Let $(1 + x + x^2)^{20}(2x + 1) = a_0 + a_1x^1 + a_2x^2 + ... + a_{41}x^{41}$,then $\frac{a_0}{1} + \frac{a_1}{2} + .... + \frac{a_{41}}{42}$ is equal to

The correct matching of List-$I$ from List-$II$ is:

Let $(1+x+x^2)^9=a_0+a_1 x+a_2 x^2 +\ldots+a_{18} x^{18}$. Then

For an integer $n \geq 2$,if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2n-3}$ is $16$,then the distance of the point $P(2n-1, n^2-4n)$ from the line $x+y=8$ is:

The fractional part of a real number $x$ is defined as $x - [x]$,where $[x]$ is the greatest integer less than or equal to $x$. Let $F_1$ and $F_2$ be the fractional parts of $(44 - \sqrt{2017})^{2017}$ and $(44 + \sqrt{2017})^{2017}$,respectively. Then,$F_1 + F_2$ lies between the numbers:

Let $R=(5 \sqrt{5}+11)^{2 n+1}$ and $f=R-[R]$,where $[x]$ denotes the greatest integer less than or equal to $x$,then $R f=$