If two liquids of same masses but densities ${\mathrm{\rho }}_1$ and ${\mathrm{\rho }}_2$ respectively are mixed, then density of mixture is given by

(1) $\mathrm{\rho }=\frac{{\mathrm{\rho }}_1+{\mathrm{\rho }}_2}{2}$                             

(2) $\mathrm{\rho }=\frac{{\mathrm{\rho }}_1+{\mathrm{\rho }}_2}{2{\mathrm{\rho }}_1{\mathrm{\rho }}_2}$

(3) $\mathrm{\rho }=\frac{2{\mathrm{\rho }}_1{\mathrm{\rho }}_2}{{\mathrm{\rho }}_1+{\mathrm{\rho }}_2}$                             

(4) $\mathrm{\rho }=\frac{{\mathrm{\rho }}_1{\mathrm{\rho }}_2}{{\mathrm{\rho }}_1+{\mathrm{\rho }}_2}$

The density $\mathrm{\rho }$ of water of bulk modulus B at a depth y in the ocean is related to the density at surface ${\mathrm{\rho }}_0$ by the relation

(1) $\mathrm{\rho }={\mathrm{\rho }}_0\left[1-\frac{{\mathrm{\rho }}_0\mathrm{gy}}{\mathrm{B}}\right]$                           

(2) $\mathrm{\rho }={\mathrm{\rho }}_0\left[1+\frac{{\mathrm{\rho }}_0\mathrm{gy}}{\mathrm{B}}\right]$

(3) $\mathrm{\rho }={\mathrm{\rho }}_0\left[1+\frac{\mathrm{B}}{{\mathrm{\rho }}_0\mathrm{gy}}\right]$                             

(4) $\mathrm{\rho }={\mathrm{\rho }}_0\left[1-\frac{\mathrm{B}}{{\mathrm{\rho }}_0\mathrm{gy}}\right]$

With rise in temperature, density of a given body changes according to one of the following relations

(a) $\mathrm{\rho }={\mathrm{\rho }}_0\left[1+\mathrm{\gamma } \mathrm{d\theta }\right]$                        (b) $\mathrm{\rho }={\mathrm{\rho }}_0\left[1-\mathrm{\gamma } \mathrm{d\theta }\right]$

(c) $\mathrm{\rho }={\mathrm{\rho }}_0\mathrm{\gamma } \mathrm{d\theta }$                                (d) $\mathrm{\rho }={\mathrm{\rho }}_0/\mathrm{\gamma } \mathrm{d\theta }$

For the figures given below, the correct observation is:
              

A given shaped glass tube having uniform cross section is filled with water and is mounted on a rotatable shaft as shown in figure. If the tube is rotated with a constant angular velocity $\mathrm{\omega }$ then

(1) Water levels in both sections A and B go up

(2) Water level in Section A goes up and that in B comes down

(3) Water level in Section A comes down and that in B it goes up

(4) Water levels remains same in both sections

An iceberg of density \(900 ~\text{kg/m}^ 3\)${\mathrm{}}^{}$ is floating in the water of density \(1000 ~\text{kg/m}^ 3.\) The percentage of the volume of ice cube outside the water is: 
1. \(20\%      \)                                 
2. \(35\%      \)
3. \(10\%      \)                                    
4. \(25\%      \)

A log of wood of mass 120 Kg floats in water. The weight that can be put on the raft to make it just sink, should be (density of wood = 600 Kg/${\mathrm{m}}^3$) 
(1) 80 Kg                                                 

(2) 50 Kg

(3) 60 Kg                                                 

(4) 30 Kg

A hemispherical bowl just floats without sinking in a liquid of density $1.2\times {10}^3$ $\mathrm{kg}/{\mathrm{m}}^3$. If the outer diameter and the density of the material of the bowl are 1 m and $2\times {10}^4$ $\mathrm{kg}/{\mathrm{m}}^3$ respectively, then the inner diameter of the bowl will be:
1. 0.94 m
2. 0.97 m
3. 0.98 m
4. 0.99 m

In making an alloy, a substance of specific gravity ${\mathrm{s}}_1$ and mass ${\mathrm{m}}_1$ is mixed with another substance of specific gravity ${\mathrm{s}}_2$ and mass ${\mathrm{m}}_2$ ; then the specific gravity of the alloy is

1. $\left(\frac{{\mathrm{m}}_1+{\mathrm{m}}_2}{{\mathrm{s}}_1+{\mathrm{s}}_2}\right)$     
               
2. $\left(\frac{{\mathrm{s}}_1{\mathrm{s}}_2}{{\mathrm{m}}_1+{\mathrm{m}}_2}\right)$

3. $\frac{{\mathrm{m}}_1+{\mathrm{m}}_2}{{\displaystyle \left(\frac{{\mathrm{m}}_1}{s_1}+\frac{m_2}{{\mathrm{s}}_2}\right)}}$ 
               
4. $\frac{{\displaystyle \left(\frac{{\mathrm{m}}_1}{{\mathrm{s}}_1}+\frac{{\mathrm{m}}_2}{{\mathrm{s}}_2}\right)}}{{\mathrm{m}}_1+{\mathrm{m}}_2}$

A concrete sphere of radius R has a cavity of radius r which is packed with sawdust. The specific gravities of concrete and sawdust are respectively 2.4 and 0.3. For this sphere to float with its entire volume submerged underwater, the ratio of mass of concrete to mass of sawdust will be: