The wavelength $\lambda$ of matter waves is given by $\lambda = \frac{h}{p}$, where $h$ is Planck's constant and $p$ is momentum. If a new quantity $Z$ is defined as $Z = \frac{h^2}{\lambda^2 p c}$, where $c$ is the speed of light, the dimensions of $Z$ are:

A physical quantity of the dimensions of length that can be formed out of $c$, $G$ and $\frac{e^2}{4\pi \varepsilon_0}$ is [where $c$ is velocity of light, $G$ is universal constant of gravitation and $e$ is charge]

In the relation y=r sin $\left( {\omega t - kx} \right),\;$, the dimensions of $\omega /k$ are

In the relation $P = \frac{\alpha }{\beta }{e^{\frac{{\alpha Z}}{{kheta }}}}$ P is pressure, Z is the distance, k is Boltzmann’s constant and $heta$ is the temperature. The dimensional formula of β will be

A rod of length L is placed on x axis between x = 0 and x = L. The linear density i.e., mass per unit length denoted by r, of this rod, varies as, r = a + bx. The dimensions of b will be

The equation $P = \frac{{mx}}{A}$ gives a relation between mass m, kept on a surface of area A and pressure P exerted on this area. The dimensions of x are

The equation of state of some gases can be expressed as $\left( {P + \frac{a}{{{V^2}}}} \right)\left( {V - b} \right) = \;RT$. Here P is the pressure, V is the volume, T is the absolute temperature and a,b,R are constants. The dimensions of ‘a’ are

The gravitational potential energy of a system is $U$. If $G$ represents the gravitational constant, what are the dimensions of $\frac{U}{G}$?

The dimensions of gravitational constant G are:

The potential energy of a particle varies with distance x from a fixed origin as $U = \left( {\frac{{A\sqrt X }}{{x + B}}} \right);$; where A and B are constants. The dimensions of AB are

For the equation $F \propto {A^a}{v^b}{d^c}$, where F is the force, A is the area v is the velocity and d is the density, the value of a,b and c are respectively