Which of the following relations cannot be deduced using dimensional analysis? [the symbols have their usual meanings]

Consider the equation $H = \frac{x^p \epsilon^q E^r}{t^s}$,where $H = \text{magnetic field}$,$E = \text{electric field}$,$\epsilon = \text{permittivity}$,$x = \text{distance}$,and $t = \text{time}$. The values of $p, q, r$,and $s$ respectively are:

Assertion $(A) :$ To check the dimensional correctness of an equation,we use the principle of homogeneity of dimensions.
Reason $(R) :$ If the dimensions of all terms in the equation are not the same,then the equation is wrong.

The $SI$ unit of energy is $J = kg \, m^{2} \, s^{-2}$; that of speed $v$ is $m \, s^{-1}$ and of acceleration $a$ is $m \, s^{-2}$. Which of the formulae for kinetic energy $(K)$ given below can you rule out on the basis of dimensional arguments ($m$ stands for the mass of the body):
$(a)$ $K = m^{2} v^{3}$
$(b)$ $K = (1/2) m v^{2}$
$(c)$ $K = m a$
$(d)$ $K = (3/16) m v^{2}$
$(e)$ $K = (1/2) m v^{2} + m a$

Consider the efficiency of Carnot's engine is given by $\eta = \frac{\alpha \beta}{\sin \theta} \log_{e} \frac{\beta x}{k T}$,where $\alpha$ and $\beta$ are constants. If $T$ is temperature,$k$ is Boltzmann constant,$\theta$ is angular displacement and $x$ has the dimensions of length. Then,choose the incorrect option.

Two quantities $A$ and $B$ have different dimensions. Which mathematical operation given below is physically meaningful?