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 _____________.

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $$2.0\,N{m^{ - 1}}$$ and the mass of the block is $$2.0$$ $$kg.$$ Ignore the mass of the spring. Initially the spring is an unstretched condition. Another block of mass $$1.0$$ $$kg$$ moving with a speed of $$2.0$$ $$m{s^{ - 1}}$$ collides elastically with the first block. The collision is such that the $$2.0$$ $$kg$$ block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is __________.

One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where $$V$$ is the volume and $$T$$ is the temperature). Which of the statements below 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 ______________.