To remove the second term of the equation \(x^4-8 x^3+x^2-x+3=0\), diminish the roots of the equation by

The points \((5,-4,5),(-3,-3,2)\) and \((-1,-6,8)\) form ......

A random variable \(X\) takes values \(0,1,2,3, \ldots\) with probability \(P(X=x)=K(x+1)\left(\frac{1}{5}\right)^x\), where \(K\) is constant, then \(P(X=0)\) is

Let \(A=\left(\begin{array}{ll}-\cot \theta & \operatorname{cosec} \theta \\ \operatorname{cosec} \theta & -\cot \theta\end{array}\right)\). If \(A^{-1}=A\) at \(\theta=\theta_1\) and \(A^{-1}\) \(+\mathrm{A}=\mathrm{O}\) at \(\theta=\theta_2\), then which one of the following is True?

In \(\triangle \mathrm{ABC}\), if \(\cos \mathrm{A} \cdot \cos \mathrm{B} \cdot \cos \mathrm{C}=\frac{1}{5}\), then \(\tan \mathrm{A} \tan \mathrm{B}+\) \(\tan \mathrm{B} \tan \mathrm{C}+\tan \mathrm{C} \tan \mathrm{A}=\)

The probability that A speaks truth is \(4 / 5\), while the probability that \(B\) speaks truth is \(3 / 4\). Then the probability that A and B contradict each other, when asked to reveal the fact is

Length of the perpendicular from the origin to the plane passing through three non-collinear points with position vectors a, \(\mathbf{b}\) and \(\mathbf{c}\) is

Water in a river \(20 \mathrm{~m}\) deep is flowing at a speed of \(10 \mathrm{~ms}^{-1}\). The shearing stress between the horizontal layers of water in the river in \(\mathrm{Nm}^{-2}\) is (Coefficient of viscosity of water \(=10^{-3} \mathrm{ \ SI}\) units)

A ball of mass \(2 \mathrm{~g}\) moving with a velocity of \(2 \mathrm{~ms}^{-1}\) collides with another ball of mass \(8 \mathrm{~g}\) which is at rest and comes to rest after collission. Then the coefficient of restitution is

A narrow slit of width 2 mm is illuminated with a monochromatic light of wavelength 500 nm. If the distance between the slit and the screen is 1 m, then first minima are separated by a distance of

Some substances are given below:
\(\mathrm{Ag}: \mathrm{CO}_2(\mathrm{~s}): \mathrm{SiO}_2 ; \mathrm{ZnS}\)
\(\mathrm{SO}_2\) (s); \(\mathrm{AlN}: \mathrm{HCl}(\mathrm{s}) ; \mathrm{H}_2 \mathrm{O}\) (s)
The number of molecular solids and network solids in the above list is respectively

In the figure shown, \(\mathrm{AB}\) is a rod of length \(30 \mathrm{~cm}\), area of cross-section \(1 \mathrm{~cm}^2\) and thermal conductivity \(336 \mathrm{SI}\) units. The ends \(\mathrm{A}\) and \(\mathrm{B}\) are at constant temperatures \(20^{\circ} \mathrm{C}\) and \(40^{\circ} \mathrm{C}\) respectively. A point \(\mathrm{C}\) of the rod is connected to ice at \(0^{\circ} \mathrm{C}\) in a thermally insulated box \(\mathrm{D}\) through a highly conducting wire of negligible heat capacity. The rate at which ice melts in the box is \(\left(\mathrm{L}_{\mathrm{ice}}=80 \mathrm{cal} \mathrm{g}^{-1}\right)\)

In an atom the difference between two energy levels is \(3.31 \mathrm{eV}\), then the wavelength of the radiation emitted when the transition takes place between these levels is nearly