1
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 ___________

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

3
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 _________.

4
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1 Let the coefficients of the middle terms in the expansion of $$\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)^{2}$$ and $$\left(1-\frac{\beta}{2} x\right)^{6}, \beta>0$$, respectively form the first three terms of an A.P. If d is the common difference of this A.P. , then $$50-\frac{2 d}{\beta^{2}}$$ is equal to __________.

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