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 ___________.

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 the sum of the coefficients of all the positive powers of x, in the Binomial expansion of ${\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}$ is 939, then the sum of all the possible integral values of n is _________.

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 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 ___________.

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

If the term without $x$ in the expansion of $\left(x^{\frac{2}{3}}+\frac{\alpha}{x^{3}}\right)^{22}$ is 7315 , then is equal to ___________.