Consider a liquid of density $\rho$ in a container that spins with angular velocity $\omega$ as shown in figure. Find relation between y and x for any point P, if liquid rises due to rotation. [This question is only for Dropper and XII batch]

1. $y=\frac{\omega x}{2g}$

2. $y=\frac{{\omega }^2{x}^2}{2g}$

3. $y=\frac{{\omega }^{2}x}{2g}$

4. $y=\frac{\omega x^2}{2g}$

The upper edge of a gate in a dam runs along water surface. The gate is 2 m high and 3 m wide and is hinged along a horizontal line through its center. Calculate the torque about hinge. [This question is only for Dropper and XII batch]

1. ${10}^4 Nm$

2. $2\times {10}^4 Nm$

3. $3\times {10}^4 Nm$

4. $4\times {10}^4 Nm$

The specific heat of a substance varies with temperature t($°C$) as
c= 0.20 + 0.14 t + 0.023 $t^2 (cal/gm °C)$. Find the heat required to raise the temperature of 2 gm of substance from 5$°C$ to 15 $°C$. [This question is only for Dropper and XII batch]

1. 41 cal

2. 82 cal

3. 80 cal

4. 40 cal

An isolated container at 127$°C$ contains an ice cube of mass 100 g at 0$°C$. The specific heat C of container varies with temperature according to relation C= a+bT, where a= 0.1 kcal/kg-K and b= 40 m cal/kg K. Find the mass of container, if the final temperature of container is 300 K.
[Take $L_F$= 80 cal/g and specific heat of water 1 cal/g K]

[This question is only for Dropper and XII batch]

1. 500 g

2. 720 g

3. 940 g

4. 1200 g

The temperature of n moles of an ideal gas is increased from $T_0 to 2T_0$ through a process $P=\frac{\alpha }{T}$ . Find the work done by the gas. [This question is only for Dropper and XII batch]

1. $\frac{nR{T}_0}{2}$

2. $nR{T}_0$

3. $\frac{3}{2}nR{T}_0$

4. 2 $nR{T}_0$

Electric field is given by E $=\frac{100}{x^2}$ . Find the potential difference between x= 10 and x= 20 m. [This question includes concepts from 12th syllabus]

1. 5 V

2. 10 V

3. 15 V

4. 20 V

Calculate the electric field on the axis of a very long uniformly charged, thin rod at a distance r from one end. The charge per unit length of the rod is $\lambda$. [This question includes concepts from 12th syllabus]

1. $\frac{2k\lambda }{r}$

2. $\frac{k\lambda }{r}$

3. $\frac{k\lambda }{2r}$

4. $\frac{k\lambda }{4r}$

Find electric field at the centre of arc which subtends '$\theta$' angle at centre

[This question is only for Dropper and XII batch]

1. $\frac{2k\lambda }{r}\sin \theta$

2. $\frac{2k\lambda }{r}\sin \frac{\theta }{4}$

3. $\frac{2k\lambda }{r}\sin \frac{\theta }{2}$

4. $\frac{2k\lambda }{r}\sin 2\theta$

The electric field vector in a region given by $\overrightarrow{E}=(3\hat{i}+4y\hat{j})V{m}^{-1}$. Calculate the potential at (1m, 1m) taking potential at origin to be zero. [This question includes concepts from 12th syllabus]

1. 5V

2. 3V

3. -1V

4. -5V

The gravitational field due to a mass distribution is given by $\overrightarrow{I}= \frac{k}{x^2}\hat{i}$, where k is a constant. Assuming the potential to be zero at infinity, find the potential at a point x = a.[This question includes concepts from Gravitation chapter]

1. $\frac{k}{a^2}$

2. $\frac{-k}{a^2}$

3. $\frac{k}{a}$

4. $\frac{-k}{a}$