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

(B) $[x]$ is odd but $[y]$ is even

(C) $[x]$ is even but $[y]$ is odd

(D) $[x]$ and $[y]$ are both odd

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Binomial Theorem

Let

x = (83 + 13)^{13}

and

y = (72 + 9)^{9}

. If [t] denotes the greatest integer

\leq t

, then

Answer: (A)

Consider

I = (13 + 83)^{13} + (13 - 83)^{13}

. Expanding both by the binomial theorem, the

3

terms cancel and

I

is an even integer. Since

13 - 83 \approx - 0.86

lies in

(- 1, 0)

, so does

(13 - 83)^{13}

, hence

x = (13 + 83)^{13} = I - (13 - 83)^{13}

exceeds the even integer

I

by a proper fraction, giving

[x] = I

, an even number. The same argument with

y = (9 + 72)^{9}

and

9 - 72 \approx - 0.90

shows

[y]

is even. Consequently

[x] + [y]

is even.

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Single correct option