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 \(x^2 + y^2 -6x+2y = 0\).

The sum of the distinct real values of \(\mu \), for which the vectors, \(\mu \hat i + \hat j + \hat k,\,\hat i + \mu \hat j + \hat k,\,\hat i + \hat j + \mu \hat k\) are co-planar, is

If \(y=y(x)\) is the solution of the differential equation \(\frac{d y}{d x}+\frac{4 x}{\left(x^2-1\right)} y=\frac{x+2}{\left(x^2-1\right)^{\frac{5}{2}}},x > 1\) such that \(y(2)=\frac{2}{9} \log _e(2+\sqrt{3})\) and \(y(\sqrt{2})=\) \(\alpha \log _e(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma}, \alpha, \beta, \gamma \in N\), then \(\alpha \beta \gamma\) is equal to \(........\).

\(\int \limits_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}} \frac{48}{\sqrt{9-4 x^2}} d x\) is equal to

The area (in sq. units) of the region \(A\,\, = \,\left\{ {\left( {x\,,\,y} \right)\,:\,{x^2}\, \le \,y\, \le \,x + 2} \right\}\) 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 ...........

If the surface area of a cube is increasing at a rate of \(3.6 cm ^{2} / sec ,\) remaining its shape; then the rate of change of its volume (in \(cm ^{3} / sec\) ), when the length of a side of the cube is \(10 cm ,\) is

In a group of 3 girls and 4 boys, there are two boys \(B_1\) and \(B_2\). The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but \(B_1\) and \(B_2\) are not adjacent to each other, is :

The greatest integer less than or equal to the sum of first \(100\) terms of the sequence \(\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots \ldots\) is equal to

Three circles of radii \(a, b, c\, ( a < b < c )\) touch each other externally. If they have \(x -\) axis as a common tangent, then

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(Ax = b\) when the vector \(b\) on the right side is equal to \(b _{1}, b _{2}\) and \(b _{3}\) respectively. If \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) \(b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],\) then the determinant of \(A\) is equal to

\(f(x)=\left\{\begin{array}{cc}\frac{\sin (x-[x])}{x-[x]} & , \quad x \in(-2,-1) \\ \max \{2 x, 3[|x|]\} & , \quad|x|<1 \\ 1 & , \quad \text { otherwise }\end{array}\right.\) where \([t]\) denotes greatest integer \(\leq t\). If \(m\) is the number of points where \(f\) is not continuous and \(n\) is the number of points where \(f\) is not differentiable, then the ordered pair \(( m , n )\) is

The projection of the line segment joining the points \((1,-1,3)\) and \((2,-4,11)\) on the line joining the points \((-1,2,3)\) and \((3,-2,10)\) is