Let $P(x) = 1 + x + x^2 + x^3 + x^4 + x^5$. What is the remainder when $P(x^{12})$ is divided by $P(x)$?

Let $\alpha = \sum_{r=0}^{n} (4r^2+2r+1) {}^{n}C_{r}$ and $\beta = \left(\sum_{r=0}^{n} \frac{{}^{n}C_{r}}{r+1}\right) + \frac{1}{n+1}$. If $140 < \frac{2\alpha}{\beta} < 281$,then the value of $n$ is...............

If $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^n) \equiv a_0 + a_1x + a_2x^2 + a_3x^3 + \dots + a_mx^m$,then $\sum_{r=0}^m a_r$ has the value equal to:

Let $(1 + x)(1 + x + x^2)(1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^{30}) = a_0 + a_1x + a_2x^2 + \dots + a_{465}x^{465}$. Then the sum $a_0 + a_2 + a_4 + \dots$ is equal to:

If the coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^6\left(1+x^2\right)^7\left(1-x^3\right)^8 ; x \neq 0$ is $\alpha$,then $|\alpha|$ equals

Let $(1+x+x^2)^{2014} = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots + a_{4028} x^{4028}$. Let $A = a_0 - a_3 + a_6 - \ldots + a_{4026}$,$B = a_1 - a_4 + a_7 - \ldots - a_{4027}$,and $C = a_2 - a_5 + a_8 - \ldots + a_{4028}$. Then,