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Two capillaries of length L and 2 L and of radii R and 2 R are connected in series. The rate of the flow of fluid through single capillary tube of length L, radius R is X$(=\mathrm{\pi } {\mathrm{PR}}^4/8 \mathrm{\eta } \mathrm{L})$. The net rate of flow fluid through the two capillaries in series is:
1. (5/6) X
2. (6/7) X
3. (7/8) X
4. (8/9) X

From a water reservoir, water is coming out through a vertical conical tube with upper end of radius R and lower end of radius r. The length of the tube is l as shown in the figure. The velocity of the water entering through the upper end of the tube is:
(ignoring the height of water column in the reservoir)

1. $\sqrt{\frac{2 \mathrm{gl}}{{(\mathrm{R}/\mathrm{r})}^2-1}}$
2. $\sqrt{\frac{2 \mathrm{gl}}{{(\mathrm{R}/\mathrm{r})}^4+1}}$
3. $\sqrt{\frac{2 \mathrm{gl}}{{(\mathrm{R}/\mathrm{r})}^4-1}}$
4. $\sqrt{\frac{2 \mathrm{gl}}{{(\mathrm{R}/\mathrm{r})}^4+1}}$

A uniformly tapering vessel shown in the figure is filled with liquid of density $900 \mathrm{kg}/{\mathrm{m}}^3$. The force that acts on the base of the vessel due to liquid is: $( \mathrm{take} \mathrm{g}=10 \mathrm{m}/{\mathrm{s}}^2)$

1. 3.6 N
2. 7.2 N
3. 9.0 N
4. 12.6 N

Two substances of densities ${\mathrm{\rho }}_1 \mathrm{and} {\mathrm{\rho }}_2$ are mixed in equal volume and the relative density of mixture is 4. When they are mixed in equal masses, the relative density of the mixture is 3. The values of ${\mathrm{\rho }}_1 \mathrm{and} {\mathrm{\rho }}_2$ are:
1. ${\mathrm{\rho }}_1=2 \mathrm{and} {\mathrm{\rho }}_2=5$
2. ${\mathrm{\rho }}_1=3 \mathrm{and} {\mathrm{\rho }}_2=6$
3. ${\mathrm{\rho }}_1=6 \mathrm{and} {\mathrm{\rho }}_2=2$
4. ${\mathrm{\rho }}_1=5 \mathrm{and} {\mathrm{\rho }}_2=4$

An open vessel containing water is given a constant acceleration a in the horizontal direction. Then the free surface of water gets sloped with the horizontal at an angle $\mathrm{\theta }$, given by
1. $\mathrm{\theta }={\cos }^{-1}\frac{\mathrm{g}}{\mathrm{a}}$
2. $\mathrm{\theta }={\tan }^{-1}\mathrm{a}/\mathrm{g}$
3. $\mathrm{\theta }={\sin }^{-1} \mathrm{a}/\mathrm{g}$
4. $\mathrm{\theta }={\tan }^{-1} \mathrm{g}/\mathrm{a}$

The density of ice s 0.9 g/c.c. and that of the sea water is 1.1 g/c.c. an ice berg of volume V is floating in sea water. The fraction of ice berg above water level is:
1. 1/11
2. 2/11
3. 3/11
4. 4/11

A candle of diameter d is floating on a liquid in a cylindrical container of diameter D (D>>d) as shown in figure. It is burning at the rate of 2 cm/hour, then the top of the candle will

1. remain at the same height
2. fall at the rate of 1 cm/h
3. fall at the rate of 2 cm/h
4. go up at the rate of 1 cm/h

Two capillary tubes of same bore diameter and lengths ${\mathrm{l}}_1 \mathrm{and} {\mathrm{l}}_2$ are fitted horizontal side by side to the bottom of a vessel containing water. The length of a single tube that can replace the two tubes such that the rate of steady flow through this tube equals the combined rate of flow through the two tubes is:
1. $({\mathrm{l}}_1+{\mathrm{l}}_2)/2$
2. $\sqrt{{\mathrm{l}}_1^2+{\mathrm{l}}_2^2}$
3. $\frac{{\mathrm{l}}_1{\mathrm{l}}_2}{({\mathrm{l}}_1+{\mathrm{l}}_2)}$
4. $\frac{2{\mathrm{l}}_1{\mathrm{l}}_2}{({\mathrm{l}}_1+{\mathrm{l}}_2)}$

A ball of mass m and radius r is released in a viscous liquid. The value of its terminal velocity is proportional to
1. $\frac{1}{\mathrm{r}}$
2. $\frac{\mathrm{m}}{\mathrm{r}}$
3. $\sqrt{\frac{\mathrm{m}}{\mathrm{r}}}$
4. m only

A large tank filled with water to a height 'h' is to be emptied through a small hole at the bottom. The ratio of time taken for the level of water to fall from h to $\frac{\mathrm{h}}{2}$ and from $\frac{\mathrm{h}}{2}$ to zero is
1. $\sqrt{2}$
2. $\frac{1}{\sqrt{2}}$
3. $\sqrt{2}-1$
4. $\frac{1}{\sqrt{2}-1}$

A clock with a metal pendulum beating seconds keeps correct time at $0°\mathrm{C}$. If it loses 12.5 s a day at $25°\mathrm{C}$, the coefficient of linear expansion of metal pendulum is
1. $\frac{1}{86400}/°C$
2. $\frac{1}{43200}/°C$
3. $\frac{1}{14400}/°C$
4. $\frac{1}{28800}/°C$

On a new scale of temperature (which is linear) and called the W scale, the freezing and boiling points of water are $39°$ W and $239°$ W respectively. what will be the temperature on the new scale, corresponding to a temperature of $39° \mathrm{C}$ on the Celsius scale
1. $200°$ W
2. $139° \mathrm{W}$
3. $78° \mathrm{W}$
4. $117°\mathrm{W}$

Five rods of same dimensions are arranged as shown in figure. They have thermal conductivities ${\mathrm{K}}_1, {\mathrm{K}}_2, {\mathrm{K}}_3, {\mathrm{K}}_4 \mathrm{and} {\mathrm{K}}_5$. When points A and B are maintained at different temperatures, no heat would flow through central rod, if

1. ${\mathrm{K}}_1{\mathrm{K}}_4={\mathrm{K}}_2{\mathrm{K}}_3$
2. ${\mathrm{K}}_1={\mathrm{K}}_4 \mathrm{and} {\mathrm{K}}_2={\mathrm{K}}_3$
3. $\frac{{\mathrm{K}}_1}{{\mathrm{K}}_4}=\frac{K_2}{K_3}$
4. ${\mathrm{K}}_1{\mathrm{K}}_2={\mathrm{K}}_3{\mathrm{K}}_4$

A slab consists of two portions of different material of same thickness and having the conductivities ${\mathrm{K}}_1 \mathrm{and} {\mathrm{K}}_2$. The equivalent thermal conductivity of the slab is
1. ${\mathrm{K}}_1+{\mathrm{K}}_2$
2. $\frac{{\mathrm{K}}_1{\mathrm{K}}_2}{{\mathrm{K}}_1+{\mathrm{K}}_2}$
3. $\frac{2{\mathrm{K}}_1{\mathrm{K}}_2}{{\mathrm{K}}_1+{\mathrm{K}}_2}$
4. $\sqrt{{\mathrm{K}}_1+{\mathrm{K}}_2}$

A hot liquid kept in a beaker cools from $80°\mathrm{C}$ to $70°\mathrm{C}$ in two minutes. If the surrounding temperature is $30°\mathrm{C}$, then the time of cooling f the same liquid from $60°\mathrm{C} \mathrm{to} 50°\mathrm{C}$ is
1. 240 s
2. 360 s
3. 480 s
4. 216 s

Three rods made of the same material and having the same cross-section have been joined as shown in Figure. Each rod is of the same length. The left and right ends are kept at $0°\mathrm{C} \mathrm{and} 90°\mathrm{C}$ respectively. The temperature of the junction of the three rods is

1. $20°\mathrm{C}$
2. $30°\mathrm{C}$
3. $45°\mathrm{C}$
4. $60°\mathrm{C}$

Two rods of lengths ${\mathrm{L}}_1 \mathrm{and} {\mathrm{L}}_2$ are made of materials having coefficients of linear expansion ${\mathrm{\alpha }}_1 \mathrm{and} {\mathrm{\alpha }}_2$ respectively. If $({\mathrm{L}}_1-{\mathrm{L}}_2)$ is independent of temperature, which of the following relations is correct?
1. ${\mathrm{L}}_1{\mathrm{L}}_2=\frac{{\mathrm{\alpha }}_1}{{\mathrm{\alpha }}_2}$
2. ${\mathrm{L}}_1{\mathrm{L}}_2={\mathrm{\alpha }}_1{\mathrm{\alpha }}_2$
3. ${\mathrm{L}}_1{\mathrm{\alpha }}_1={\mathrm{L}}_2{\mathrm{\alpha }}_2$
4. None of these