If the sum of the coefficients of all even powers of $x$ in the product $(1+x+x^{2}+\ldots+x^{2n})(1-x+x^{2}-x^{3}+\ldots+x^{2n})$ is $61$,then $n$ is equal to

The number of natural numbers $n$ in the interval $[1005, 2010]$ for which the polynomial $1+x+x^2+x^3+\ldots+x^{n-1}$ divides the polynomial $1+x^2+x^4+x^6+\ldots+x^{2010}$ is

If $T_r = ^{2016}C_r x^{2016-r}$ for $r = 0, 1, 2, \dots, 2016$,then $(T_0 - T_2 + T_4 - \dots + T_{2016})^2 + (T_1 - T_3 + T_5 - \dots - T_{2015})^2$ is equal to-

Let $C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+x)^{10}$. If $\alpha, \beta \in R$,and $C_{1}+3 \cdot 2 C_{2}+5 \cdot 3 C_{3}+\ldots$ (up to $10$ terms) $= \frac{\alpha \times 2^{11}}{2^{\beta}-1} \left( C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\ldots \right.$ (up to $10$ terms) $)$,then the value of $\alpha+\beta$ is equal to:

The remainder when the polynomial $1+x^2+x^4+x^6+\ldots+x^{22}$ is divided by $1+x+x^2+x^3+\ldots+x^{11}$ is

If $1 + (2 + {}^{49}C_{1} + {}^{49}C_{2} + \dots + {}^{49}C_{49})({}^{50}C_{2} + {}^{50}C_{4} + \dots + {}^{50}C_{50})$ is equal to $2^{n} \cdot m$,where $m$ is odd,then $n + m$ is equal to.