The force $F$ is given in terms of time $t$ and displacement $x$ by the equation $F = A \cos(Bx) + C \sin(Dt).$ The dimensional formula of $D/B$ is

Match List-$I$ with List-$II$.

List-$I$	List-$II$
$A$. Meter $(L)$	$I$. $\sqrt{\frac{hc}{G}}$
$B$. Second $(S)$	$II$. $\sqrt{\frac{Gh}{c^5}}$
$C$. Kilogram $(M)$	$III$. $\sqrt{\frac{L^2c^3}{Gh}}$
$D$. Kelvin $(K)$	$IV$. $\sqrt{\frac{Gh}{c^3}}$

where $h$ (Planck's constant),$G$ (gravitational constant) and $c$ (speed of light in vacuum) are fundamental units. Choose the correct answer from the options given below:

To find the distance $d$ over which a signal can be seen clearly in foggy conditions,a railway engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog,intensity (power/area) $S$ of the light from the signal,and its frequency $f$. The engineer finds that $d$ is proportional to $S^{1/n}$. The value of $n$ is:

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.

Density of a liquid in $CGS$ system is $0.625 \ g/cm^3$. What is its magnitude in $SI$ system?

If speed $V$,area $A$,and force $F$ are chosen as fundamental units,then the dimension of Young's modulus will be: