A car moving with a certain velocity, jumps from an inclined plane placed at one bank of a river and reaches the other bank by attained a maximum height of \(80 \mathrm{~m}\). If the same car, moving with the same velocity jumps from another inclined plane having different angle of inclination and reaches the same point on the other bank by attaining maximum height of \(45 \mathrm{~m}\), then the width of the river is

The horizontal acceleration that should be given to a smooth inclined plane of angle \(\sin ^{-1}\left(\frac{1}{l}\right)\) to keep an object stationary on the plane, relative to the inclined plane is :

The efficiency of a Carnot's heat engine is \(\frac{1}{3}\). If the temperature of the source is decreased by \(50^{\circ} \mathrm{C}\) and the temperature of the sink is increased by \(25^{\circ} \mathrm{C}\), the efficiency of the engine becomes \(\frac{3}{16}\). The initial temperature of the sink is

Two points charges are kept in air with a separation between them. The force between them is \(F_1\), if half of the space between the charges is filled with a dielectric constant 4 and the force between them is \(F_2\). If \(\frac{1}{3} \mathrm{rd}\) of the space between the charges is filled with dielectric of dielectric constant 9. Then \(\frac{F_1}{F_2}\) is

The ratio of speed of electrons in the first excited state of hydrogen atom to the speed of light in vacuum is [Given, Planck's constant \(=6.625 \times 10^{-34} \mathrm{Js}\) and permittivity of free space is \(8.85 \times 10^{-12} \mathrm{Fm}^{-1}\) ]

A proton moving with a velocity \(2.5 \times 10^7 \mathrm{~m} / \mathrm{s}\), enters a magnetic field of intensity \(2.5 \mathrm{~T}\) making an angle \(30^{\circ}\) with the magnetic field. The force on the proton is

A particle is projected from the ground with an initial speed of \(v\) at an angle of projection \(\theta\). The average velocity of the particle between its time of projection and time it reaches highest point of trajectory is

If the quadratic equation formed by eliminating \(x\) from \(x^2+\alpha x+\beta=0\) and \(x y+l(x+y)+m=0\) has the same roots as that of the given quadratic equation, then the set of values of \(\beta\) is

Which one of the following has highest dipole moment?

The standard deviation of the following distribution is
\(\begin{array}{|l|c|c|c|c|}
\hline \begin{array}{l}
\text { Class interval } \\
\end{array} & 0-10 & 10-20 & 20-30 & 30-40 \\
\hline \begin{array}{l}
\text { Frequency } \\
\end{array} & 1 & 3 & 4 & 2 \\
\hline
\end{array}\)

A short bar magnet produces a magnetic field of \(6.4 \times 10^{-5} \mathrm{~T}\) at a distance of \(20 \mathrm{~cm}\) from the centre of the magnet on the normal bisector of the magnet. The magnetic field produced by this magnet at a distance of \(40 \mathrm{~cm}\) from the centre of the magnet on the axis, is

Two balls of charge \(q_1\) and \(q_2\) initially have a velocity of the same magnitude and direction. After a uniform electric field has been applied during a certain time, the direction of the velocity of the first ball changes by \(60^{\circ}\), and the velocity magnitude is reduced by half. The direction of the velocity of the second ball changes thereby by \(90^{\circ}\).
In what proportion will the velocity of the second ball change? Determine the magnitude of the charge-to-mass ratio for the second ball if it is equal to \(k_1\) for the first ball. The electrostatic interaction between the balls should be neglected.

If a solid sphere is rolling without slipping on a horizontal plane, then the ratio of its rotational and total kinetic energies is