1
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Question

# If p=(8+3√7)n and f=p−[p], where [⋅] denotes the greatest integer function, then the value of p(1−f) is

A
1
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B
2
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C
2n
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D
22n
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Solution

## The correct option is A 1Given p=(8+3√7)nand p=[p]+f (where[.]=GIF)Sol. : LetN=(8−3√7)n⇒ Now, (a+b)n+(a−b)n=2[nC0an+nC2an−2−b2+...bn]⇒ Take a=8 & b=3√7∴p+N=2[nC08n+nC28n−2−(3√7)2+....] =2.K (where k &I)Now, 0<f<1 & 0<N<1 ∵N=(√64−√63)n∴p+N=[p]+f+N=2K[P] is always integer & p+N is an integer =f+N must be an integer.But, 0<(f+N)<2 →f+N=1⇒p(1−f)=p.N=(8+3√7)n(8−3√7)n⇒(82+(3√7)2)n=1n=1.Hence, the answer is 1.

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