1
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
$$30 \sqrt{3}$$
2
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
3
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1

If the coefficient of $$x^{15}$$ in the expansion of $$\left(\mathrm{a} x^{3}+\frac{1}{\mathrm{~b} x^{1 / 3}}\right)^{15}$$ is equal to the coefficient of $$x^{-15}$$ in the expansion of $$\left(a x^{1 / 3}-\frac{1}{b x^{3}}\right)^{15}$$, where $$a$$ and $$b$$ are positive real numbers, then for each such ordered pair $$(\mathrm{a}, \mathrm{b})$$ :

A
a = 3b
B
ab = 1
C
ab = 3
D
a = b
4
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1
Out of Syllabus

The coefficient of $${x^{301}}$$ in $${(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}$$ is :

A
$${}^{500}{C_{300}}$$
B
$${}^{501}{C_{200}}$$
C
$${}^{500}{C_{301}}$$
D
$${}^{501}{C_{302}}$$
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