A compass needle of oscillation magnetometer oscillates \(20\) times per minute at a place \(P\) of dip \(30^{\circ}\). The number of oscillations per minute become \(10\) at another place \(Q\) of \(60^{\circ} dip\). The ratio of the total magnetic field at the two places \(\left(B_{Q}: B_{p}\right)\) is.

Match List-I with List-II.

List-I (Relation)		List-II (Law)
A.	\(\oint \overrightarrow{ E } \cdot \overrightarrow{ d } l =-\frac{ d }{ dt } \oint \overrightarrow{ B } \cdot \overrightarrow{ d a}\)	I.	Ampere's circuital law.
B.	\(\oint\) B. \(\vec{d} l=\mu_0\left(1+\epsilon_0 \frac{d \phi_E}{d t}\right)\)	II.	Faraday's laws of electromagnetic induction.
C.	\(\oint \overrightarrow{ E } . \overrightarrow{ d a}=\frac{1}{\epsilon_0} \int \rho dv\)	III.	Ampere-Maxwell law
D.	\(\oint \overrightarrow{ B } \cdot \overrightarrow{ d } l =\mu_0 I\)	IV.	Gauss's law of electrostatics

Choose the correct answer from the options given below :

As per the given figure, two blocks each of mass \(250\,g\) are connected to a spring of spring constant \(2\,Nm ^{-1}\). If both are given velocity \(V\) in opposite directions, then maximum elongation of the spring is:

A conducting circular loop is placed in a uniform magnetic field of \(0.4\,T\) with its plane perpendicular to the field. Somehow, the radius of the loop starts expanding at a constant rate of \(1\,mm / s\). The magnitude of induced emf in the loop at an instant when the radius of the loop is \(2\,cm\) will be \(...........\,\mu V\).

A cubical block of density \(\rho_{ b }=600 kg / m ^3\) floats in a liquid of density \(\rho_{ e }=900 kg / m ^3\). If the height of block is \(H=8.0 cm\) then height of the submerged part is _________ cm .

Let a total charge \(2Q\) be distributed in a sphere of radius \(R\), with the charge density given by \(\rho(r) = kr\), where \(r\) is the distance from the centre. Two charges \(A\) and \(B\), of \(-Q\) each, are placed on diametrically opposite points, at equal distance, \(a\), from the centre. If \(A\) and \(B\) do not experience any force, then

A system of three polarizers \(P_1, P_2, P_3\) is set up such that the pass axis of \(P_3\) is crossed with respect to that of \(P_1\). The pass axis of \(P_2\) is inclined at \(60^o\) to the pass axis of \(P_3\). When a beam of unpolarized light of intensity \(I_0\) is incident on \(P_1\), the intensity of light transmitted by the three polarizers is \(I\). The ratio \((I_0/I)\) equals (nearly)

The area of cross section of the rope used to lift a load by a crane is \(2.5 \times 10^{-4} m ^{2}\). The maximum lifting capacity of the crane is \(10\) metric tons. To increase the lifting capacity of the crane to \(25\) metric tons, the required area of cross section of the rope should be.\(.........\times 10^{-4} \,m ^{2}\) (take \(g =10\, ms ^{-2}\) )

For a given transistor amplifier circuit in \(CE\) configuration \(V_{C C}=1 V , R_c=1 k \Omega, R_b=100 k \Omega\) and \(\beta=100\). Value of base current \(I_b\) is

A transmitting station releases waves of wavelength \(960\, m\). A capacitor of \(2.56\, \mu F\) is used in the resonant circuit. The self inductance of coil necessary for resonance is ............. \(-\times 10^{-8} H\)

There is a small source of light at some depth below the surface of water (refractive index \(=\frac{4}{3}\) ) in a tank of large cross sectional surface area. Neglecting any reflection from the bottom and absorption by water, percentage of light that emerges out of surface is (nearly)...........\(\%\) [Use the fact that surface area of a spherical cap of height \(\mathrm{h}\) and radius of curvature \(\mathrm{r} \text { is } 2 \pi \mathrm{rh}]\)

An engine is attached to a wagon through a shock absorber of length \(1.5\, {m}\). The system with a total mass of \(40,000\, {kg}\) is moving with a speed of \(72\, {kmh}^{-1}\) when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by \(1.0\, {m}\). If \(90\, \%\) of energy of the wagon is lost due to friction, the spring constant is \(....\, \times 10^{5}\, {N} / {m}\)