If two vectors \(\vec{P}=\hat{i}+2 m \hat{j}+m \hat{k}\) and \(\vec{Q}=4 \hat{i}-2 \hat{j}+ mk\) are perpendicular to each other. Then, the value of \(m\) will be :

There are \(10^{10}\) radioactive nuclei in a given radioactive element, Its half-life time is \(1\, minute.\) How many nuclei will remain after \(30\, seconds?\) \((\sqrt{2}=1.414)\)

A real gas within a closed chamber at \(27^{\circ} \mathrm{C}\) undergoes the cyclic process as shown in figure. The gas obeys \(P V^3=\mathrm{RT}\) equation for the path \(A\) to \(B\). The net work done in the complete cycle is _______.(assuming \(R=8 \mathrm{~J} / \mathrm{molK}\) )

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle \(\theta\) with each other. When suspended in water the angle remains the same. If density of the material of the sphere is \(1.5 \mathrm{~g} / \mathrm{cc}\), the dielectric constant of water will be _______. (Take density of water \(=1 \mathrm{~g} / \mathrm{cc}\) )

Two point charges \(2 q\) and \(q\) are placed at vertex \(A\) and centre of face CDEF of the cube as shown in figure. The electric flux passing through the cube is :

A large number of water drops, each of radius \(r\), combine to have a drop of radius \(R\). If the surface tension is \(T\) and mechanical equivalent of heat is \(J ,\) the rise in heat energy per unit volume will be

The mean intensity of radiation on the surface of the Sun is about \(10^{8}\,W/m^2.\) The \(rms\) value of the corresponding magnetic field is closet to

For an ideal heat engine, the temperature of the source is \(127\,^{\circ} C\). In order to have \(60\, \%\) efficiency the temperature of the sink should be \(........\,{ }^{\circ} C\). (Round off to the Nearest Integer)

The position vector of the centre of mass \(\vec r\, cm\) of an asymmetric uniform bar of negligible area of cross-section as shown in figure is

The electric field in a region is given by \(\overrightarrow{\mathrm{E}}=(2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}) \times 10^3 \mathrm{~N} / \mathrm{C}\). The flux of the field through a rectangular surface parallel to \(x-z\) plane is \(6.0 \mathrm{Nm}^2 \mathrm{C}^{-1}\). The area of the surface is ________ \(\mathrm{cm}^2\).

A \(0.07\,{H}\) inductor and a \(12 \,\Omega\) resistor are connected in series to a \(220 \,{V}, 50 \,{Hz}\) ac source. The approximate current in the circuit and the phase angle between current and source voltage are respectively. [Take \(\pi\) as \(\left.\frac{22}{7}\right]\)