Let $\lim_{n \rightarrow \infty} ( \frac{n}{\sqrt{n^4+1}} - \frac{2n}{(n^2+1)\sqrt{n^4+1}} + \frac{n}{\sqrt{n^4+16}} - \\ \frac{8n}{(n^2+4)\sqrt{n^4+16}} + \dots + \frac{n}{\sqrt{n^4+n^4}} - \frac{2n \cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}} ) \\ = \frac{\pi}{k},$ using only the principal values of the inverse trigonometric functions. Then $\mathrm{k}^2$ 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 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 ___________.

If the circles ${x^2} + {y^2} + 6x + 8y + 16 = 0$ and , , touch internally at the point , then is equal to ________________.

Let the point $(p, p+1)$ lie inside the region $E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^{2}}, 0 \leq x \leq 3\right\}$. If the set of all values of is the interval , then 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 __________.

A circle with centre (2, 3) and radius 4 intersects the line $x+y=3$ at the points P and Q. If the tangents at P and Q intersect at the point $S(\alpha,\beta)$, then is equal to ___________.

If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and , where , then the distance of the point from the line is __________.

Let $w=z \bar{z}+k_{1} z+k_{2} i z+\lambda(1+i), k_{1}, k_{2} \in \mathbb{R}$. Let be the circle of radius 1 in the first quadrant touching the line and the -axis. If the curve intersects at and , then is equal to __________

If the area enclosed by the parabolas $\mathrm{P_1:2y=5x^2}$ and is equal to the area enclosed by and , then is equal to ____________.