The area of the region enclosed between the circles \( x^{2}+y^{2}=4 \) and \( x^{2}+(y-2)^{2}=4 \) is:

Let the sixth term in the binomial expansion of \(\left(\sqrt{2^{\log _2}\left(10-3^x\right)}+\sqrt[5]{2^{(x-2) \log _2 3}}\right)^m\), in the increasing powers of \(2^{(x-2) \log _2 3}\), be \(21\) . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an \(A.P.\), then the sum of the squares of all possible values of \(x\) is \(.........\).

If the system of equations \(x+y+a z=b\) \(2 x+5 y+2 z=6\) \(x+2 y+3 z=3\) has infinitely many solutions, then \(2 a+3 b\) is equal to \(...........\).

Let the solution curve of the differential equation \(x d y=\left(\sqrt{x^{2}+y^{2}}+y\right) d x, x>0\), intersect the line \(x =1\) at \(y =0\) and the line \(x =2\) at \(y =\alpha\). Then the value of \(\alpha\) is.

Let \(\alpha \) and \(\beta \) be the roots of equation \(p{x^2} + qx + r = 0\) ( where \(p \ne 0\)) . If \(p,q,r\) are in \(A.P.\) and \(\frac{1}{\alpha } + \frac{1}{\beta } = 4\) , then the value of \(\left| {\alpha - \beta } \right|  \) is

If \(P = \left[ {\begin{array}{*{20}{c}}
{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\
{ - \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}
\end{array}} \right],\,A = \,\left[ {\begin{array}{*{20}{c}}
1&1\\
0&1
\end{array}} \right]\) and \(Q=PAP^T,\) then \(P^T\) \(Q^{2015}\) \(P\) is 

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______

A rectangle is inscribed in a circle with a diameter lying along the line \(3y = x + 7\) . If the two adjacent vertices of the rectangle are \((-8, 5)\) and \((6, 5)\) then the area of the rectangle (in \(sq. units\)) is

Locus of the image of point \( (2,3)\) in the line \(\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,k \in R\) is  a:

If \(x^{2}+9 y^{2}-4 x+3=0, x, y \in R\), then \(x\) and \(y\) respectively lie in the intervals:

Consider the function \(\mathrm{f}:(0,2) \rightarrow \mathrm{R}\) defined by \(f(x)=\frac{x}{2}+\frac{2}{x}\) and the function\( g(x)\) defined by \(g(x)=\left\{\begin{array}{cc}\min \{f(t)\}, & 0 < t \leq x \text { and } 0 < x \leq 1 \\ \frac{3}{2}+x, & 1 < x< 2\end{array}\right.\). Then

If the curves, \(x^{2}-6 x+y^{2}+8=0\) and \(\mathrm{x}^{2}-8 \mathrm{y}+\mathrm{y}^{2}+16-\mathrm{k}=0,(\mathrm{k}>0)\) touch each other at a point, then the largest value of \(\mathrm{k}\) is

Let \(\mathrm{Q}\) and \(\mathrm{R}\) be the feet of perpendiculars from the point \(\mathrm{P}(\mathrm{a}, \mathrm{a}, \mathrm{a})\) on the lines \(\mathrm{x}=\mathrm{y}, \mathrm{z}=1\) and \(\mathrm{x}=-\mathrm{y}\), \(\mathrm{z}=-1\) respectively. If \(\angle \mathrm{QPR}\) is a right angle, then \(12 \mathrm{a}^2\) is equal to