If the orbital velocity of a planet is given by $v = G^a M^b R^c$,then:

The expressions below give current $I$ through an electronic component as a function of applied potential $V$. $I_0$ and $V_0$ are constants having dimensions of current and potential respectively. Which of the following are dimensionally incorrect?
$(A)$ $I=I_0\left(e^{\frac{2 V}{V_0}}+1\right)$
$(B)$ $I=I_0\left(e^{\frac{V}{2 V_0}}-1\right)$
$(C)$ $I=I_0 V_0\left(e^{\frac{V}{V_0}}-1\right)$
$(D)$ $I=I_0\left(\frac{V}{V_0}\right)\left(e^{\frac{V}{V_0}}-1\right)$

If the time period $T \propto P^a d^b E^c$,where $P$ is pressure,$d$ is density,and $E$ is energy,then the value of $c$ is:

Which of the following relations cannot be deduced using dimensional analysis? [the symbols have their usual meanings]

The energy $(E)$, angular momentum $(L)$, and universal gravitational constant $(G)$ are chosen as fundamental quantities. The dimensions of the universal gravitational constant in the dimensional formula of Planck's constant $(h)$ is

Force $F$ is given in terms of time $t$ and distance $x$ by $F = a \sin(ct) + b \cos(dx)$. Then the dimension of $a/b$ is: