Greatest Term in Binomial Expansion

Find the greatest term of (considering numerical value):

${\left(\frac{2a}{x}+\frac{x}{a}\right)}^{10}$ where $x=\frac{1}{3},a=\frac{1}{2}$.

Find the greatest term of (considering numerical value):

${\left(5+\frac{x^2}{2}\right)}^6,x=3$

Find the greatest term of (considering numerical value):

${\left(\frac{1}{2}+2x\right)}^7,x=\frac{2}{3}$

Find the greatest term of (considering numerical value):

${\left(1+\frac{3}{4}\right)}^{13}$

Find the greatest coefficient of ${\left(x+\frac{1}{2}\right)}^4$.

Find the greatest coefficient of ${\left(2+3y^2\right)}^7$.

Find the greatest coefficient of ${\left(1+3x\right)}^6$.

Numerically the greatest term in the binomial expansion of ${\left(2a-3b\right)}^{19}$ when $a=\frac{1}{4}$ and $b=\frac{2}{3}$ is

The greatest term from the beginning in the expansion of ${\left(3+4x\right)}^9$, when $x=\frac{4}{3}$ is :

If the largest coefficient in the expansion of $(1+\mathrm{x}{)}^{30}$ is ${}^n\mathrm{C}_r$, then $n+r=$ _____

The condition for which the greatest term in the expansion of ${\left(1+x\right)}^n$ may have the greatest coefficient also, if $n$ is even positive integer, is:

The greatest value of the term independent of x in the expansion of ${\left(x\sin \alpha +\frac{\cos \alpha }{x}\right)}^{10}$ where $\alpha \in R$, is $\mathrm{\lambda }$, then $\left[\mathrm{\lambda }\right]$ is equal to, (where $[·]$ represents greatest integer function)

${\left(x\sin \alpha +\frac{\cos \alpha }{x}\right)}^{10}$ के प्रसार में x से स्वतंत्र पद का महत्तम मान $\mathrm{\lambda }$ है, जहाँ $\alpha \in R$, तब $\left[\mathrm{\lambda }\right]$ का मान है जहाँ [·] महत्तम पूर्णांक फलन को दर्शाता है।

If the middle term in the expansion of ${\left(1+x\right)}^{2n}$ is the greatest term, then $x$ lies in the interval

If the middle term of ${\left(1+x\right)}^{2n}(x>0, n\in N)$ is the greatest term of the expansion then the interval in which $x$ lies, is

The largest term in the expansion ${\left(\frac{b}{2}+\frac{b}{2}\right)}^{100}$ is

The value of the greatest term in the expansion of $\sqrt{3}{\left(1+\frac{1}{\sqrt{3}}\right)}^{20}$ is

The greatest term in expansion of ${\left(4+3x\right)}^7$, when $x=\frac{2}{3}$ is

The greatest binomial coefficient in the expansion of ${\left(x+\frac{1}{x}\right)}^{16}$ is

The last digit of the number $3^{400}$ is equal to

The interval in which $x$ must lie so that the greatest term in the expansion of ${(1+x)}^{2n}$ has the greatest coefficient, is