A body is executing simple harmonic motion. At a displacement ' \(x\) ' its potential energy is, \(E_1\) and at a displacement ' \(y\) ' its potential energy is \(E_2\), The potential energy E at a displacement \((x+y)\) is

If a microscope is placed in air, the minimum separation of two objects seen as distinct is 6 pm . If the same is placed in a medium of refractive index 1.5 , then the minimum separation of the two objects to see as distinct is

A man running at a speed of \(5 \mathrm{~km} / \mathrm{h}\), find that the rain falls vertically. When he stops running, he finds that the rain is falling at an angle of \(60^{\circ}\) with the horizontal. The velocity of rain with respect to running man is

At the moment \(\mathrm{t}=0\), a time dependent force \(\mathrm{F}=\) at (where a is constant equal to \(1 \mathrm{Ns}^{-1}\) ) is applied to a body of mass \(1 \mathrm{~kg}\) resting on a smooth horizontal plane as shown in the figure. If the direction of this force makes an angle \(45^{\circ}\) with the horizontal, then the velocity of the body at the moment it leaves the plane is
(acceleration due to gravity \(=10 \mathrm{~m} \mathrm{~s}^{-2}\) )

Water of mass 5 kg in a closed vessel is at a temperature of \(20^{\circ} \mathrm{C}\). If the temperature of the water when heated for a time of 10 minutes becomes \(30^{\circ} \mathrm{C}\), then the increase in the internal energy of the water is
(Specific heat capacity of water \(=4200 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\) )

The ratio of radii of two wires is \(1: 2\) and the density of their materials are in the ratio \(1: 4\). If same tension is applied to both the wires then the ratio of the speed of transverse waves produced in them is

The radius of an atomic nucleus of mass number $64$ is $4.8$ fermi. Then the mass number of another atomic nucleus of radius $6$ fermi is

A Carnot heat engine absorbs 600 J of heat from a source at a temperature of \(127^{\circ} \mathrm{C}\) and rejects 400 J of heat to a sink in each cycle. The temperature of the sink is

To which group and period does the element belong if the electronic configuration of an element in its -2 oxidation state is \(1 s^2 2 s^2 2 p^6 3 s^2 3 p^6 ?\)

If the slope of one line of the pair of lines \(2 x^2+h x y+6 y^2\) \(=0\) is thrice the slone of the other line, then \(\mathrm{h}=\)

Match the following lists.
\begin{array}{|c|c|c|c|}
\hline & List I & & List II \\
\hline A & \begin{array}{l}\text { Zeroth law of } \\
\text { thermodynamics }\end{array} & 1 & \begin{array}{l}\text { Direction of flow of } \\
\text { heat }\end{array} \\
\hline B & \begin{array}{l}\text { First law of } \\
\text { thermodynamics }\end{array} & \| & Work done is zero \\
\hline\underline{\mathrm{C}} & Free expansion of a gas & III & Thermal equilibrium \\
\hline D & \begin{array}{l}\text { Second law of } \\
\text { Thermodynamics }\end{array} & IV & \begin{array}{l}\text { Law of conservation } \\
\text { of energy }\end{array} \\
\hline
\end{array}
The correct answer is
\(\begin{array}{lccccccccc} & \text { A } & \text { B } & \text { C } & \text { D } \end{array}\)

If the determinant \(\left|\begin{array}{ccc}\cos 2 x & \sin ^2 x & \cos 2 x \\ \sin ^2 x & \cos 2 x & \cos ^2 x \\ \cos 2 x & \cos ^2 x & \cos 2 x\end{array}\right|\) is expanded in powers of \(\cos x\), then the constant term in the expansion is

A metallic wire loop of side \((l) 0.1 \mathrm{~m}\) and resistance of \(1 \Omega\) is moved with a constant velocity in a uniform magnetic field of \(2 \mathrm{Wm}^{-2}\) as shown in the figure. The magnetic field is perpendicular to the plane of the loop. The loop is connected to a network of resistors. The velocity of loop so as to have a steady current of 1 mA in loop is