The dimensional formula of the coefficient of thermal conductivity is :

1.  $\left[{\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-3}{\mathrm{K}}^{-1}\right]$

2.  $\left[{\mathrm{M}}^3{\mathrm{L}}^{-1}{\mathrm{T}}^{-3}\mathrm{K}\right]$

3.  $\left[{\mathrm{M}}^1{\mathrm{L}}^{-3}{\mathrm{T}}^{-1}{\mathrm{K}}^{-1}\right]$

4.  $\left[{\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-3}\mathrm{K}\right]$

The measurements are made as $18.425$ cm, $7.21$ cm, and $5.0$ cm. The addition of these measurements should be written as:

1. $30.635$ cm

2. $30.64$ cm

3. $30.63$ cm

4. $30.6$ cm

A physical parameter '$a$' can be determined by measuring the parameters $b$, $c$, $d$ and $e$ using the relation, $a= \dfrac{b^{\alpha}c^{\beta}}{d^{\gamma}e^{\delta}}.$ If the maximum errors in the measurement of $b, ~c, ~d,~\text{and}~e$ are $b_1\%,~c_1\%,~d_1\%~\text{and}~e_1\%$, then the maximum error in the value of '$a$' determined by the experiment is:
1. $(b_1+c_1+d_1+e_1)\%$
2. $(b_1+c_1-d_1-e_1)\%$
3. $(\alpha b_1+\beta c_1-\gamma d_1-\delta e_1)\%$
4. $(\alpha b_1+\beta c_1+\gamma d_1+\delta e_1)\%$

The unit of angular acceleration in the SI system is 

1. $Nk{g}^{-1}$

2. $m{s}^{-2}$

3. $rad{s}^{-2}$

4. $mk{g}^{-1}K$

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. $\frac{m{r}^2}{6\pi \eta }$

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

3. $\frac{m}{6\pi \eta rv}$

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 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

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. $1, 2, \frac{1}{3}$