A wheel is rotaing freely with an angular speed \(\omega\) on a shaft. The moment of inertia of the wheel is \(I\) and the moment of inertia of the shaft is negligible. Another wheel of momet of inertia \(3I\) initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is :

A capacitor of \(10 \mu \mathrm{F}\) capacitance whose plates are separated by \(10 \mathrm{~mm}\) through air and each plate has area \(4 \mathrm{~cm}^2\) is now filled equally with two dielectric media of \(\mathrm{K}_1=2, \mathrm{~K}_2=3\) respectively as shown in figure. If new force between the plates is \(8 \mathrm{~N}\). The supply voltage is _______ \(\times 10^{-4}\)V.

A conducting wire of length \( l\) area of cross-section \(A\) and electric resistivity \(\rho\) is connected between the terminals of a battery. \(A\) potential difference \(V\) is developed between its ends, causing an electric current.If the length of the wire of the same material is doubled and the area of cross-section is halved, the resultant current would be

A light ray is incident on a glass slab of thickness \(4 \sqrt{3} \mathrm{~cm}\) and refractive index \(\sqrt{2}\). The angle of incidence is equal to the critical angle for the glass slab with air. The lateral displacement of ray after passing through glass slab is _______ \(\mathrm{cm}\). \(\left(\right.\) Given \(\left.\sin 15^{\circ}=0.25\right)\)

A bar magnet has total length \(2 l=20\) units and the field point \(P\) is at a distance \(\mathrm{d}=10\) units from the centre of the magnet. If the relative uncertainty of length measurement is \(1 \%\), then uncertainty of the magnetic field at point P is :

A current carrying rectangular loop PQRS is made of uniform wire. The length \(PR = QS =5\,cm\) and \(PQ = RS =100\,cm\). If ammeter current reading changes from I to \(2 I\), the ratio of magnetic forces per unit length on the wire \(P Q\) due to wire RS in the two cases respectively \(f_{ PQ }^{ I }: f_{ PQ }^{2 I }\) is :

From a tower of height \(H\), a particle is thrown vertically upwards with a speed \( u\). The time taken by the particle, to hit the ground, is \(n\) times that taken by it to reach the highest point of its path. The relation between \(H, u\) and \(n\) is:

If \({L_1}\) is the line of interseciton of the planes \(2x - 2y + 3z - 2 = 0,\) \(x - y + z + 1 = 0\) and \({L_2}\) is the line of intersection of the planes  \(x + 2y - z - 3 = 0,\) \(3x - y + 2z - 1 = 0\) , then the distance of the origin from the plane , containing the line \({L_1}\) and \({L_2}\)  is:

The value of \(k\) for which the function \(f\left( x \right) = \left\{ \begin{gathered} {\left( {\frac{4}{5}} \right)^{\frac{{\tan \,4x}}{{\tan \,5x}}}},\,\,\,\,0 < x < \frac{\pi }{2} \hfill \\  k + \frac{2}{5}\,\,\,,\,\,\,\,\,\,\,\,\,\,\,x = \frac{\pi }{2} \hfill \\ \end{gathered}  \right.\) is continuous at  \(x\,= \frac{\pi}{2}\) is

The temperature \(\mathrm{T}(\mathrm{t})\) of a body at time \(\mathrm{t}=0\) is \(160^{\circ}\) \(\mathrm{F}\) and it decreases continuously as per the differential equation \(\frac{\mathrm{dT}}{\mathrm{dt}}=-\mathrm{K}(\mathrm{T}-80)\), where \(\mathrm{K}\) is positive constant. If \(\mathrm{T}(15)=120^{\circ} \mathrm{F}\), then \(\mathrm{T}(45)\) is equal to ...........

The distance of the line \(\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}\) from the point \((1,4,0)\) along the line \(\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}\) is :

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R .
Assertion A : If oxygen ion \(\left(\mathrm{O}^{-2}\right)\) and Hydrogen ion \(\left(\mathrm{H}^{+}\right)\)enter normal to the magnetic field with equal momentum, then the path of \(\mathrm{O}^{-2}\) ion has a smaller curvature than that of \(\mathrm{H}^{+}\).
Reason R : A proton with same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.
In the light of the above statements, choose the correct answer from the options given below :

A calorimeter of water equivalent \(20\, g\) contains \(180\, g\) of water at \(25^{\circ} C\). '\(m\)' grams of steam at \(100^{\circ} C\) is mixed in it till the temperature of the mixure is \(31^{\circ} C\). The value of \('m'\) is close to (Latent heat of water \(=540\) cal \(g ^{-1}\), specific heat of water \(=1\) cal \(g^{-1}{ }^{\circ} C ^{-1}\) )