If the speed of light $(c)$, acceleration due to gravity $(g)$ and pressure $(p)$ are taken as the fundamental quantities, then the dimension of gravitational constant is

The equation of a wave is given by$Y = A\sin \omega \left( {\frac{x}{v} - k} \right)$where $\omega $ is the angular velocity and $v$ is the linear velocity. The dimension of $k$ is

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

With the usual notations, the following equation ${S_t} = u + \frac{1}{2}a(2t - 1)$ 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 dimensionally consistent relation for the volume V of a liquid of coefficient of viscosity ' $\eta$ ' flowing per second, through a tube of radius $r$ and length / and having a pressure difference $P$ across its ends, is