The tangent to the circle \(C_1 : x^2 + y^2 - 2x- 1\, = 0\) at the point \((2, 1)\) cuts off a chord of length \(4\) from a circle \(C_2\) whose centre is \((3, - 2)\). The radius of \(C_2\) is

Let \(S\) be the set of all values of \(\lambda\), for which the shortest distance between the lines \(\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{z+6}{1}\) and \(\frac{x+\lambda}{3}=\frac{y}{-4}=\frac{z-6}{0}\) is \(13\) Then \(8\left|\sum_{\lambda \in S} \lambda\right|\) is equal to

The area enclosed by \(y ^{2}=8 x\) and \(y=\sqrt{2} x\) that lies outside the triangle formed by \(y=\sqrt{2} x, x=\) \(1, y=2 \sqrt{2}\), is equal to

If the sum of series \(\frac{1}{1 \cdot(1+d)}+\frac{1}{(1+d)(1+2 d)}+\ldots \ldots+\frac{1}{(1+9 d)(1+10 d)}\) is equal to \(5\) , then \(50 \mathrm{~d}\) is equal to :

If \(y = y ( x ), x \in\left(0, \frac{\pi}{2}\right)\) be the solution curve of the differential equation \(\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=\)\(2 e ^{-4 x }(2 \sin 2 x +\cos 2 x )\), with \(y \left(\frac{\pi}{4}\right)= e ^{-\pi}\), then \(y \left(\frac{\pi}{6}\right)\) is equal to.

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation \(2 \sin ^{-1} x+3 \cos ^{-1} x=\frac{2 \pi}{5}\), is .........

Let \(A\) be the area of the region \(\left\{(x, y): y \geq x^2, y \geq(1-x)^2, y \leq 2 x(1-x)\right\}\) Then \(540\,A\) is equal to

Let \(A=\left\{n \in N \mid n^{2} \leq n+10,000\right\}, B=\{3 k+1 \mid k \in N\}\) and \(C=\{2 k \mid k \in N\}\), then the sum of all the elements of the set \(A \cap(B-C)\) is equal to \(.....\)

Let \([\cdot]\) denote the greatest integer function. If the domain of the function \(f(x) = \sin^{-1}\left(\dfrac{x+[x]}{3}\right)\) is \([\alpha, \beta)\), then \(\alpha^2 + \beta^2\) is equal to:

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

If two vertices of an equilateral triangle are \(A (- a, 0)\) and \(B ( a, 0),\,a > 0,\) and the third vertex \(C\) lies above \(x-\) axis then the equation of the circumcircle of \(\Delta ABC\) is

Let \(\overrightarrow{ a }=\hat{ i }+2 \hat{ j }-\hat{ k }, \overrightarrow{ b }=\hat{ i }-\hat{ j }\) and \(\overrightarrow{ c }=\hat{ i }-\hat{ j }-\hat{ k }\) be three given vectors. If \(\overrightarrow{ r }\) is a vector such that \(\overrightarrow{ r } \times \overrightarrow{ a }=\overrightarrow{ c } \times \overrightarrow{ a }\) and \(\overrightarrow{ r } \cdot \overrightarrow{ b }=0,\) then \(\overrightarrow{ r } \cdot \overrightarrow{ a } \quad\) is equal to ...........

The value of \(\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3\) is