Suppose $2-p, p, 2-\alpha, \alpha$ are the coefficient of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2+6 \alpha+2 p$ equals

If the coefficients of ${p^{th}}$, ${(p + 1)^{th}}$ and ${(p + 2)^{th}}$ terms in the expansion of ${(1 + x)^n}$ are in $A.P.$, then

Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2+(1+x)^3+\cdots+(1+x)^{49}+(1+m x)^{50}$ is $(3 n+1)^{51} C_3$ for some positive integer $n$. Then the value of $n$ is

If the greatest value of the term independent of $^{\prime}x^{\prime}$ in the expansion of $\left(x \sin \alpha+a \frac{\cos \alpha}{x}\right)^{10}$ is $\frac{10 !}{(5 !)^{2}}$, then the value of $' a^{\prime}$ is equal to:

The term independent of $x$ in ${\left[ {\sqrt{\frac{ x }{3}} + \frac{{\sqrt 3 }}{{{x^2}}}} \right]^{10}}$ is

If the ${(r + 1)^{th}}$ term in the expansion of ${\left( {\sqrt[3]{{\frac{a}{{\sqrt b }}}} + \sqrt {\frac{b}{{\sqrt[3]{a}}}} } \right)^{21}}$ has the same power of $a$ and $b$, then the value of $r$ is