Two balls of same mass and carrying equal charge are hung from a fixed support of length \(l\). At electrostatic equilibrium, assuming that angles made by each thread is small, the separation, \(x\) between the balls is proportional to

A tennis ball (treated as hollow spherical shell) starting from \(O\) rolls down a hill. At point \(A\) the ball becomes air borne leaving at an angle of \(30^o\) with the horizontal. The ball strikes the ground at \(B\). What is the value of the distance \(AB\) ? (Moment of inertia of a spherical shell of mass \(m\) and radius \(R\) about its diameter .......... \(m\). \(= \frac {2}{3}\,mR^2\))

A poly-atomic molecule ( \(C_V=3 R, C_P=4 R\), where \(R\) is gas constant) goes from phase space point \(\mathrm{A}\left(\mathrm{P}_{\mathrm{A}}=10^5 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{A}}=4 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{B}\left(\mathrm{P}_{\mathrm{B}}=5 \times 10^4 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{B}}=6 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{C}\left(\mathrm{P}_{\mathrm{C}}=10^4\right.\) \(\left.\mathrm{Pa}, \mathrm{V}_{\mathrm{C}}=8 \times 10^{-6} \mathrm{~m}^3\right)\). A to B is an adiabatic path and \(B\) to \(C\) is an isothermal path. The net heat absorbed per unit mole by the system is :

Consider a frame that is made up of two thin massless rods \(AB\) and \(AC\) as shown in the figure. \(A\) vertical force \(\overrightarrow{ P }\) of magnitude \(100 \;N\) is applied at point \(A\) of the frame. Suppose the force is \(\overrightarrow{ P }\) resolved parallel to the arms \(AB\) and \(AC\) of the frame. The magnitude of the resolved component along the arm \(AC\) is \(xN\). The value of \(x\), to the nearest integer, is ............ [Given : \(\sin \left(35^{\circ}\right)=0.573, \cos \left(35^{\circ}\right)=0.819\) \(\left.\sin \left(110^{\circ}\right)=0.939, \cos \left(110^{\circ}\right)=-0.342\right]\)  

A bob of mass ' \(m\) ' is suspended by a light string of length ' \(L\) '. It is imparted a minimum horizontal velocity at the lowest point \(A\) such that it just completes half circle reaching the top most position B. The ratio of kinetic energies \(\frac{(\text { K.E. })_A}{(\text { K.E. })_B}\) is _______.

If \({R}_{{E}}\) be the radius of Earth, then the ratio between the acceleration due to gravity at a depth \(' {r} '\) below and a height \(' r '\) above the earth surface is: (Given : \(\left.{r}<{R}_{{E}}\right)\)

For a diatomic gas, if \(\gamma_1=\left(\frac{C p}{C v}\right)\) for rigid molecules and \(\gamma_2=\left(\frac{C p}{C v}\right)\) for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct ? (Cp and Cv are specific heats of the gas at constant pressure and volume)

Let \(z\) be a complex number such that the real part of \(\frac{z-2 i}{z+2 i}\) is zero. Then, the maximum value of \(|\mathrm{z}-(6+8 \mathrm{i})|\) is equal to :

A ball having kinetic energy KE, is projected at an angle of \(60^{\circ}\) from the horizontal. What will be the kinetic energy of ball at the highest point of its flight ?

Let \(A\) be a \(3 \times 3\) invertible matrix. If \(|adj (24 A ) \mid=\) \(\operatorname{adj}(3 \operatorname{adj}(2 A )) \mid\), then \(\mid A ^{2}|=\dots\dots\dots\)  is equal to

A block takes \(t\) time to slide down a plane inclined at \(45°\) to the horizontal. If the surface is made smooth (frictionless), the block takes time \(\dfrac{t}{2}\) to slide down the plane. The coefficient of friction between the block and the inclined plane is \(\left(\dfrac{\alpha}{100}\right)\). The value of \(\alpha\) is __________.

Consider a mixture of gas molecule of types \(A, B\) and \(C\) having masses \(m_{A}\,<\,m_{B}\,<\,m_{C}\) ratio of their root mean square speeds at normal temperature and pressure is :

A ray of light passing through the point \(P (2,3)\) reflects on the \(x-\)axis at point \(A\) and the reflected ray passes through the point \(Q(5,4)\). Let \(R\) be the point that divides the line segment \(AQ\) internally into the ratio \(2: 1\). Let the co-ordinates of the foot of the perpendicular \(M\) from \(R\) on the bisector of the angle \(PAQ\) be \((\alpha, \beta)\). Then, the value of \(7 \alpha+3 \beta\) is equal to.......