A ring and disc are initially at rest, side by side, at the top of an inclined plane which makes an angle $${60^ \circ }$$ with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $$\left( {2 - \sqrt 3 } \right)/\sqrt {10} \,\,s,$$ then the height of the top of the inclined plane, in metres is ______________ . Take $$g = 10\,\,m{s^{ - 2}}.$$

In the figure below, the switches $${S_1}$$ and $${S_2}$$ are closed simultaneously at $$t=0$$ and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current $$I$$ in the middle wire reaches its maximum magnitude $${I_{\max }}$$ at time $$t = \tau $$ . Which of the following statements is (are) true?

In the $$xy$$-plane, the region $$y > 0$$ has a uniform magnetic field $${B_1}\widehat k$$ and the region $$y < 0$$ has another uniform magnetic field $${B_2}\widehat k.$$ A positively charged particle is projected from the origin along the positive $$y$$-axis with speed $${v_0} = \pi \,m{s^{ - 1}}$$ at $$t=0,$$ as shown in the figure. Neglect gravity in this problem. Let $$t=T$$ be the time when the particle crosses the $$x$$-axis from below for the first time. If $${B_2} = 4{B_1},$$ the average speed of the particle, in $$m{s^{ - 1}},$$ along the $$x$$-axis in the time interval $$T$$ is ___________.

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $${T_1} = 300\,K$$ and $${T_2} = 100\,K$$, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller once and the thermal conductivities of the materials of the smaller and the larger cylinders are $${K_1}$$ and $${K_2}$$ respectively. If the temperature at the junction of the two cylinders in the steady state is $$200$$ $$K,$$ then $${K_1}/{K_2} = $$ ______________.