A spherical body of mass m and radius r is allowed to fall in a medium of viscosity $\eta$. The time in which the velocity of the body increases from zero to 0.63 times the terminal velocity $\left(v\right)$ is called time constant $\left(\tau \right)$. Dimensionally $\tau$ can be represented by 

(1) $\frac{m{r}^2}{6\pi \eta }$

(b) $\sqrt{\left(\frac{6\pi mr\eta }{g^2}\right)}$

(c) $\frac{m}{6\pi \eta rv}$

(4) None of the above

The frequency of vibration f of a mass m suspended from a spring of spring constant K is given by a relation of this type $f=C{m}^x{K}^y$; where C is a dimensionless quantity. The value of x and y are 

1. $x=\frac{1}{2},y=\frac{1}{2}$

2. $x=-\frac{1}{2},y=-\frac{1}{2}$

3. $x=\frac{1}{2},y=-\frac{1}{2}$

4. $x=-\frac{1}{2},y=\frac{1}{2}$

The quantities A and B are related by the relation, m = A/B, where m is the linear density and A is the force. The dimensions of B are of

1. Pressure

2. Work

3. Latent heat

4. None of the above

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) $ML{T}^2$