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Unit 12
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Binomial
Expansions
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Unit 12 Binomial expansions
0606 Additional Mathematics
2
Mathematical Formulae
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1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
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b − 4 ac
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a ()
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Binomial Theorem
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(a + b)n = an + 1 an–1 b + 2 an–2 b2 + … + r an–r br + … + bn,
n
n!
where n is a positive integer and r =
(n – r)!r!
2. TRIGONOMETRY
Identities
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
Formulae for ∆ABC
a
b
c
sin A = sin B = sin C
a2 = b2 + c2 – 2bc cos A
∆=
1
bc sin A
2
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2! of !12
0606/12/F/M/15
Unit 12 Binomial expansions
0606 Additional Mathematics
7
0606/21/M/J/14
6
(a) Find the coefficient of x 5 in the expansion of ^3 - 2xh8 .
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(b) (i) Write down the first three terms in the expansion of ^1 + 2xh6 in ascending powers of x. [2]
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(ii) In the expansion of ^1 + axh^1 + 2xh6 , the coefficient of x 2 is 1.5 times the coefficient of x.
Find the value of the constant a.
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3! of !12
0606/21/M/J/14
[Turn over
Unit 12 Binomial expansions
0606 Additional Mathematics
6
0606/22/M/J/14
5
(i) Find and simplify the first three terms of the expansion, in ascending powers of x, of ^1 - 4xh5 .
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(ii) The first three terms in the expansion of ^1 - 4xh 1 + ax + bx
value of each of the constants a and b.
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are 1 - 23x + 222x 2 . Find the
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4! of !12
0606/22/M/J/14
Unit 12 Binomial expansions
0606 Additional Mathematics
6
0606/13/M/J/14
5
(i) The first three terms in the expansion of (2 - 5x) 6 , in ascending powers of x, are p + qx + rx 2 .
Find the value of each of the integers p, q and r.
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(ii) In the expansion of (2 - 5x) (a + bx) , the constant term is equal to 512 and the
coefficient of x is zero. Find the value of each of the constants a and b.
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© UCLES 2014
5! of !12
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Unit 12 Binomial expansions
0606 Additional Mathematics
7
0606/11/O/N/14
6
(i) Given that the coefficient of x 2 in the expansion of (2 + px) 6 is 60, find the value of the positive
constant p.
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(ii) Using your value of p, find the coefficient of x 2 in the expansion of (3 - x) (2 + px) 6 .
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6! of !12
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Unit 12 Binomial expansions
0606 Additional Mathematics
11
0606/13/O/N/14
9
(a) Given that the first 3 terms in the expansion of ^5 - qxh p are 625 - 1500x + rx 2 , find the value of
each of the integers p, q and r.
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(b) Find the value of the term that is independent of x in the expansion of
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1 N
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0606/13/O/N/14
7! of !12
[Turn over
Unit 12 Binomial expansions
0606 Additional Mathematics
6
0606/12/F/M/15
4
(i) Write down, in ascending powers of x, the first 3 terms in the expansion of (3 + 2x) 6 .
Give each term in its simplest form.
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(ii) Hence find the coefficient of x 2 in the expansion of (2 - x) (3 + 2x) 6 .
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8! of !12
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Unit 12 Binomial expansions
0606 Additional Mathematics
8
0606/22/M/J/15
7
In the expansion of ^1 + 2xhn , the coefficient of x 4 is ten times the coefficient of x 2 . Find the value of
the positive integer, n.
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9! of !12
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Unit 12 Binomial expansions
0606 Additional Mathematics
5
0606/13/M/J/15
3
^2 + x 2h6
(i) Find the first 4 terms in the expansion of
in ascending powers of x.
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(ii) Find the term independent of x in the expansion of
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^2 + x 2h6 c1 - 3 m .
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! of !12
10
0606/13/M/J/15
[Turn over
Unit 12 Binomial expansions
0606 Additional Mathematics
4
0606/22/O/N/15
2
(i) Find,inthesimplestform,thefirst3termsoftheexpansionof (2 - 3x) 6 ,inascendingpowers
ofx.
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(ii) Findthecoefficientof x 2 intheexpansionof(1 + 2x) (2 - 3x) 6 .
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11
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Unit 12 Binomial expansions
0606 Additional Mathematics
10
0606/13/O/N/15
8
(a) Given that the first 4 terms in the expansion of (2 + kx) 8 are 256 + 256x + px 2 + qx 3 , find the
valueofk,ofpandofq.
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(b) Findthetermthatisindependentof xintheexpansionof c x -
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