The roots of the equation $x^4 - 4x^3 + 6x^2 - 4x + 1 = 0$ are

Let $n$ be a positive integer. Let $A = \sum_{k=0}^{n} (-1)^{k} {}^{n}C_{k} \left[ \left(\frac{1}{2}\right)^{k} + \left(\frac{3}{4}\right)^{k} + \left(\frac{7}{8}\right)^{k} + \left(\frac{15}{16}\right)^{k} + \left(\frac{31}{32}\right)^{k} \right]$. If $63A = 1 - \frac{1}{2^{30}}$,then $n$ is equal to ......

The coefficient of $x^6$ in the power series expansion of $\frac{x^4-12x^2+7}{(x^2+1)^3}$ is

Consider the data on $x$ taking the values $2^0, 2^1, 2^2, \ldots, 2^n$ with frequencies ${}^nC_0, {}^nC_1, {}^nC_2, \ldots, {}^nC_n$ respectively. If the mean of this data is $\frac{728}{2^n}$,then $n$ is equal to

Which is larger: $(1.01)^{1000000}$ or $10,000$?

The expression $(2 + \sqrt{2})^4$ has a value lying between