Suppose F is the fractional part of  M = (13+11)6then value of M (1– F)  is

# Suppose $F$ is the fractional part of  = $\left(\sqrt{13}+\sqrt{11}{\right)}^{6}$then value of   is

1. A

128

2. B

64

3. C

32

4. D

16

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### Solution:

Let $N=\left(\sqrt{13}-\sqrt{11}{\right)}^{6}$

As $\left(\sqrt{13}-\sqrt{11}\right)=\frac{2}{\sqrt{13}+\sqrt{11}}$we get

Also, $M+N=\left(\sqrt{13}+\sqrt{11}{\right)}^{6}+\left(\sqrt{13}-\sqrt{11}{\right)}^{6}$

is an integer, say, $J$

Let,  where is the greatest integer contained in

We have

As $0 we get,

Also,  is an integer

Thus $M\left(1-F\right)=\left(\sqrt{13}+\sqrt{11}{\right)}^{6}\left(\sqrt{13}-\sqrt{11}{\right)}^{6}={2}^{6}=64$

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