The velocity of water waves v may depend upon their wavelength $\lambda$, the density of water $\rho$ and the acceleration due to gravity g. The method of dimensions gives the relation between these quantities as: 

1. ${\mathrm{v}}^2\propto {\mathrm{g\lambda }}^{-1}{\mathrm{\rho }}^{-1}$

2. ${\mathrm{v}}^2\propto \mathrm{g\lambda \rho }$

3. ${\mathrm{v}}^2\propto \mathrm{g\lambda }$

4. ${\mathrm{v}}^2\propto {\mathrm{g}}^{-1}{\mathrm{\lambda }}^{-3}$

The dimensions of resistivity in terms of \(M\), \(L\), \(T\), and \(Q\) where \(Q\) stands for the dimensions of charge, will be:
1. \(\left[M L^3 T^{-1} Q^{-2}\right]\)
2. \(\left[M L^3 T^{-2} Q^{-1}\right]\)
3. \(\left[M L^2 T^{-1} Q^{-1}\right]\)
4. \(\left[M L T^{-1} Q^{-1}\right]\)

The dimensions of Farad are , where Q represents electric charge [This question includes concepts from 12th syllabus]

1. $M^{-1}{L}^{-2}{T}^2{Q}^2$

2. $M^{-1}{L}^{-2}TQ$

3. $M^{-1}{L}^{-2}{T}^{-2}Q$

4. $M^{-1}{L}^{-2}T{Q}^2$

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

1. LT

2. T

3. $T^{-1}$

4. T²

Dimensional formula of capacitance is 

1. $M^{-1}{L}^{-2}{T}^4{A}^2$

2. $M{L}^2{T}^4{A}^{-2}$

3. $ML{T}^{-4}{A}^2$

4. $M^{-1}{L}^{-2}{T}^{-4}{A}^{-2}$

Dimensional formula of heat energy is  

1. $M{L}^2{T}^{-2}$

2. $ML{T}^{-1}$

3. $M^0{L}^0{T}^{-2}$

4. None of these

If C and L denote capacitance and inductance respectively, then the dimensions of LC are 

1. $M^0{L}^0{T}^0$

2. $M^0{L}^0{T}^2$

3. $M^2{L}^0{T}^2$

4. $ML{T}^2$

Which of the following quantities has the same dimensions as that of energy?

1. Power

2. Force

3. Momentum

4. Work

The dimensions of "time constant" $\frac{L}{R}$ during growth and decay of current in all inductive circuit is same as that of  

1. Constant

2. Resistance

3. Current

4. Time

The period of a body under SHM is presented by $\mathrm{T}={\mathrm{P}}^{\mathrm{a}}{\mathrm{D}}^{\mathrm{b}}{\mathrm{S}}^{\mathrm{c}}$; where P is pressure, D is density and S is surface tension. The value of a, b and c are:

1. $-\frac{3}{2},\frac{1}{2},1$

2. $-1,-2,3$

3. $\frac{1}{2},-\frac{3}{2},-\frac{1}{2}$

4. $1, 2, \frac{1}{3}$