A body of mass \(5 \mathrm{~kg}\) moving with a uniform speed \(3 \sqrt{2} \mathrm{~ms}^{-1}\) in \(\mathrm{X}-\mathrm{Y}\) plane along the line \(\mathrm{y}=\mathrm{x}+4\).The angular momentum of the particle about the origin will be _______ \(\mathrm{kg}\  \mathrm{m} \mathrm{s}^{-1}\).

Mass per unit area of a circular disc of radius \(\mathrm{a}\) depends on the distance \(\mathrm{r}\) from its centre as \(\sigma(\mathrm{r})=\mathrm{A}+\mathrm{Br} .\) The moment of inertia of the disc about the the axis, perpendicular to the plane and passing through its centre is

Two identical long current carrying wires are bent into the shapes shown in the following figures. If the magnitude of magnetic fields at the centres P and Q of a semicircular arc are \(B_1\) and \(B_2\) respectively, then the ratio \(\dfrac{B_1}{B_2}\) is _______.

A mass of \(50\,g\) of water in a closed vessel, with surroundings at a constant temperature takes \(2\, minutes\) to cool from \(30\,^oC\) to \(25\,^oC\). A mass of \(100\,g\) of another liquid in an identical vessel with identical surroundings takes the same time to cool from \(30\,^oC\) to \(25\,^o C\). The specific heat of the liquid is .......... \(kcal/kg\) (The water equivalent of the vessel is \(30\,g\).)

A biconvex lens of refractive index \(1.5\) has a focal length of \(20 \mathrm{~cm}\) in air. Its focal length when immersed in a liquid of refractive index \(1.6\) will be:

Three particles of masses \(50\, g\), \(100\, g\) and \(150\, g\) are placed at the vertices of an equilateral triangle of side \(1\, m\) (as shown in the figure). The \((x, y)\) coordinates of the centre of mass will be

Two bar magnets oscillate in a horizontal plane in earth's magnetic field with time periods of \(3\,s\) and \(4\,s\) respectively. If their moments of inertia are in the ratio of \(3: 2\) then the ratio of their magnetic moments will e.

A block takes \(t\) time to slide down a plane inclined at \(45°\) to the horizontal. If the surface is made smooth (frictionless), the block takes time \(\dfrac{t}{2}\) to slide down the plane. The coefficient of friction between the block and the inclined plane is \(\left(\dfrac{\alpha}{100}\right)\). The value of \(\alpha\) is __________.

Match List \(− I\) (Event) with List \(−II \) (Order of the time interval for happening of the event) and select the correct option from the options given below the lists

List \(- I\)	List \(- II\)
\((1)\) Rotation period of earth	\((i)\) \(10^5\, s\)
\((2)\) Revolution period of earth	\((ii)\) \(10^7\, s\)
\((3)\) Period of light wave	\((iii)\) \(10^{-15}\, s\)
\((4)\) Period of sound wave	\((iv)\) \(10^{-3}\, s\)

If \(\overrightarrow{\mathrm{L}}\) and \(\overrightarrow{\mathrm{P}}\) represent the angular momentum and linear momentum respectively of a particle of mass ' m ' having position vector \(\overrightarrow{\mathrm{r}}=\mathrm{a}(\hat{\mathrm{i}} \cos \omega \mathrm{t}+\hat{\mathrm{j}} \sin \omega \mathrm{t})\). The direction of force is

A particle of mass \(\mathrm{m}\) is dropped from a height \(h\) above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of \(\sqrt{2 \mathrm{gh}} .\) If they collide head-on completely inelastically, the time taken for the combined mass to reach the ground, in units of \(\sqrt{\frac{\mathrm{h}}{\mathrm{g}}}\) is