09 Binomial Expansion 1
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7/31/2019 09 Binomial Expansion 1
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7/31/2019 09 Binomial Expansion 1
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JJC/05/P1/Q1
6)Expand
3 8
1
x+in ascending powers ofx up to and including the term in
2x . [3]
MJC/05/P2/Q5
7)Express ( )xf as partial fractions, where ( )
( )( )121661
f2
2
+
+=
xx
xxx .
[4]
Hence, or otherwise, obtain the expansion of ( )xf in ascending powers ofxup to and including the term inx
3. [4]
State the range of values ofx for which the expansion is valid. [1]
NYJC/05/P1/Q1
8)Find the first four terms of
2
)1(
x , stating the range ofx for which thisexpansion is valid.
By choosing a suitable value ofx, find
=
112
3
rr
r.
[5]
RJC/05/P1/Q5
9)Expand ( )2
1
4 y+ in ascending powers ofy up to and including the term in
.3y Simplify the coefficients. [3]
In the expansion of ( )21
284 kxx ++ , where k is a constant, the coefficient
of3
x is zero. By writing2
8 kxx + as y, find the value of k. [2]
TJC/05/P1/Q7
10)Express
( )( )xxxx
2112
11232
2
+
+in the form
x
C
x
BAx
2112
+
+
+whereA,B and Care
constants. [3]
Hence or otherwise, expand( )( )xx
xx
2112
11232
2
+
+in ascending powers ofx up to
and including the term inx2. State the values ofx for which the expansion is
valid. [4]
TPJC/05/P1/Q1
11) Find the binomial expansion of x21+ up to and including the term in x3,simplifying the coefficient. State the values ofx for which this expansion is
valid. [4]
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VJC/05/P1/Q8
12) i)Express
( )( )22
21
7
+
++
xx
xxin partial fractions.
[4]
ii) Show that the expansion of 22
1
+ x in ascending powers ofx up to
and including thex2
term is
+
2
4
31 xx .
[2]
iii) Given that ( ) K+++= 21 11 xxx , find the expansion of( )( )2
2
21
7
+
++
xx
xxin ascending powers ofx up to and including the 2x
term. [2]
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Possible solutions
1)
Using sine rule
C
AB
x
ACBC
sin
6sin
4sin
=
+
=
( )
( )( )
( )
+
++
+
+
+
+
+=
++=
++=
++
+
=
+
=
+
=
+
=
2
22
222
22
12
12
12
2
2
7312
32
312
23
2312
23
!2
111
2312
2312
2312
3
2
1
2
sin2
3cos
2
12
1
sin6
coscos6
sin2
1
6sin
4sin
xx
xx
x
xxxx
xx
xx
xx
xx
xx
xx
xx
xAC
BC
LLL
L
L
LL
A
B
C4
x+
6
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2)( )( )121
3222
122
124
)12)(1(
1242
2
23
23
2
23
+
++=
+
+=
+
+
xx
xx
xxx
xx
xx
xx
( )( ) ( )( ) ( )( )112322
121121322
22
22
2
+++=+
+
++
=
+
+
xSRxxQxx
xSRx
xQ
xxxx
When 1=x ,( )
1
12322
=
+=+
Q
Q
When 0=x ,( ) ( )( )
2
1101300
=
++=+
S
S
When 0=x ,( ) ( )( )
0
1222181348
=
++=+
R
R
Therefore( ) ( )12
21
12)12)(1(
12422
23
+
+=
+
+
xxxxxx . 2,0,1,2 ==== SRQP
( ) ( )
( ) ( )
( ) ( )
( )( )( )( )
( )
( )( )( )( )
( )( )[ ][ ] ( )
L
LL
L
L
++=
+++++=
++
+
+
++
=
+=
++=
+
+=
+
+
322
232
12
3
21
121
121
22
23
31
21212
2112
!3
21111!2
11111
2
21212
21212
12
2
1
12
)12)(1(
124
xxx
xxxx
x
x
xx
xx
xx
xxxx
xx
The series is valid for 1
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3)( )( )
( )( ) ( )
( )
( )
L
L
L
L
L
++++=
++++
++++
=
+++++=
+
+
+
+
+=
+=
+
32
32
32
32
3
2
2
1
42
521
23
2
5
2
31
2
5
2
311
2!3
22
11
2
1
2
1
2!2
1
2
1
2
1
2211
1
21121
1
xxx
xxx
xxx
xxxx
x
xx
x
xxx
x
The series is valid for2
112
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4i) ( )
( )( )( ) ( )
( )( )111
11
1
1
11
1f
2
2
23
2
23
3
2
3
+
+=
+
+=
+
=
+
=
xx
x
xxx
x
xxx
xx
xx
xxx
( )( ) ( )( ) ( )22
2
2
2
1111111 ++
+
+
=
+
=
+
x
D
x
C
x
B
xx
x
xx
x
( ) ( )( ) ( )1111 22 ++++= xDxxCxBx
When 1=x , ( )4
121
2== BB
When 1=x , ( )2
1111 == BD
When 0=x , ( ) ( ) ( )4
31
2
111
2
10
2=++= CC
( )
( )( ) ( ) ( ) ( )22
3
12
1
14
3
14
11
11
1f
+
+
+
=
+
=
xxxxx
xxx
ii)
( )( ) ( ) ( )
( ) ( ) ( )
( ) ( )
[ ]
L
L
L
L
L
LL
++=
++
+++
++
=
+++++
+++++++=
++++=
+
+
+
=
32
32
32
32
32
3232
211
2
1
22
3
2
14
3
4
3
4
3
4
34
1
4
1
4
1
4
1
1
4321
2
1
14
31
4
11
12
11
4
31
4
11
12
1
14
3
14
11f
xx
xxx
xxx
xxx
xxx
xxxxxx
xxx
xxxx
The series is valid for 1
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5)( )
( )
( )( )
( )( )( )
L
L
L
+++=
++++=
+
+
+
+
=
=
32
32
3
21
2
22
8
1
16
3
4
1
2
1
4
31
4
1
2!3
22122
2!2
122
221
4
1
2122
xx
xxx
x
xx
xx
The series is valid for 21
2
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7)Let
( )( ) 121121661
22
2
+
+
+=
+
+
x
C
x
BAx
xx
xx
( )( ) ( )11266122+++=+
xCxBAxxx
When2
1=x ,
2
14
1
4
16
2
161
=
++=
+
C
C
When 0=x ,1
21
=
+=
B
B
When 1=x ,( )( ) ( )
4
112121661
=
+++=+
A
A
( )
( )( ) 12
2
1
41
121
661f
22
2
+
+
=
+
+=
xx
x
xx
xxx
( )
( )( ) ( )( )[ ] ( ) ( )[ ]
L
LL
LL
+=
+++=
+++++=
+=
+
+
=
32
3232
322
112
2
12941
16842441
22212141
212141
12
2
1
41f
xxx
xxxxxx
xxxxx
xxx
xx
xx
The series is valid for2112and12
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9)( )
L
L
L
+++=
+++=
+
+
+
+=
+=+
32
32
32
2
1
2
1
512
1
64
1
4
12
1024
1
128
1
8
112
4!3
22
1
12
1
2
1
4!2
12
1
2
1
42
112
4124
yyy
yyy
yyy
yy
The expansion is valid for 414
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10)( )( )
( ) ( )( ) ( )22
22
2
1211123
2
1
2112112
1123
xCxBAxxx
x
C
x
BAx
xx
xx
+++=+
+
+
+=
+
+
When2
1=x ,
2
1
4
11
4
111
2
123
2
1
=
++=
+
C
C
When 0=x ,( ) ( )
1
12
11
2
3
=
+=
B
B
When 1=x ,( ) ( )( ) ( )
3
112
12111123
2
1
=
+++=+
A
A
( )( ) ( )xxx
xx
xx
212
1
1
13
2112
112322
2
+
+
+=
+
+
( )( ) ( )
( )( ) ( )
( )( ) ( )
L
LL
LL
+++=
++++++=
++++++=
+++=
+
+
+=
+
+
2
22
22
112
22
2
42
3
22
113
4212
1113
212
1113
212
1
1
13
2112
1123
xx
xxxx
xxxx
xxx
xx
x
xx
xx
The expansion is valid for2
112xand12
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12)i) Let
( )( ) ( )222
22121
7
+
+
+
+
=
+
++
x
C
x
B
x
A
xx
xx
( ) ( )( ) ( )xCxxBxAxx ++++=++ 12127 22
When 1=x , ( ) 121711 2 =+=++ AA When 2=x , ( ) 321724 =+=+ CC
When 0=x , ( )( ) 032147 =++= BB
( )( ) ( )222
2
3
1
1
21
7
+
+
=
+
++
xxxx
xx
ii)
( )( )( )
L
L
++=
+
+
+=
+
2
22
4
31
2!2
122
221
21
xx
xxx
The expansion is valid for 212
