Let R+f=(55+11)2x+1and F=R−[R] where [.] denotes greater integer less than or equal to R and if f=(55+11)2x+1List – IList - II a) [R]1) 0b) (R+F)f2) 42x+1c) f−F3) Odd integerd) R+F−f4) Even integer 5) 42x−1

R+F−f=(55+11)2n−(55−112)2n+1=2[C1 2n+1(55)2n(11)+C3 2n+1(55)2n−2(11)2]∴R+F−f=2K (even)∴[R]=2K is an even integer

Let R+f=(55+11)2x+1and F=R−[R] where [.] denotes greater integer less than or equal to R and if f=(55+11)2x+1List – IList - II a) [R]1) 0b) (R+F)f2) 42x+1c) f−F3) Odd integerd) R+F−f4) Even integer 5) 42x−1

R+F−f=(55+11)2n−(55−112)2n+1=2[C1 2n+1(55)2n(11)+C3 2n+1(55)2n−2(11)2]∴R+F−f=2K (even)∴[R]=2K is an even integer

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Let $R + f = {(5 \sqrt{5} + 11)}^{2 x + 1}$ and $F = R - [R]$ where [.] denotes greater integer less than or equal to R and if $f = {(5 \sqrt{5} + 11)}^{2 x + 1}$
List – I
List - II

a) $[R]$ 1) 0
b) $(R + F) f$ 2) $4^{2 x + 1}$
c) $f - F$ 3) Odd integer
d) $R + F - f$ 4) Even integer
5) $4^{2 x - 1}$

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$A \to 1, B \to 5, C \to 2, D \to 3$

$A \to 4, B \to 2, C \to 1, D \to 4$

$A \to 3, B \to 1, C \to 2, D \to 4$

$A \to 2, B \to 3, C \to 1, D \to 5$

answer is D.

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Detailed Solution

$R + F - f = {(5 \sqrt{5} + 11)}^{2 n} - {(5 \sqrt{5} - 11^{2})}^{2 n + 1}$

$= 2 [{}^{2 n + 1}C_{1} {(5 \sqrt{5})}^{2 n} (11) + {}^{2 n + 1}C_{3} {(5 \sqrt{5})}^{2 n - 2} {(11)}^{2}]$

$∴ R + F - f = 2 K (e v e n)$

$∴ [R] = 2 K$ is an even integer

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List – I	List - II
a) $[R]$	1) 0
b) $(R + F) f$	2) $4^{2 x + 1}$
c) $f - F$	3) Odd integer
d) $R + F - f$	4) Even integer
	5) $4^{2 x - 1}$

Let R+f=(55+11)2x+1and F=R−[R] where [.] denotes greater integer less than or equal to R and if f=(55+11)2x+1List – IList - II a) [R]1) 0b) (R+F)f2) 42x+1c) f−F3) Odd integerd) R+F−f4) Even integer 5) 42x−1

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