The value of ${\sum }_{\mathrm{r}=0}^{20} {}^{50-\mathrm{r}}{\mathrm{C}}_6$ is equal to:

If the maximum value of the term independent of t in the expansion of ${\left(t^2{x}^{\frac{1}{5}}+\frac{{\left(1-x\right)}^{\frac{1}{10}}}{t}\right)}^{15}$ <!-- [if gte mso 9]><xml> </xml><![endif]-->, x $\ge$ <!-- [if gte mso 9]><xml> </xml><![endif]-->, is K, then 8 K is equal to…………..

If the constant term in the expansion of ${\left(3x^3-2x^2+\frac{5}{x^5}\right)}^{10}is{2}^k.\mathcal{l},$ where $\mathcal{l}$ is an odd integer, then the value of k is equal to:

  ${\displaystyle \sum _{k=1}^{10}\frac{k}{k^4+k^2+1}=\frac{m}{n};}$ where m and n are co-prime, then m + n is equal to……..

Let the coefficients of the middle terms in the expansion of ${\left(\frac{1}{\sqrt[]{6}}+\beta x\right)}^4,{\left(1-3\beta x\right)}^2$ 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 $-\frac{2d}{{\beta }^2}$ is equal to………….

Let A₁, A₂, A₃,…. be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = $\frac{1}{1296}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> and A₂ + A₄ = $\frac{7}{36},$ <!-- [if gte mso 9]><xml> </xml><![endif]--> then, the value of A₆ + A₈ + A₁₀ is equal to