The amount of heat energy $Q$ used to heat up a substance depends on its mass $m$,its specific heat capacity $s$,and the change in temperature $\Delta T$ of the substance. Using the dimensional method,find the expression for $s$. (Given that $[s] = [L^2 T^{-2} K^{-1}]$)

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

The time period of revolution of a satellite $(T)$ around the earth depends on the radius of the circular orbit $(R)$,mass of the earth $(M)$,and universal gravitational constant $(G)$. The expression for $T$,using dimensional analysis is ($K$ is a constant of proportionality):

The entropy of any system is given by
$S = \alpha^{2} \beta \ln \left[\frac{\mu k R}{J \beta^{2}} + 3\right]$
Where $\alpha$ and $\beta$ are constants. $\mu, J, k$ and $R$ are the number of moles,mechanical equivalent of heat,Boltzmann constant,and gas constant respectively. [Take $S = \frac{dQ}{T}$].
Choose the incorrect option from the following:

In the relation $y = a \cos (\omega t - kx)$,the dimensional formula for $k$ is

The dimensional formula for a physical quantity $x$ is $[M^{-1} L^{3} T^{-2}]$. The errors in measuring the quantities $M, L$ and $T$ respectively are $2\%, 3\%$ and $4\%$. The maximum percentage of error that occurs in measuring the quantity $x$ is (in $\%$)