Let \( A=\{2,3,5,7,9\} \). Let R be the relation on A defined by \(x\) Ry if and only if \( 2x\le3y \). Let \(l\) be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then \( l+m \) is equal to :

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

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the work in 24 days less than mason B alone. Then mason A alone will complete the work in:

Let \(S_n\) denote the sum of first \(n\) terms an arithmetic progression. If \(S_{20}=790\) and \(S_{10}=145\), then \(S_{15}-\) \(S_5\) is :

Statement \(1\) : The only circle having radius \(\sqrt {10} \) and a diameter along line \(2x + y = 5\) is \(x^2 + y^2 - 6x +2y = 0\).
Statement \(2\) : \(2x + y = 5\) is a normal to the circle \(x^2 + y^2 -6x+2y = 0\).

Let \(A\) be a \(2 \times 2\) matrix with \(\operatorname{det}(A)=-1\) and det \((( A + I )(\operatorname{Adj}( A )+ I ))=4\). Then the sum of the diagonal elements of \(A\) can be.

The sum of all the solutions of the equation \((8)^{2 x}-16 \cdot(8)^x+48=0\) is :

Two circles with equal radii intersecting at the points \((0, 1)\) and \((0, -1).\) The tangent at the point \((0, 1)\) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is

If \(\int {{x^5}\,{e^{ - {x^2}}}\,dx\, = \,g\,(x)\,{e^{ - {x^2}}} + \,c,} \) where \(c\) is a constant of integration, then \(g(-1)\) is equal to

If \(a, b\) and \(c\) are the greatest value of \(^{19} \mathrm{C}_{\mathrm{p}},^{20} \mathrm{C}_{\mathrm{q}}\) and \(^{21 }\mathrm{C}_{\mathrm{r}}\) respectively, then

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

If \(\int {\frac{{log\left( {t + \sqrt {1 + {t^2}} } \right)}}{{\sqrt {1 + {t^2}} }}dt = \frac{1}{2}{{\left( {g\left( t \right)} \right)}^2} + C} \) , where \(C\) is a constant, then \(g(2)\) is equal to

Let a line having direction ratios \(1,-4,2\) intersect the lines \(\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}\) and \(\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}\) at the point \(A\) and \(B\). Then \(( AB )^{2}\) is equal to