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Question

If (1y)′′′(1+y)′′=1+a1y+a2y2+...+am+nym+n where mN and a1=a2=10, then (m,n) is

A
(20,45)
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B
(35,20)
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C
(45,35)
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D
(35,45)
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Solution

The correct option is C (35,45)
Given (1y)m(1+y)n=1+a1y+a2y2+......+am+nym+n
a1=a2=10
Differentiate on both sides w.r.t y
m(1y)m1(1+y)n+n(1y)m(1+y)n1=a1+2a2y+......+(m+n)am+nym+n1 ....... (1)
Put y=0,
nm=a1=10
nm=10
Differentiate (1) w.r.t y, we get
m(m1)(1y)m2(1+y)n2mn(1y)m1(1+y)n1+(n1)(n)(1y)m(1+y)n2
=2a2+6a3y+.....+(m+n)(m+n1)am+nym+n2
Put y=0,
m(m1)2mn+n(n1)=2a2
(mn)2(m+n)=20
m+n=80
nm=10
m+n=80
n=45,m=35
(m,n)=(35,45)


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