A circular disc \(D_1\) of mass \(M\) and radius \(R\) has two identical discs \(D_2\) and \(D_3\) of the same mass \(M\) and radius \(R\) attached rigidly as its opposite ends (see figure). The moment of inertia of the system about the axis \(OO\)’, passing through the centre of \(D_1\) as shown in the figure, will

A guitar string of length \(90\,cm\) vibrates with a fundamental frequency of \(120\,Hz.\) The length of the string producing a fundamental frequency of \(180\,Hz\) will be \(...........cm\).

Consider a n-type semiconductor in which \(\mathrm{n}_{\mathrm{e}}\) and \(\mathrm{n}_{\mathrm{h}}\) are number of electrons and holes, respectively.
(A) Holes are minority carriers
(B) The dopant is a pentavalent atom
(C) \(\mathrm{n}_{\mathrm{e}} \mathrm{n}_{\mathrm{h}} \neq \mathrm{n}_{\mathrm{i}}^2\)
(where \(n_i\) is number of electrons or holes in semiconductor when it is intrinsic form)
(D) \(\mathrm{n}_{\mathrm{e}} \mathrm{n}_{\mathrm{h}} \geq \mathrm{n}_{\mathrm{i}}^2\)
(E) The holes are not generated due to the donors Choose the correct answer from the options given below :

A thin prism with angle \( 5^{\circ} \) of refractive index 1.72 is combined with another prism of refractive index 1.9 to produce dispersion without deviation. The angle of second prism is ___________.

A simple pendulum of length \(L\) is placed between the plates of a parallel plate capacitor having electric field \(E,\) as shown in figure. Its bob has mass \(m\) and charge \(q.\) the time period of the pendulum is given by

An object of mass \(1 kg\) is taken to a height from the surface of earth which is equal to three times the radius of earth. The gain in potential energy of the object will be \(....MJ\)[If, \(g =10\,ms ^{-2}\) and radius of earth \(=6400\,km\) ]

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R.\) Assertion \(A:\) An electric fan continues to rotate for some time after the current is switched off. Reason \(R:\) Fan continuous to rotate due to inertia of motion. In the light of above statements, choose the most appropriate answer from the options given below.

Six point charges are kept \( 60^{\circ} \) apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the centre of the circle is __________ .
(\( \epsilon \) is permittivity of free space)

The distance travelled by an object in time \(t\) is given by \(s =(2 \cdot 5) t ^2\). The instantaneous speed of the object at \(t =5\,s\) will be \(....\,ms ^{-1}\)

A capacitor has capacitance \(5 \mu F\) when it's parallel plates are separated by air medium of thickness \(d\). A slab of material of dielectric constant \(1.5\) having area equal to that of plates but thickness \(\frac{ d }{2}\) is inserted between the plates. Capacitance of the capacitor in the presence of slab will be \(..........\mu F\)

A block of ice of mass \(120\,g\) at temperature \(0^{\circ} C\) is put in \(300\,gm\) of water at \(25^{\circ} C\). The \(xg\) of ice melts as the temperature of the water reaches \(0^{\circ} C\). The value of \(x\) is [Use: Specific heat capacity of water \(=4200\) \(J\,kg ^{-1} K ^{-1}\), Latent heat of ice \(\left.=3.5 \times 10^{5} J\,kg ^{-1}\right]\)

A plano-convex lens having radius of curvature of first surface 2 cm exhibits focal length of \(f_1\) in air. Another plano-convex lens with first surface radius of curvature 3 cm has focal length of \(\mathrm{f}_2\) when it is immersed in a liquid of refractive index 1.2 . If both the lenses are made of same glass of refractive index 1.5 , the ratio of \(f_1\) and \(f_2\) will be _______.

A cylindrical vessel containing a liquid is rotated about its axis so that the liquid rises at its sides as shown in the figure. The radius of vessel is \(5\, cm\) and the angular speed of rotation is  \(\omega\; rad \,s^{-1}\). The difference in the height, \(h(\) in \(cm )\) of liquid at the centre of vessel and at the side will be