JEE MAIN - Mathematics (2023 - 30th January Evening Shift - No. 13)

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

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

$[x]$ and $[y]$ are both odd

$[x]+[y]$ is even

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

If $${I_1} + f = {(8\sqrt 3 + 13)^{13}},f' = {(8\sqrt 3 - 13)^{13}}$$

$${I_1} + f - f'=$$ Even

$${I_1} = $$ Even

$${I_2} + f - f' = {(7\sqrt 2 + 9)^9} + {(7\sqrt 2 - 9)^9}$$

= Even

$${I_2} = $$ Even