If $1 + (2 + {}^{49}{C_1} + {}^{49}{C_2} + \,\,...\,\, + \,\,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4} + \,\,...\,\, + \,\,{}^{50}{C_{50}})$ is equal to $2^{\mathrm{n}} \cdot \mathrm{m}$, where $\mathrm{m}$ is odd, then $\mathrm{n}+\mathrm{m}$ is equal to __________.

The coefficient of $x^{18}$ in the expansion of $\left(x^{4}-\frac{1}{x^{3}}\right)^{15}$ is __________.

Let the mirror image of a circle $c_{1}: x^{2}+y^{2}-2 x-6 y+\alpha=0$ in line be . If is the radius of circle , then is equal to ________.

Let $f(x)=\frac{x}{\left(1+x^{n}\right)^{\frac{1}{n}}}, x \in \mathbb{R}-\{-1\}, n \in \mathbb{N}, n > 2$.

Let $\alpha = 8 - 14i,A = \left\{ {z \in c:{{\alpha z - \overline \alpha \overline z } \over {{z^2} - {{\left( {\overline z } \right)}^2} - 112i}}=1} \right\}$ and . Then is equal to ____________.

Let $\mathrm{S}=\left\{z \in \mathbb{C}-\{i, 2 i\}: \frac{z^{2}+8 i z-15}{z^{2}-3 i z-2} \in \mathbb{R}\right\}$. If , then is equal to _________.

In the expansion of $(1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$, the sum of the coefficients of and is equal to __________.

For $\alpha, \beta, z \in \mathbb{C}$ and $\lambda > 1$, if is the radius of the circle , then is equal to __________.

$\begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\ &=33 \cdot \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right] \end{aligned}$, then the integral value of is equal to _____________

Let the centre of a circle, passing through the points $(0,0),(1,0)$ and touching the circle $x^2+y^2=9$, be . Then for all possible values of the coordinates of the centre is equal to __________.

Let C_r denote the binomial coefficient of x^r in the expansion of ${(1 + x)^{10}}$. If for $\alpha$, R, ....... upto 10 terms then the value of + is equal to ___________.