If velocity of light $c$,Planck's constant $h$,and gravitational constant $G$ are taken as fundamental quantities,then express length in terms of dimensions of these quantities.

The frequency of vibration $f$ of a mass $m$ suspended from a spring of spring constant $K$ is given by a relation of the type $f = C\,{m^x}{K^y}$,where $C$ is a dimensionless quantity. The values of $x$ and $y$ are:

$A$ system has basic dimensions as density $[D]$,velocity $[V]$,and area $[A]$. The dimensional representation of force in this system is:

If $C$ is the velocity of light,$h$ is Planck's constant,and $G$ is the gravitational constant,and these are taken as fundamental quantities,then the dimensional formula of mass is:

$A$ book with many printing errors contains four different formulas for the displacement $y$ of a particle undergoing a certain periodic motion:
$(a) \; y = a \sin \left(\frac{2 \pi t}{T}\right)$
$(b) \; y = a \sin v t$
$(c) \; y = \left(\frac{a}{T}\right) \sin \frac{t}{a}$
$(d) \; y = (a \sqrt{2}) \left(\sin \frac{2 \pi t}{T} + \cos \frac{2 \pi t}{T}\right)$
($a =$ maximum displacement of the particle,$v =$ speed of the particle,$T =$ time-period of motion). Rule out the wrong formulas on dimensional grounds.

$A$ quantity $f$ is given by $f = \sqrt{\frac{hc^5}{G}}$,where $c$ is the speed of light,$G$ is the universal gravitational constant,and $h$ is Planck's constant. The dimension of $f$ is that of: