Two short bar magnets of moments ' \(\mathrm{M}\) ' and ' \(\sqrt{3} \mathrm{M}\) ' are joined like a cross (+). This cross is suspended from its centre with its plane horizontal in the earth's magnetic field. When the cross comes to equilibrium, angle made by the weaker magnet with the earth's magnetic field \(B_H\) is

A diatomic gas consisting of rigid molecules is at a temperature of \(87^{\circ} \mathrm{C}\). If the moment of inertia of the rotating diatomic rigid molecule is \(2.76 \times 10^{-39} \mathrm{gcm}^2\), then the rms angular speed of the molecule is (Boltzmann constant \(=1.38 \times 10^{-23} \mathrm{JK}^{-1}\) )

The logic gates in which all the inputs must be low to get a high output are

Three particles of each charge \(q\) are placed at the vertices of an equilateral triangle of side \(L\). The work to be done to decrease the side of the triangle to \(\frac{L}{2}\) 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 \(\left.=10 \mathrm{~m} \mathrm{~s}^{-2}\right]\)

When the angle of inclination of an inclined plane is \(\theta\), an object slides down with uniform velocity. If the same object is pushed up with a initial velocity \(u\) on the same inclined plane; it goes up the plane and stops at a certain distance on the plane. Thereafter the body :

The truth table given below corresponds to logic gate. $\begin{array}{|ccc|}\hline \mathbf{A}& \mathbf{B}& \mathbf{X}\\ 0& 0& 0\\ 0& 1& 1\\ 1& 0& 1\\ 1& 1& 1\\ \hline\end{array}$

Electric field lines of force of a positive point charge are

The pair of xenon compounds which have same number of lone pairs of electrons on the central atom is

If the slope of one of the lines represented by \(a x^2-6 x y+\) \(y^2=0\) is the square of the other, then the value of \(a\) is

If \(\mathrm{A}(\cos \alpha, \sin \alpha), \mathrm{B}(\sin \alpha,-\cos \alpha), \mathrm{C}(1,2)\) are the vertices of a \(\triangle \mathrm{ABC}\) then the locus of its centroid is

The angle made by the vector $2\overline{i}-\overline{j}+\overline{k}$ with the plane represented by $\overline{r}·(\overline{i}+\overline{j}+2\overline{k})=7$ is

A solid sphere of radius $R$ carries a positive charge $Q$ distributed uniformly throughout its volume. A very thin hole is drilled through it's center. A particle of mass $m$ and charge $-q$ performs simple harmonic motion about the center of the sphere in this hole. The frequency of oscillation is