to 4
decimal places. State the value of π₯ , which you use in your expansion, and show all your
working.
Question 8 Categorisation: As above.
[Edexcel C4 June 2018 Q1b Edited]
It can be shown that
β4 β 9π₯ β 2 β9
4π₯ β
81
64π₯2 , |π₯| <
4
9
Use this expansion, with a suitable value of π₯ , to find an approximate value for β310 Give
your answer to 3 decimal places.
..........................
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Question 9 Categorisation: Rewrite more complication expressions (involving fractions) as a
product of two Binomial expansions.
[OCR C4 June 2012 Q3ii Edited]
It can be shown that 1+π₯2
β1+4π₯β 1 β 2π₯ + 7π₯2 β 22π₯3
State the set of values of π₯ for which this expansion is valid.
..........................
Question 10 Categorisation: As per Q5, but involving further problem solving.
[OCR C4 June 2016 Q7]
Given that the binomial expansion of (1 + ππ₯)π is 1 β 6π₯ + 30π₯2 +β―. find the values of π
and π .
State the set of values of π₯ for which this expansion is valid.
π = ..........................
π = ..........................
|π₯| < ..........................
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Question 11 Categorisation: Multiply a Binomial expansion by a further bracket.
[Edexcel C4 June 2014(R) Q1b Edited] Given that
1
β9 β 10π₯=1
3+
5
27π₯ +
25
162π₯2 +β―
where |π₯| <9
10 , find the expansion of
3 + π₯
β9 β 10π₯
|π₯| <9
10 , in ascending powers of π₯ , up to and including the term in π₯2 .
Give each coefficient as a simplified fraction.
..........................
Question 12 Categorisation: As per Q9.
[OCR C4 June 2011 Q6]
Find the coefficient of π₯2 in the expansion in ascending powers of π₯ of
β1 + ππ₯
4 β π₯
giving your answer in terms of π .
..........................
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Question 13 Categorisation: As above.
[Edexcel C4 June 2013 Q2a Edited]
Find the binomial expansion of
β1 + π₯
1 β π₯
where |π₯| < 1 up to and including the term in π₯2 .
..........................
Question 14 Categorisation: Use partial fractions to find a Binomial expansion.
[Edexcel C4 June 2010 Q5b Edited]
It is given that:
2π₯2 + 5π₯ β 10
(π₯ β 1)(π₯ + 2)β‘ 2 β
1
π₯ β 1+
4
π₯ + 2
Hence, or otherwise, expand 2π₯2+5π₯β10
(π₯β1)(π₯+2) in ascending powers of π₯ , as far as the term in π₯2 .
Give each coefficient as a simplified fraction.
..........................
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Question 15
Categorisation: Determine the Binomial expansion of (π + πππ)π
, i.e. involving a
more general power of π within the bracket.
[Edexcel C4 June 2011 Q2]
π(π₯) =1
β9 + 4π₯2
where |π₯| <3
2 .
Find the first three non-zero terms of the binomial expansion of π(π₯) in ascending powers of
π₯ . Give each coefficient as a simplified fraction.
..........................
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Answers
Question 1
Question 2
Question 3
Question 4
Question 5
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Question 6
Question 7
Question 8
Question 9
Question 10
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Question 11
1 +8
9π₯ +
35
54π₯2
Question 12
β1
16π2 +
1
32π +
3
256
Question 13
1 + π₯ +1
2π₯2
Question 14
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Question 15