A monochromatic light is travelling in a medium of refractive index $$n=1.6.$$ It enters a stack of glass layers from the bottom side at an angle $$\theta = {30^ \circ }.$$ The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $${n_m} = n - m\Delta n,$$ where $${n_m}$$ is the refractive index of the $${m^{th}}$$ slab and $$\Delta n = 0.1$$ (see the figure). The ray is refracted out parallel to the interface between the $${\left( {m - 1} \right)^{th}}$$ and $${m^{th}}$$ slabs from the right side of the stack. What is the value of $$m$$?

A stationary source emits sound of frequency $${f_0} = 492\,Hz.$$ The sound is reflected by a large car approaching the source with a speed of $$2\,m{s^{ - 1.}}$$ The reflected signal is received by the source and superposed with the original.

What will be the beat frequency of the resulting signal in $$Hz$$? (Given that the speed of sound in air is $$330\,m{s^{ - 1}}$$ and the car reflects the sound at the frequency it has received).

$${}^{131}{\rm I}$$ is an isotope of Iodine that $$B$$ decays to an isotope of Xenon with a half-life of $$8$$ days. A small amount of a serum labelled with $${}^{131}{\rm I}$$ is injected into the blood of a person. The activity of the amount of $${}^{131}{\rm I}$$ injected was $$2.4 \times {10^5}$$ Becquerel $$(Bq).$$ It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After $$11.5$$ hours, $$2.5$$ ml of blood is drawn from person's body, and gives an activity of $$115$$ $$Bq$$. The total volume of blood in the person's body, in liters is approximately (you may use $${e^x} \approx 1 + x\,\,$$ for $$\left| x \right| < < 1$$ and $$\ln 2 \approx 0.7).$$

A charged particle (electron or proton) is introduced at the origin (x=0,y=0,z=0) with a given initial velocity $$\overrightarrow v .$$ A uniform electric field $$\overrightarrow E $$ and a uniform magnetic field $$\overrightarrow B $$ exist everywhere. The velocity $$\overrightarrow v ,$$ electric field $$\overrightarrow E $$ and magnetic field $$\overrightarrow B $$ are given in column $$1,2$$ and $$3,$$ respectively. The quantities $${E_0},{B_0}$$ are positive in magnitude.

Column 1		Column 2		Column 3
(I)	Electron with $$\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$$	(i)	$$\overrightarrow E = {E_0}\widehat z$$	(P)	$$\overrightarrow B = - {B_0}\widehat x$$
(II)	Electron with $$\overrightarrow v = {{{E_0}} \over {{B_0}}}\widehat y$$	(ii)	$$\overrightarrow E = - {E_0}\widehat y$$	(Q)	$$\overrightarrow B = {B_0}\widehat x$$
(III)	Proton with $$\overrightarrow v = 0$$	(iii)	$$\overrightarrow E = - {E_0}\widehat x$$	(R)	$$\overrightarrow B = {B_0}\widehat y$$
(IV)	Proton with $$\overrightarrow v = 2{{{E_0}} \over {{B_0}}}\widehat x$$	(iv)	$$\overrightarrow E = {E_0}\widehat x$$	(S)	$$\overrightarrow B = {B_0}\widehat z$$

In which case will the particle describe a helical path with axis along the positive $$z$$ direction?