Let x = ( 8 3 + 13 ) 13 and y = ( 7 2 + 9 ) 9 . If [ t ] denotes the greatest integer t , then [2023]

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Solution

Q.
Let $x = {(8 \sqrt{3} + 13)}^{13}$ and $y = {(7 \sqrt{2} + 9)}^{9} .$ If $[t]$ denotes the greatest integer $\leq t$ , then [2023]

1 [x] is even but [y] is odd

2 [x] + [y] is even

3 [x] and [y] are both odd

4 [x] is odd but [y] is even

Ans.
(2)

$x = {(8 \sqrt{3} + 13)}^{13}$ $= {}^{13}C_{0} \cdot {(8 \sqrt{3})}^{13} + {}^{13}C_{1} {(8 \sqrt{3})}^{12} {(13)}^{1} + \dots$

And $x' = {(8 \sqrt{3} - 13)}^{13}$ $= {}^{13}C_{0} {(8 \sqrt{3})}^{13} - {}^{13}C_{1} {(8 \sqrt{3})}^{12} {(13)}^{1} + \dots$

$∴$ $x - x' = 2 [{}^{13}C_{1} \cdot {(8 \sqrt{3})}^{12} {(13)}^{1} + {}^{13}C_{3} {(8 \sqrt{3})}^{10} \cdot {(13)}^{3} + \dots]$

Thus, $x - x'$ is an even integer, hence $[x]$ is even.

Now, $y = {(7 \sqrt{2} + 9)}^{9} = {}^{9}C_{0} {(7 \sqrt{2})}^{9} + {}^{9}C_{1} {(7 \sqrt{2})}^{8} {(9)}^{1} + {}^{9}C_{2} {(7 \sqrt{2})}^{7} {(9)}^{2} + \dots$

And $y' = {(7 \sqrt{2} - 9)}^{9} = {}^{9}C_{0} {(7 \sqrt{2})}^{9} - {}^{9}C_{1} {(7 \sqrt{2})}^{8} {(9)}^{1} + {}^{9}C_{2} {(7 \sqrt{2})}^{7} {(9)}^{2} \dots$

$∴$ $y - y' = 2 [{}^{9}C_{1} {(7 \sqrt{2})}^{8} {(9)}^{1} + {}^{9}C_{3} {(7 \sqrt{2})}^{6} {(9)}^{3} + \dots]$

Thus, $y - y'$ is an even integer, hence $[y]$ is even.

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