The number of particles crossing a unit area perpendicular to the \(x\)-axis in unit time is given by $$\(n= -D\frac{n_2-n_1}{x_2-x_1}\), where \(n_1\) and \(n_2\) are the number of particles per unit volume for the value of \(x\) equal to \(x_1\) and \(x_2\) respectively. The dimensions of \(D,\) known as the diffusion constant, will be:
1. \(\left[M^0LT^{2}\right]\)
2. \(\left[M^0L^2T^{-4}\right]\)
3. \(\left[M^0LT^{-3}\right]\)
4. \(\left[M^0L^2T^{-1}\right]\)

With the usual notations, the following equation $S_t=u+\frac{1}{2}a(2t-1)$ is

(1) Only numerically correct

(2) Only dimensionally correct

(3) Both numerically and dimensionally correct

(4) Neither numerically nor dimensionally correct

If the dimensions of length are expressed as $G^x{c}^y{h}^z$; where G, c and h are the universal gravitational constant, speed of light and Planck's constant respectively, then  

(1) $x=\frac{1}{2},y=\frac{1}{2}$

(2) $x=\frac{1}{2},z=\frac{1}{2}$

(3) $y=\frac{1}{2},z=\frac{3}{2}$

(4) $y=-\frac{3}{2},z=-\frac{1}{2}$

A highly rigid cubical block A of small mass M and side L is fixed rigidly onto another cubical block B of the same dimensions and of low modulus of rigidity $\eta$ such that the lower face of A completely covers the upper face of B. The lower face of B is rigidly held on a horizontal surface. A small force F is applied perpendicular to one of the side faces of A. After the force is withdrawn block A executes small oscillations. The time period of which is given by 

(1) $2\pi \sqrt{\frac{M\eta }{L}}$

(2) $2\pi \sqrt{\frac{L}{M\eta }}$

(3) $2\pi \sqrt{\frac{ML}{\eta }}$

(4) $2\pi \sqrt{\frac{M}{\eta L}}$

The pair(s) of physical quantities that do not have the same dimensions, is (are) 

(1) Reynolds number and coefficient of friction

(2) Latent heat and gravitational potential

(3) Curie and frequency of a light wave

(4) Planck's constant and torque

The speed of light (c), gravitational constant (G) and Planck's constant (h) are taken as the fundamental units in a system. The dimension of time in this new system should be 

(1) $G^{1/2}{h}^{1/2}{c}^{-5/2}$

(2) $G^{-1/2}{h}^{1/2}{c}^{1/2}$

(3) $G^{1/2}{h}^{1/2}{c}^{-3/2}$

(4) $G^{1/2}{h}^{1/2}{c}^{1/2}$

If the constant of gravitation (G), Planck's constant (h) and the velocity of light (c) be chosen as fundamental units. The dimension of the radius of gyration is 

(1) $h^{1/2}{c}^{-3/2}{G}^{1/2}$

(2) $h^{1/2}{c}^{3/2}{G}^{1/2}$

(3) $h^{1/2}{c}^{-3/2}{G}^{-1/2}$

(4) $h^{-1/2}{c}^{-3/2}{G}^{1/2}$

X = 3YZ² find dimension of Y in (MKSA) system, if X and Z are the dimension of capacity and magnetic field respectively

(1) $M^{-3}{L}^{-2}{T}^{-4}{A}^{-1}$

(2) ML^–2

(3) $M^{-3}{L}^{-2}{T}^4{A}^4$

(4) $M^{-3}{L}^{-2}{T}^8{A}^4$

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

(1) $\left[M^0{L}^2{T}^0\right]$

(2) $\left[M^1{L}^2{T}^1\right]$

(3) $\left[M^1{L}^0{T}^{-1}\right]$

(4) $\left[M^0{L}^2{T}^{-1}\right]$

The frequency of vibration of string is given by $\nu =\frac{p}{2l}{\left[\frac{F}{m}\right]}^{1/2}$. Here p is number of segments in the string and l is the length. The dimensional formula for m will be

(1) $\left[M^{0}L{T}^{-1}\right]$

(2) $\left[M{L}^0{T}^{-1}\right]$

(3) $\left[M{L}^{-1}{T}^0\right]$

(4) $\left[M^0{L}^0{T}^0\right]$