If the coefficient of $x$ in the expansion of $(ax^{2}+bx+c)(1-2x)^{26}$ is $-56$ and the coefficients of $x^{2}$ and $x^{3}$ are both zero, then $a+b+c$ 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 ${T_0}, {T_1}, {T_2}, \dots, {T_n}$ represent the terms in the expansion of ${(x + a)^n}$,then $({T_0} - {T_2} + {T_4} - \dots)^2 + ({T_1} - {T_3} + {T_5} - \dots)^2 = $

If the $6^{th}$ term in the expansion of the binomial $[\sqrt{2^{\log(10 - 3^x)}} + \sqrt[5]{2^{(x - 2)\log 3}}]^m$ is equal to $21$ and it is known that the binomial coefficients of the $2^{nd}$,$3^{rd}$ and $4^{th}$ terms in the expansion represent respectively the first,third and fifth terms of an $A.P.$ (the symbol $\log$ stands for logarithm to the base $10$),then $x = $

Let the smallest value of $k \in N$, for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + \dots + (1+x)^{99} + (1+kx)^{100}, x \neq 0$, is $(43n + \frac{101}{4}) ({}^{100}C_3)$ for some $n \in N$, be $p$. Then the value of $p+n$ is:

If $(1 + x - 3x^2)^{2145} = a_0 + a_1x + a_2x^2 + \dots$,then $a_0 - a_1 + a_2 - a_3 + \dots$ ends with: