Even if a physical quantity depends upon three quantities,out of which two are dimensionally same,then the formula cannot be derived by the method of dimensions. This statement

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):
$(a)$ $K = m^{2} v^{3}$
$(b)$ $K = (1/2) m v^{2}$
$(c)$ $K = m a$
$(d)$ $K = (3/16) m v^{2}$
$(e)$ $K = (1/2) m v^{2} + m a$

Write and explain the principle of homogeneity. Check the dimensional consistency of the given equation: $x = x_{0} + v_{0}t + \frac{1}{2}at^{2}$.

If the gravitational constant $(G)$,Planck's constant $(h)$,and the velocity of light $(c)$ are chosen as fundamental units,the dimension of the radius of gyration is:

From the equation $\tan \theta = \frac{rg}{v^2}$,one can obtain the angle of banking $\theta$ for a cyclist taking a curve (the symbols have their usual meanings). Then say,it is

In an experiment,four quantities $a, b, c,$ and $d$ are measured with percentage errors of $1\%, 2\%, 3\%,$ and $4\%$ respectively. The quantity $P$ is calculated as $P = \frac{a^3 b^2}{cd}$. The percentage error in $P$ is ........ $\%$