Newtonian Mechanics and Forces

The main quantity that measures the motion of a body is momentum. An object's momentum increases with the object's mass and with its velocity.

Gradient of a quantity $Q$ is given as $\dfrac{\Delta Q}{\Delta x}$
Thus, a gradient will be unitless if its numerator has same dimensions as denominator, i.e. $x$ (which has the dimension $L$).
Thus, out of the options, Displacement has the dimension $L$ and hence its gradient will be dimensionless.

Angular velocity and angular frequency have the same dimension  $[T^{-1}]$.

The gradient of any physical quantity is computed to as per unit length. This means it is a rate with respect to length. So, the Displacement gradient will also be length per unit length. Because a displacement gradient is the ratio of two quantities having dimensions of "length". That is, a displacement gradient has units of a meter per meter for example. 

The length dimensions will be cancelled, and we have a unitless quantity. While others will be having certain units. Hence it is a unitless quantity.

Thus option B is correct.

Kinetic energy is defined as the energy possessed by an object due to its motion.

When an object is floating, the net force on it will be zero. This happens when the volume of the object submerged displaces an amount of liquid whose weight is equal to the weight of the object.

We know that potential energy is the energy possesd by the virtue of its position.
So, $F = mgh $
Where, $ m = $ mass
So, From above equation we can deduce that potential energy depends on the mass of the substance.

Potential energy of a body is defined as energy possessed by a body by  virtue of its position at a height.
$ P.E = mgh$

Angular momentum is conserved when the net external torque on a system is zero. Torque, not force or pressure, is what changes angular momentum. The absence of net external torque means angular momentum remains constant.

The law of conservation of linear momentum is generally true for isolated systems, but in the presence of external forces (which are ubiquitous), it is not always conserved for a specific body. However, in the context of fundamental physics, all listed conservation laws are highly robust, but linear momentum is often cited as the one most easily 'broken' by external forces.

Under the action of a central force, there is a conservation ofAngular momentum and mechanical energy.

Angular momentum is conserved when the net external torque acting on a system is zero. This is the rotational analog to the conservation of linear momentum, which occurs when the net external force is zero.