A particle of mass $$m$$ is initially at rest at the origin. It is subjected to a force and starts moving along the $$x$$-axis. Its kinetic energy $$K$$ changes with time as $$dK/dt = \gamma t,$$ where $$\gamma $$ is a positive constant of appropriate dimensions. Which of a positive constant of appropriate dimensions. Which of the following statement is (are) true?

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $$m=0.4$$ $$kg$$ is at rest on this surface. An impulse of $$1.0$$ $$Ns$$ is applied to the block at time $$t=0$$ so that it starts moving along the $$x$$-axis with a velocity $$v\left( t \right) = {v_0}{e^{ - t/\tau }},$$ where $${v_0}$$ is a constant and $$\tau = 4s.$$ The displacement of the block, in metres, at $$t = \tau $$ is ______________ Take $${e^{ - 1}} = 0.37.$$

A particle, of mass $${10^{ - 3}}$$ $$kg$$ and charge $$1.0$$ $$C,$$ is initially at rest. At time $$t=0,$$ the particle comes under the influence of an electric field $$\overrightarrow E \left( t \right) = {E_0}\sin \,\,$$ $$\omega t\widehat i,$$ where $${E_0} = 1.0\,N{C^{ - 1}}$$ and $$\omega = 10{}^3\,rad\,{s^{ - 1}}.$$ Consider the effect of only the electrical force on the particle. Then the maximum speed, in $$m{s^{ - 1}},$$ attained by the particle at subsequent times is _______________.

In the List-$${\rm I}$$ below, four different paths of a particle are given as functions of time. In these functions, $$\alpha $$ and $$\beta $$ are positive constants of appropriate dimensions and $$\alpha \ne \beta $$ In each case, the force acting on the particle is either zero or conservative. In List-$${\rm I}{\rm I}$$, five physical quantities of the particle are mentioned $$\overrightarrow p $$ is the linear momentum, $$\overrightarrow L $$ is the angular momentum about the origin, $$K$$ is the kinetic energy, $$U$$ is the potential energy and $$E$$ is the total energy. Match each path in List-$${\rm I}$$ with those quantities in List-$${\rm II}$$, which are conserved for that path.

LIST - I		LIST - II
P.	$$\overrightarrow r $$(t)=$$\alpha $$ $$t\,\widehat i + \beta t\widehat j$$	1.	$$\overrightarrow p $$
Q.	$$\overrightarrow r \left( t \right) = \alpha \cos \,\omega t\,\widehat i + \beta \sin \omega t\,\widehat j$$	2.	$$\overrightarrow L $$
R.	$$\overrightarrow r \left( t \right) = \alpha \left( {\cos \omega t\,\widehat i + \sin \omega t\widehat j} \right)$$	3.	K
S.	$$\overrightarrow r \left( t \right) = \alpha t\,\widehat i + {\beta \over 2}{t^2}\widehat j$$	4.	U
		5.	E