Two identical metallic spheres \(A\) and \(B\) when placed at certain distance in air repel each other with a force of \(F\). Another identical uncharged sphere \(C\) is first placed in contact with \(A\) and then in contact with \(B\) and finally placed at midpoint between spheres \(A\) and \(B\). The force experienced by sphere \(C\) will be.

A particle is moving with speed \(v= b\sqrt x\) along positive \(x-\) axis. Calculate the speed of the particle at time \(t = \tau\) (assume that the particle is at origin at \(t = 0\) ).

A proton and an electron are associated with same de-Broglie wavelength. The ratio of their kinetic energies is _______. (Assume \(\mathrm{h}=6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s}, \mathrm{~m}_{\mathrm{e}}=9.0 \times 10^{-31} \mathrm{~kg}\) and \(\mathrm{m}_{\mathrm{p}}=1836\) times \(\mathrm{m}_{\mathrm{e}}\) )

The one division of main scale of vernier callipers reads \(1\,mm\) and \(10\) divisions of Vernier scale is equal to the \(9\) divisions on main scale. When the two jaws of the instrument touch each other the \(zero\) of the Vernier lies to the right of \(zero\) of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the \(zero\) of the Vernier scale lies in between \(4.1\,cm\) and \(4.2\,cm\) and \(6^{\text {th }}\) Vernier division coincides with a main scale division. The diameter of the bob will be \(.............10^{-2}\,cm\)

Match List \(I\) with List \(II.\)

List \(I\)	List \(II\)
\((a)\) Rectifier	\((i)\) Used either for stepping up or stepping down the \(a.c.\) voltage
\((b)\) Stabilizer	\((ii)\) Used to convert \(a.c.\) voltage into \(d.c.\) voltage
\((c)\) Transformer	\((iii)\) Used to remove any ripple in the rectified output voltage
\((d)\) Filter	\((iv)\) Used for constant output voltage even when the input voltage or load current change

Choose the correct answer from the options given below:

An ideal fluid of density \(800 \; kgm ^{-3}\), flows smoothly through a bent pipe (as shown in figure) that tapers in cross-sectional area from \(a\) to \(\frac{ a }{2}\). The pressure difference between the wide and narrow sections of pipe is \(4100 \; Pa\). At wider section, the velocity of fluid is \(\frac{\sqrt{x}}{6} \; ms ^{-1}\) for \(x = \dots\) \(\left(\right.\) Given \(g =10 \; m ^{-2}\) )

An upright object is placed at a distance of \(40\, cm\) in front of a convergent lens of focal length \(20\, cm\). A convergent mirror of focal length \(10\, cm\) is placed at a distance of \(60\, cm\) on the other side of the lens. The position and size of the final image will be

A satellite of mass \(\mathrm{m}\) is launched vertically upwards with an initial speed \(u\) from the surface of the earth. After it reaches height \(\mathrm{R}\) (\(R =\) radius of the earth), it ejects a rocket of mass \(\frac{\mathrm{m}}{10}\) so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is (\(G\) is the gravitational constant: \(\mathrm{M}\) is the mass of the earth)

Number of ways of arranging \(8\) identical books into \(4\) identical shelves where any number of shelves may remain empty is equal to

Consider the following logic circuit.

The output is \(\mathrm{Y}=0\) when :

Let \(x=x(y)\) be the solution of the differential equation \(y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y\gt0\) and \(x(1)=\frac{\pi}{2}\). Then \(\cos (x(2))\) is equal to :

A hall has a square floor of dimension \(10\, \mathrm{~m} \times 10\, \mathrm{~m}\) (see the figure) and vertical walls. If the angle \(GPH\) between the diagonals \(\mathrm{AG}\) and \(\mathrm{BH}\) is \(\cos ^{-1} \frac{1}{5}\), then the height of the hall (in \(meters\)) is :

A charge \(Q\) is placed at a distance \(a/2\) above the centre of the square surface of edge \(a\) as shown in the figure. The electric flux through the square surface is