JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

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

Choose the correct answer:

Explanation

Equations:

$x = (8\sqrt{3} + 13)^{13}$
$y = (7\sqrt{2} + 9)^9$

Steps for $x$ :

Let $f = (8\sqrt{3} - 13)^{13}$ , where $0 < f < 1$ .
$x - f = (8\sqrt{3} + 13)^{13} - (8\sqrt{3} - 13)^{13}$
$x - f = 2 \times (\text{Integer}) = \text{Even}$
Isliye, $[x]$ is even.

Steps for $y$ :

Let $f' = (7\sqrt{2} - 9)^9$ , where $0 < f' < 1$ .
$y - f' = (7\sqrt{2} + 9)^9 - (7\sqrt{2} - 9)^9$
$y - f' = 2 \times (\text{Integer}) = \text{Even}$
Isliye, $[y]$ is even.

Conclusion:

$[x] + [y] = \text{Even} + \text{Even} = \text{Even}$

Sahi Option: (2)