An ice cube has a bubble inside. When viewed from one side the apparent distance of the bubble is \(12\,cm\). when viewed from the opposite side, the apparent distance of the bubble is observed as \(4\,cm\). If the side of the ice cube is \(24\,cm\), the refractive index of the ice cube is \(.....\).

A convex lens of refractive index \(1.5\) and focal length \(18 cm\) in air is immersed in water. The change in focal length of the lens will be \(...........cm\).\(\left(\right.\) Given refractive index of water \(\left.=\frac{4}{3}\right)\)

A galvanometer of resistance \(G\) is converted into a voltmeter of range \(0-1\, V\) by connecting a resistance \(R _{1}\) in series with it. The additional resistance that should be connected in series with \(R _{1}\) to increase the range of the voltmeter to \(0-2\, V\) will be

The distance of centre of mass from end \(A\) of a one dimensional rod \(( AB )\) having mass density \(\rho=\rho_{0}\left(1-\frac{ x ^{2}}{ L ^{2}}\right)\,kg / m\) and length \(L\) (in meter) is \(\frac{3 L }{\alpha} m\). The value of \(\alpha\) is \(\ldots \ldots \ldots . .\) (where \(x\) is the distance form end \(A\) )

Four particles, each of mass \(M\) and equidistant from each other, move along a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle is

Two open organ pipes of length \(60 \mathrm{~cm}\) and \(90 \mathrm{~cm}\) resonate at \(6^{\text {th }}\) and \(5^{\text {th }}\) harmonics respectively. The difference of frequencies for the given modes is _______ \(\mathrm{Hz}\). (Velocity of sound in air \(=333 \mathrm{~m} / \mathrm{s}\) )

Consider a finite insulated, uncharged conductor placed near a finite positively charged conductor. The uncharged body must have a potential

Given below are two statements: Statement \(I :\) Pressure in a reservoir of water is same at all points at the same level of water. Statement \(II :\) The pressure applied to enclosed water is transmitted in all directions equally. In the light of the above statements, choose the correct answer from the options given below:

A uniform rod of length \('l'\) is pivoted at one of its ends on a vertical shaft of negligible radius When the shaft rotates at angular speed \(\omega\) the rod makes an angle \(\theta\) with it (see figure). To find \(\theta\) equate the rate of change of angular momentum (direction going into the paper ) \(\frac{ m \ell^{2}}{12} \omega^{2} \sin \theta \cos \theta\) about the centre of mass \((CM)\) to the torque provided by the horizontal and vertical forces \(F_{H}\) and \(F_{V}\) about the CM. The value of \(\theta\) is then such that:

Correct formula for height of a satellite from earths surface is _______.

Match the List\(-I\) with List\(-II\) 

List \(I\)	List \(II\)
\(A\) Microwaves	\(I\) Radio active decay of the nucleus
\(B\) Gamma rays	\(II\) Rapid acceleration and deceleration of electron in aerials
\(C\) Radio waves	\(III\) Inner shell electrons
\(D\) \(X\)-rays	\(IV\) Klystron valve

Choose the correct answer from the options given below