1
JEE Main 2023 (Online) 24th January Evening Shift
Numerical
+4
-1 Let the sum of the coefficients of the first three terms in the expansion of $${\left( {x - {3 \over {{x^2}}}} \right)^n},x \ne 0.~n \in \mathbb{N}$$, be 376. Then the coefficient of $$x^4$$ is __________.

2
JEE Main 2023 (Online) 24th January Morning Shift
Numerical
+4
-1 Suppose $$\sum\limits_{r = 0}^{2023} {{r^2}{}~^{2023}{C_r} = 2023 \times \alpha \times {2^{2022}}}$$. Then the value of $$\alpha$$ is ___________

3
JEE Main 2022 (Online) 29th July Evening Shift
Numerical
+4
-1 $$\text { If } \sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000 L \text {, then } L \text { is equal to }$$ ________.

4
JEE Main 2022 (Online) 29th July Morning Shift
Numerical
+4
-1 Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $$\left(\sqrt{2}+\frac{1}{\sqrt{3}}\right)^{\mathrm{n}}$$, in the increasing powers of $$\frac{1}{\sqrt{3}}$$ be $$\sqrt{6}: 1$$. If the sixth term from the beginning is $$\frac{\alpha}{\sqrt{3}}$$, then $$\alpha$$ is equal to _________.

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