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 $$g = 10\,\,m{s^{ - 2}}.$$

Two infinitely long straight wires lie in the $$xy$$-plane along the lines $$x = \pm R.$$ The wire located at $$x = + R$$ carries a constant current $${I_1}$$ and the wire located at $$x=-R$$ carries a constant current $${I_2}.$$ A circular loop of radius $$R$$ is suspended with its center at $$\left( {0,0,\sqrt 3 R} \right)$$ and in a plane parallel to the $$xy$$-plane. This loop carries a constant current $$I$$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $$ + \widehat j$$ direction. which of the following statements regarding the magnetic field $$\overrightarrow B $$ is (are) true?

Two vectors $$\overrightarrow A $$ and $$\overrightarrow B $$ are defined as $$\overrightarrow A $$ $$=$$ $$a\widehat i$$ and $$\overrightarrow B = a$$ $$\left( {\cos \,\omega T\widehat i + \sin \,\omega t\,\widehat j} \right),$$ where $$a$$ is a constant and $$\omega = \pi /6\,\,rad{s^{ - 1}}.$$ If $$\left| {\overrightarrow A + \overrightarrow B } \right| = \sqrt 3 \left| {\overrightarrow A - \overrightarrow B } \right|$$ at time $$t = \tau $$ for the first time, the value of $$\tau ,$$ in second, is ______________.

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed $$1.0\,m{s^{ - 1}}$$ and the man behind walks at a speed $$2.0\,m{s^{ - 1}}.$$ A third man in standing at a height $$12$$ $$m$$ above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency $$1430$$ $$Hz$$. The speed of sound in air is $$330\,m{s^{ - 1}}.$$ At the instant, when the moving men are $$10$$ $$m$$ apart, the stationary man is equidistant from them. The frequency of beats in $$Hz,$$ heard by the stationary man at this instant, is _____________.