When an air bubble of radius \(r\) rises from the bottom to the surface of a lake, its radius becomes \(\frac{{5r}}{4}\).Taking the atmospheric pressure to be equal to \(10\,m\) height of water column, the depth of the lake would approximately be ....... \(m\) (ignore the surface tension and the effect of temperature)

When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by __________ %.

The electromagnetic waves travel in a medium at a speed of \(2.0 \times 10^{8}\, m / s\). The relative permeability of the medium is \(1.0.\) The relative permittivity of the medium will be

A light of energy \(12.75 \;eV\) is incident on a hydrogen atom in its ground state. The atom absorbs the radiation and reaches to one of its excited states. The angular momentum of the atom in the excited state is \(\frac{x}{\pi} \times 10^{-17} \;eVs\). The value of \(x\) is \(........\) (use \(h=4.14 \times 10^{-15} \;eVs , c=3 \times 10^8\;\left.ms ^{-1}\right)\)

The force required to stretch a wire of crosssection \(1 cm ^{2}\) to double its length will be ........ \( \times 10^{7}\,N\) (Given Yong's modulus of the wire \(=2 \times 10^{11}\,N / m ^{2}\) )

In a Young's double slit experiment, the source is white light. One of the slits is covered by red filter and another by a green filter. In this case...

A bullet of mass \(200\,g\) having initial kinetic energy \(90\,J\) is shot inside a long swimming pool as shown in the figure. If it's kinetic energy reduces to \(40\,J\) within 1s, the minimum length of the pool, the bullet has a to travel so that it completely comes to rest is \(.....m\)

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 : \(\sin \left(35^{\circ}\right)=0.573, \cos \left(35^{\circ}\right)=0.819\) \(\left.\sin \left(110^{\circ}\right)=0.939, \cos \left(110^{\circ}\right)=-0.342\right]\)  

Let \(S=\{z \in C :|z-3| \leq 1\) and \(z(4+3 i)+\bar{z}(4-3 i) \leq 24\}\). If \(\alpha+i \beta\) is the point in \(S\) which is closest to \(4 i\), then \(25(\alpha+\beta)\) is equal to

An electric appliance supplies \(6000\, {J} / {min}\) heat to the system. If the system delivers a power of \(90\, {W}\). How long (in \(sec\)) it would take to increase the internal energy by \(2.5 \times 10^{3}\, {J}\) ?

If mass is written as \(\mathrm{m}=\mathrm{kc}^{\mathrm{p}} \mathrm{G}^{-1 / 2} \mathrm{~h}^{1 / 2}\) then the value of \(P\) will be _______. (Constants have their usual meaning with \(\mathrm{k}\) a dimensionless constant)

A variable plane passes through a fixed point \((3,2,1)\) and meets \(x, y\) and \(z\) axes at \(A, B\) and \(C\) respectively. A plane is drawn parallel to \(yz-\) plane through \(A\), a second plane is drawn parallel \(zx -\)  plane through \(B\) and a third plane is drawn parallel to \(xy -\) plane through \(C\). Then the locus of the point of intersection of these three planes, is

The number of seven digit positive integers formed using the digits \(1,2,3\) and \(4\) only and sum of the digits equal to \(12\) is \(...........\).