If \(\alpha, \beta, \gamma\) are the roots of \(\left|\begin{array}{ccc}1-x & -2 & 1 \\ -2 & 4-x & -2 \\ 1 & -2 & 1-x\end{array}\right|=0\), then \(\alpha \beta+\beta \gamma+\gamma \alpha=\)

The point to which the origin should be shifted in order to eliminate the $x$ and $y$ terms from the equation $9x^2+4y^2+10x+12y+1=0$ is

Let \(A=\left[\begin{array}{ccc}b^2+c^2 & a^2 & a^2 \\ b^2 & c^2+a^2 & b^2 \\ c^2 & c^2 & a^2+b^2\end{array}\right]\). If
\(a=\sin \frac{\pi}{6}, b=\cos \frac{\pi}{4}\) and \(c=\cot \frac{\pi}{2}\), then \(A\) is

If \(\operatorname{Tanh}^{-1} \mathrm{x}=\operatorname{Coth}^{-1} \mathrm{y}=\log \sqrt{5}\), then \(\operatorname{Tan}^{-1}(\mathrm{xy})=\)

If \(A>0, B>0\) and \(A+B=\frac{\pi}{3}\), then the maximum value of \(\tan A \tan B\) is

In a binomial distribution the probability of getting success is \(\frac{1}{4}\) and the standard deviation is 3 . Then, its mean is

If \(\int \frac{\cos 4 x+1}{\cot x-\tan x} d x=k \cos 4 x+c\), then \(k\) is

A wire of length 0.5 m and area of cross-section \(4 \times 10^{-6} \mathrm{~m}^2\) at a temperature of \(100^{\circ} \mathrm{C}\) is suspended vertically by fixing its upper end to the ceiling. The wire is then cooled to \(0^{\circ} \mathrm{C}\), but is prevented from contracting, by attaching a mass at the lower end. If the mass of the wire is negligible, then the value of the mass attached to the wire is
[Young's modulus of material of the wire \(=10^{11} \mathrm{~N} \mathrm{~m}^{-2}\); coefficient of linear expansion of the material of the wire \(=10^{-5} \mathrm{~K}^{-1}\) and acceleration due to gravity \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)

The elevation in the boiling point of an aqueous solution of $\mathrm{NaCl}$ is $0.01°\mathrm{C}$. If its van't Hoff factor is $1.92,$ the molality of $\mathrm{NaCl}$ solution is ___$({\mathrm{K}}_{\mathrm{b}}$ for water $=0.52 \mathrm{K} \mathrm{kg} {\mathrm{mol}}^{-1})$

In a plane electromagnetic wave, the maximum value of the electric field component is \(4.4 \mathrm{Vm}^{-1}\). The intensity of the wave is nearly

Two small spheres each having equal positive charge \(Q\) (Coulomb) on each are suspended by two insulating strings of equal length \(L\) (metre) from a rigid hook (shown in figure). The whole set up is taken into satellite where there is no gravity. The two balls are now held by electrostatic forces in horizontal position, the tension in each string is then

If the position vectors of \(A, B, C, D\) are \(\bar{i}+2 \bar{j}+2 \bar{k}, 2 \bar{i}-\bar{j}, \bar{i}+\bar{j}+3 \bar{k}\) and \(4 \overline{\mathrm{j}}+5 \overline{\mathrm{k}}\) respectively, then the quadrilateral ABCD is a

The equation of the plane bisecting the line segment joining the points \((2,0,6)\) and \((-6,2,4)\) and perpendicular to it, is