The number of roots of the equation, $(81)^{\sin ^2 x}+(81)^{\cos ^2 x}=30$ in the interval $[0, \pi]$, is equal to

If $\quad \int(2 x+4) \sqrt{x-1} d x=a(x-1)^{5 / 2}+b(x-1)^{3 / 2}+c$ where $c$ is a constant of integration, then the value of $(2 a+b)$ is

Let $\mathrm{X} \sim \mathrm{B}\left(6, \frac{1}{2}\right)$, then $\mathrm{P}[|x-4| \leqslant 2]$ is

The mean of $n$ observations is $\bar{x}$. If three observations $\mathrm{n}+1, \mathrm{n}-1,2 \mathrm{n}-1$ are added such that mean remains same, then value of $n$ is