1
JEE Main 2022 (Online) 29th July Evening Shift
Numerical
+4
-1

$$\text { If } \sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000 L \text {, then } L \text { is equal to }$$ ________.

2
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 $$\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{\mathrm{n}}$$, in the increasing powers of $$\frac{1}{\sqrt[4]{3}}$$ be $$\sqrt[4]{6}: 1$$. If the sixth term from the beginning is $$\frac{\alpha}{\sqrt[4]{3}}$$, then $$\alpha$$ is equal to _________.

3
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 __________.

4
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1

If $$1 + (2 + {}^{49}{C_1} + {}^{49}{C_2} + \,\,...\,\, + \,\,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4} + \,\,...\,\, + \,\,{}^{50}{C_{50}})$$ is equal to $$2^{\mathrm{n}} \cdot \mathrm{m}$$, where $$\mathrm{m}$$ is odd, then $$\mathrm{n}+\mathrm{m}$$ is equal to __________.

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