Q. A cylindrical vessel of base radius R and height H has a narrow neck of height h and radius r at one end(see figure). The vessel is filled with water(density ${\rho }_w$) and its neck is filled with immiscible oil (density ${\rho }_o$). Then the pressure at :

1. M is $gH{\rho }_w$
2. M is $g(h{\rho }_o+H{\rho }_w)$
3. N is $g(h{\rho }_o+H{\rho }_w)\frac{r^2}{R^2}$
4. N is $g\frac{{\rho }_{w}H{R}^2+{\rho }_{o}h{r}^2}{R^2+r^2}$

Q. The number of completely filled shells for the element ${}_{16}{S}^{32}$ is?

1. $3$
2. $2$
3. $1$
4. $4$

Q. The refractive index of water in a biology laboratory tank varies as $1.33+0.002/{\lambda }^2$, where $\lambda$ is the wavelength of light. Small pieces of organic matter of different colours are seen at the bottom of the tank using a travelling microscope. Then the image of the organic matter appears.

1. Shallower for the blue pieces than the orange ones
2. Deeper for the violet pieces than the green ones
3. At the same depth for both the blue and orange pieces
4. Deeper for the green pieces than the red ones

Q. A total charge q is divided as $q_1$ and $q_2$ which are kept at two of the vertices of an equilateral triangle of side a. The magnitude of the electric field E at the third vertex of the triangle is to be depicted schematically as a function of $x=q_1/q$. Choose the correct figure.

1. 
   
2. 
   
3. 
   
4. 
   

Q. A mirror is placed at an angle of ${30}^o$ with respect to y-axis(see figure). A light rat travelling in the negative y-direction strikes the mirror. The direction of the reflected ray is given by the vector.

1. $\hat{i}$
2. $\hat{i}-2\hat{j}$
3. $\hat{i}-\sqrt{3}\hat{j}$
4. $\sqrt{3}\hat{i}-\hat{j}$

Q. Particles used in the Rutherford's scattering experiment to deduce the structure of atoms.

1. Had atomic number $2$ and were fully ionised
2. Had atomic number $2$ and were neutral
3. Had atomic number $4$ and were fully ionised
4. Had atomic number $4$ and were neutral

Q. A uniform metallic wire of length L is mounted in two configurations. In configuration $1$(triangle), it is an equilateral triangle and a voltage V is applied to corners A and B. In configuration $2$(circle), it is bent in the form of a circle, and the potential V is applied at diametrically opposite points P and Q. The ratio of the power dissipated in configuration $1$ to configuration $2$ is?

1. $2/3$
2. $5/4$
3. $9/8$
4. $7/8$

Q. Six objects are placed at the vertices of a regular hexagon. The geometric center of the hexagon is at the origin with objects $1$ and $4$ on the x-axis(see figure). The mass of the $k^{th}$ object is $m_k=k^{i}M|\cos {\theta }_k|$ where i is an integer, M is a constant with dimension of mass, and ${\theta }_k$ is the angular position of the $k^{th}$ vertex measured from the positive x-axis in the counter-clockwise sense. If the net gravitational force on a body at the centroid vanishes, the value of i is?

1. $1$
2. $0$
3. $2$
4. $3$

Q. Two students P and Q perform an experiment to verify Ohm's law for a conductor with resistance R. They use a current source and a voltmeter with least counts of $0.1$mA and $0.1$mV, respectively. The plots of the variation of voltage drop(V) across R with current(i) for both are shown. the statement which is most likely to be correct is?

1. P has only random error(s)
2. Q has only systematic error(s)
3. Q has both random and systematic errors
4. P has both random and systematic errors

Q. In an experiment on simple pendulum to determine the acceleration due to gravity, a student measures the length of the thread as $63.2$cm and diameter of the pendulum bob as $2.256$cm. The student should take the length of the pendulum to be.

1. $64.328$cm
2. $64.3$cm
3. $65.456$cm
4. $65.5$cm