A-PDF Merger DEMO : Purchase from www.A-PDF.com to remove the watermark Example : 1 (i)
⎛ 4x 5 ⎞ − ⎟ Find the 7th term in the expansion of ⎜ ⎝ 5 2x ⎠
(ii)
⎛ 2 1 ⎞ ⎟ Find the coefficient of x in ⎜ ax + bx ⎠ ⎝
9
11
7
Solution (i)
⎛ 4x 5 ⎞ − ⎟ In the expansion of ⎜ ⎝ 5 2x ⎠
9
⎛ 4x ⎞ ⎟ The general terms is Tr+1 = Cr ⎜ ⎝ 5 ⎠
9 −r
⎛ 5 ⎞ ⎜− ⎟ ⎝ 2x ⎠
9
r
For 7th term (T7), Put r = 6 9−6
⇒
⎛ 4x ⎞ ⎟ T7 = T6+1 = C6 ⎜ ⎝ 5 ⎠
⇒
T7 =
9×8×7 ⎛ 4⎞ 3 ⎛ 5⎞ ⎜ ⎟ x ⎜− ⎟ 3! ⎝5⎠ ⎝ 2⎠
⇒
T7 =
9×8×7 3 1 5 3 3! x
⇒
T7 =
⎛ 5 ⎞ ⎜− ⎟ ⎝ 2x ⎠
9
3
(ii)
6
6
1 x6
10500
⎛ 2 1 ⎞ ⎟ In ⎜ ax + bx ⎠ ⎝
x3 11
general term is Tr+1 = 11Cr a11–r b–r x22–3r
for term involving x7 , 22 – 3r = 7 ⇒ r=5 Hence T5+1 or the 6th term will contain x7. 5
11
2 11–5
T6 = C5 (ax )
⎛ 1 ⎞ 11× 10 × 9 × 8 × 7 a 6 7 462a 6 7 ⎜ ⎟ = x = x 5! b5 b5 ⎝ bx ⎠
Hence the coefficient of x7 is
462a 6 b5
Example : 2
⎛ 3x 2 1 ⎞⎟ − Find the term independent of x in ⎜⎜ 3x ⎟⎠ ⎝ 2
9
Solution
⎛ 3x 2 ⎞ ⎟ 9 Tr+1 = Cr ⎜⎜ ⎟ ⎝ 2 ⎠
9 −r
r
⎛ 1 ⎞ ⎜− ⎟ = 9Cr ⎝ 3x ⎠
⎛ 3x 2 ⎞ ⎜ ⎟ ⎜ 2 ⎟ ⎝ ⎠
9 −r
r
⎛ 1 ⎞ 18–3r ⎜− ⎟ x ⎝ 3x ⎠
for term independent of x, 18 – 3r = 0 ⇒ r=6 Hence T6+1 or 7th term is independent of x.
⎛ 3x 2 ⎞ ⎟ T7 = C6 ⎜⎜ ⎟ ⎝ 2 ⎠ 9
9 −6
6
9×8×7 ⎛3⎞ ⎛ 1 ⎞ ⎜− ⎟ = ⎜ ⎟ 3 x 3! ⎝ ⎠ ⎝2⎠
3
6
7 ⎛ 1⎞ ⎜− ⎟ = 3 18 ⎝ ⎠
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