If $$x,$$ $$v$$ and $$a$$ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $$T,$$ then, which of the following does not change with time?

Three sound waves of equal amplitudes have frequencies $$\left( {v - 1} \right),\,v,\,\left( {v + 1} \right).$$ They superpose to give beats. The number of beats produced per second will be :

A motor cycle starts from rest and accelerates along a straight path at $$2m/{s^2}.$$ At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the siren at $$94\% $$ of its value when the motor cycle was at rest? (Speed of sound $$ = 330\,m{s^{ - 1}}$$)

Let $$P\left( r \right) = {Q \over {\pi {R^4}}}r$$ be the change density distribution for a solid sphere of radius $$R$$ and total charge $$Q$$. For a point $$'p'$$ inside the sphere at distance $${r_1}$$ from the center of the sphere, the magnitude of electric field is :