A small quantity of water of mass ' m ' at temperature \(\theta^{\circ} \mathrm{C}\) is mixed with a large mass ' M ' of ice which is at its melting point. If ' s ' is specific heat capacity of water and ' \(L\) ' is the Latent heat of fusion of ice, then the mass of ice melted is

In a crossed field, the magnetic field induction is \(2.0 \mathrm{~T}\) and electric field intensity is \(20 \times 10^3 \mathrm{~V} / \mathrm{m}\). At which velocity the electron will travel in a straight line without the effect of electric and magnetic fields?

Two concentric coils of 20 turns each are placed in same plane. Their radii are \(30 \mathrm{~cm}\) and \(60 \mathrm{~cm}\) and carry \(0.4 \mathrm{~A}\) and \(0.6 \mathrm{~A}\) currents respectively in opposite directions. The magnetic induction at the centre in tesla is ....... .

When a pure resistor is connected to an AC source, the phase difference between the voltage and the current through the resistor is

Hydrogen gas is undergoing a process given by \(\mathrm{PV}^2=\) constant. The ratio of work done by the gas to the change in its internal energy is

A large vessel with a small hole at the bottom is filled with water and kerosine with kerosine floating on water. The length of water column is \(20 \mathrm{~cm}\) and that of kerosine is \(25 \mathrm{~cm}\). The velocity with which water flows out of the hole is (density of kerosine \(=0.8 \mathrm{~g} \mathrm{~cm}^{-3}\), neglect viscous force).

The radius ' \(r\) ' of a wire is given by \(r=\sqrt{\frac{64 \mathrm{LA}}{\pi \mathrm{Bv}}}\), where \(\mathrm{I}\) is electric current, \(\mathrm{B}\) is magnetic field and ' \(v\) ' is velocity. Then the parameter A in the formula represents ____.

If \(x\) and \(y\) are acute angles such that \(\cos x+\cos y=\frac{3}{2}\) and \(\sin x+\sin y=\frac{3}{4}\), then \(\sin (x+y)\) equals to

Let ' \(\mathrm{O}\) ' be the origin, \(\mathrm{A}\) and \(\mathrm{B}\) be two points with position vectors \(-3 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(4 \hat{i}-4 \hat{j}-3 \hat{k}\) respectively. Let \(P\) be a point such that the line drawn through \(P\) parallel to \(\overrightarrow{\mathrm{OB}}\) meets \(\mathrm{OA}\) in \(\mathrm{L}\) and another line through \(\mathrm{P}\) parallel to \(\overrightarrow{\mathrm{OA}}\) meets \(\mathrm{OB}\) in \(\mathrm{M}\). If \(\mathrm{L}\) divides \(\mathrm{OA}\) in the ratio \(2: 3\) and \(\mathrm{M}\) divides \(\mathrm{OB}\) in the ratio \(3: 2\), then the distance from 0 to \(\mathrm{P}\) is

Let \(y=Y(x)\) be the solution of the differential equation \(\frac{d y}{d x}+y \tan x=2 x+x^2 \tan x\), \(x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)\), such that \(Y(0)=1\), then

Let \(P\) be a real number and \(|P| \geq 2\). If \(A, B, C\) are variable angles such that
\[
\left(\sqrt{P^2-4}\right) \tan A+P \tan B+\left(\sqrt{P^2+4}\right) \tan C=6 P \text {, }
\]
then the minimum value of
\[
\tan ^2 A+\tan ^2 B+\tan ^2 C=
\]

The total number of body centred lattices possible among the 14 bravais lattices is

\(\cos 6^{\circ} \sin 24^{\circ} \cos 72^{\circ}=\)