1
JEE Main 2023 (Online) 6th April Evening Shift
+4
-1

Among the statements:

(S1): $$2023^{2022}-1999^{2022}$$ is divisible by 8

(S2) : $$13(13)^{n}-11 n-13$$ is divisible by 144 for infinitely many $$n \in \mathbb{N}$$

A
both (S1) and (S2) are incorrect
B
only (S1) is correct
C
only (S2) is correct
D
both (S1) and (S2) are correct
2
JEE Main 2023 (Online) 6th April Morning Shift
+4
-1

If $${ }^{2 n} C_{3}:{ }^{n} C_{3}=10: 1$$, then the ratio $$\left(n^{2}+3 n\right):\left(n^{2}-3 n+4\right)$$ is

A
$$27: 11$$
B
$$2: 1$$
C
$$35: 16$$
D
$$65: 37$$
3
JEE Main 2023 (Online) 6th April Morning Shift
+4
-1

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $$\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{\mathrm{n}}$$ is $$\sqrt{6}: 1$$, then the third term from the beginning is:

A
$$30 \sqrt{2}$$
B
$$60 \sqrt{3}$$
C
$$60 \sqrt{2}$$
D
$$30 \sqrt{3}$$
4
JEE Main 2023 (Online) 30th January Evening Shift
+4
-1
Let $x=(8 \sqrt{3}+13)^{13}$ and $y=(7 \sqrt{2}+9)^9$. If $[t]$ denotes the greatest integer $\leq t$, then :
A
$[x]$ is odd but $[y]$ is even
B
$[x]$ and $[y]$ are both odd
C
$[x]+[y]$ is even
D
$[x]$ is even but $[y]$ is odd
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