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Question

If a0n+1+a1n+a2n1++an12+an=0, then the equation a0xn+a1xn1++an1x+an=0 has, in the interval (0,1)

A
Exactly one root
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B
Atleast one root
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C
Atmost one root
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D
No root
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Solution

The correct option is B Atleast one root
Consider the function
f(x)=a0xn+1n+1+a1xnn+...an1x22+anx
Then f(0)=0 ...(i)
And
f(1)=a0n+1+a1n+...an12+an
=0 ...(given).
Hence
f(0)=f(1).
Then applying Rolle's theorem, there exists atleast one 'c' 0<c<1 such that
f(c)=0
Or
[a0xn+a1xn1+a2xn2+...an1x+an]x=c=0
Where 0<c<1.

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