A thin glass plate of thickness is $\frac { 2500 } { 3 } \lambda$ ($\lambda$ is wavelength of light used) and refractive index $\mu = 1.5$ is inserted between one of the slits and the screen in Young's double slit experiment. At a point on the screen equidistant from the slits, the ratio of the intensities before and after the introduction of the glass plate is:
(1) $2 : 1$
(2) $1 : 4$
(3) $4 : 1$
(4) $4 : 3$

The source that illuminates the double-slit in 'double-slit interference experiment' emits two distinct monochromatic waves of wavelength 500 nm and 600 nm, each of them producing its own pattern on the screen. At the central point of the pattern when path difference is zero, maxima of both the patterns coincide and the resulting interference pattern is most distinct at the region of zero path difference. But as one moves out of this central region, the two fringe systems are gradually out of step such that maximum due to one wavelength coincides with the minimum due to the other and the combined fringe system becomes completely indistinct. This may happen when path difference in nm is:
(1) 2000
(2) 3000
(3) 1000
(4) 1500

The focal length of the objective and the eyepiece of a telescope are 50 cm and 5 cm respectively. If the telescope is focussed for distinct vision on a scale distant 2 m from its objective, then its magnifying power will be:
(1) $-4$
(2) $-8$
(3) $+8$
(4) $-2$

A ray of light of intensity I is incident on a parallel glass slab at point A as shown in diagram. It undergoes partial reflection and refraction. At each reflection, $25 \%$ of incident energy is reflected. The rays AB and $\mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime }$ undergo interference. The ratio of $\mathrm { I } _ { \text {max } }$ and $\mathrm { I } _ { \text {min } }$ is :
(1) $49 : 1$
(2) $7 : 1$
(3) $4 : 1$
(4) $8 : 1$

The image of an illuminated square is obtained on a screen with the help of a converging lens. The distance of the square from the lens is 40 cm. The area of the image is 9 times that of the square. The focal length of the lens is:
(1) 36 cm
(2) 27 cm
(3) 60 cm
(4) 30 cm

A person lives in a high-rise building on the bank of a river 50 m wide. Across the river is a well lit tower of height 40 m . When the person, who is at a height of 10 m , looks through a polarizer at an appropriate angle at light of the tower reflecting from the river surface, he notes that intensity of light coming from distance X from his building is the least and this corresponds to the light coming from light bulbs at height ' Y ' on the tower. The values of X and Y are respectively close to (refractive index of water $\simeq \frac { 4 } { 3 }$ )
(1) $25 \mathrm {~m} , 10 \mathrm {~m}$
(2) $13 \mathrm {~m} , 27 \mathrm {~m}$
(3) $22 \mathrm {~m} , 13 \mathrm {~m}$
(4) $17 \mathrm {~m} , 20 \mathrm {~m}$

In the Bohr model an electron moves in a circular orbit around the proton. Considering the orbiting electron to be a circular current loop, the magnetic moment of the hydrogen atom, when the electron is in $n ^ { \text {th} }$ excited state, is :
(1) $\left( \frac { e } { 2 m } \frac { n ^ { 2 } h } { 2 \pi } \right)$
(2) $\left( \frac { e } { m } \right) \frac { n h } { 2 \pi }$
(3) $\left( \frac { e } { 2 m } \right) \frac { n h } { 2 \pi }$
(4) $\left( \frac { e } { m } \right) \frac { n ^ { 2 } h } { 2 \pi }$

Orbits of a particle moving in a circle are such that the perimeter of the orbit equals an integer number of de Broglie wavelengths of the particle. For a charged particle moving in a plane perpendicular to a magnetic field, the radius of the $n^{\text{th}}$ orbital will therefore be proportional to:
(1) $n^2$
(2) $n$
(3) $n^{1/2}$
(4) $n^{1/4}$

The wave number of the first emission line in the Balmer series of H-Spectrum is: ($R =$ Rydberg constant):
(1) $\frac{5}{36}R$
(2) $\frac{9}{400}R$
(3) $\frac{7}{6}R$
(4) $\frac{3}{4}R$

By how many folds the temperature of a gas would increase when the root mean square velocity of the gas molecules in a container of fixed volume is increased from $5 \times 10 ^ { 4 } \mathrm {~cm} / \mathrm { s }$ to $10 \times 10 ^ { 4 } \mathrm {~cm} / \mathrm { s }$ ?
(1) Two
(2) Three
(3) Six
(4) Four

In reaction $\mathrm { A } + 2 \mathrm {~B} \rightleftharpoons 2 \mathrm { C } + \mathrm { D }$, initial concentration of B was 1.5 times of $[ \mathrm { A } ]$, but at equilibrium the concentrations of A and B became equal. The equilibrium constant for the reaction is :
(1) 8
(2) 4
(3) 12
(4) 6

An element having an atomic radius of 0.14 nm crystallizes in an $f _ { c c }$ unit cell. What is the length of a side of the cell ?
(1) 0.56 nm
(2) 0.24 nm
(3) 0.96 nm
(4) 0.4 nm

12 g of a nonvolatile solute dissolved in 108 g of water produces the relative lowering of vapour pressure of 0.1. The molecular mass of the solute is :
(1) 80
(2) 60
(3) 20
(4) 40

If a polythene sample contains two monodisperse fractions in the ratio $2 : 3$ with degree of polymerization 100 and 200 , respectively, then its weight average molecular weight will be :
(1) 4900
(2) 4600
(3) 4300
(4) 5200

Consider: Statement-I: $(p \wedge \sim q) \wedge (\sim p \wedge q)$ is a fallacy. Statement-II: $(p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)$ is a tautology.
(1) Statement-I is true; Statement-II is false.
(2) Statement-I is false; Statement-II is true.
(3) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
(4) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.

From a sphere of mass $M$ and radius R, a smaller sphere of radius $\frac{\mathrm{R}}{2}$ is carved out such that the cavity made in the original sphere is between its centre and the periphery. For the configuration in the figure where the distance between the centre of the original sphere and the removed sphere is $3R$, the gravitational force between the two spheres is:
(1) $\frac{41\mathrm{GM}^2}{3600\mathrm{R}^2}$
(2) $\frac{41\mathrm{GM}^2}{450\mathrm{R}^2}$
(3) $\frac{59\mathrm{GM}^2}{450\mathrm{R}^2}$
(4) $\frac{\mathrm{GM}^2}{225\mathrm{R}^2}$

Two hypothetical planets of masses $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ are at rest when they are infinite distance apart. Because of the gravitational force they move towards each other along the line joining their centres. What is their speed when their separation is '$d$'? (Speed of $\mathrm{m}_{1}$ is $\mathrm{v}_{1}$ and that of $\mathrm{m}_{2}$ is $\mathrm{v}_{2}$)
(1) $v_{1}=v_{2}$
(2) $$\mathrm{v}_{1}=\mathrm{m}_{2}\sqrt{\frac{2\mathrm{G}}{\mathrm{d}(\mathrm{m}_{1}+\mathrm{m}_{2})}}$$ $$\mathrm{v}_{2}=\mathrm{m}_{1}\sqrt{\frac{2\mathrm{G}}{\mathrm{d}(\mathrm{m}_{1}+\mathrm{m}_{2})}}$$ (3) $$\mathrm{v}_{1}=\mathrm{m}_{1}\sqrt{\frac{2\mathrm{G}}{\mathrm{d}(\mathrm{m}_{1}+\mathrm{m}_{2})}}$$ $$\mathrm{v}_{2}=\mathrm{m}_{2}\sqrt{\frac{2\mathrm{G}}{\mathrm{d}(\mathrm{m}_{1}+\mathrm{m}_{2})}}$$ (4) $$\mathrm{v}_{1}=\mathrm{m}_{2}\sqrt{\frac{2\mathrm{G}}{\mathrm{m}_{1}}}$$ $$\mathrm{v}_{2}=\mathrm{m}_{2}\sqrt{\frac{2\mathrm{G}}{\mathrm{m}_{2}}}$$

The angular frequency of the damped oscillator is given by, $\omega = \sqrt{\left(\frac{\mathrm{k}}{\mathrm{m}} - \frac{\mathrm{r}^2}{4\mathrm{~m}^2}\right)}$ where k is the spring constant, m is the mass of the oscillator and $r$ is the damping constant. If the ratio $\frac{r^2}{\mathrm{mk}}$ is $8\%$, the change in time period compared to the undamped oscillator is approximately as follows:
(1) increases by $1\%$
(2) increases by $8\%$
(3) decreases by $1\%$
(4) decreases by $8\%$

Which of the following expressions corresponds to simple harmonic motion along a straight line, where $x$ is the displacement and $\mathrm{a},\mathrm{b},\mathrm{c}$ are positive constants?
(1) $a+bx-cx^{2}$
(2) $bx^{2}$
(3) $a-bx+cx^{2}$
(4) $-bx$

An object is located in a fixed position in front of a screen. Sharp image is obtained on the screen for two positions of a thin lens separated by 10 cm. The size of the images in two situations are in the ratio $3:3$. What is the distance between the screen and the object?
(1) 124.5 cm
(2) 144.5 cm
(3) 65.0 cm
(4) 99.0 cm

In a compound microscope the focal length of objective lens is 1.2 cm and focal length of eye piece is 3.0 cm. When object is kept at 1.25 cm in front of objective, final image is formed at infinity. Magnifying power of the compound microscope should be:
(1) 200
(2) 100
(3) 400
(4) 150

Two monochromatic light beams of intensity 16 and 9 units are interfering. The ratio of intensities of bright and dark parts of the resultant pattern is:
(1) $\frac{16}{9}$
(2) $\frac{4}{3}$
(3) $\frac{7}{1}$
(4) $\frac{49}{1}$

In an experiment of single slit diffraction pattern, first minimum for red light coincides with first maximum of some other wavelength. If wavelength of red light is $6600\,\AA$, then wavelength of first maximum will be:
(1) $3300\,\AA$
(2) $4400\,\AA$
(3) $5500\,\AA$
(4) $6600\,\AA$

A photon of wavelength $\lambda$ is scattered from an electron, which was at rest. The wavelength shift $\Delta\lambda$ is three times of $\lambda$ and the angle of scattering $\theta$ is $60^\circ$. The angle at which the electron recoiled is $\phi$. The value of $\tan\phi$ is: (electron speed is much smaller than the speed of light)
(1) 0.16
(2) 0.22
(3) 0.25
(4) 0.28

A beam of light has two wavelengths of $4972\,\AA$ and $6216\,\AA$ with a total intensity $3.6\times10^{-3}\,\mathrm{Wm}^{-2}$ equally distributed among the two wavelengths. The beam falls normally on an area of $1\,\mathrm{cm}^{2}$ of a clean metallic surface of work function 2.3 eV. Assume that there is no loss of light by reflection and that each capable photon ejects one electron. The number of photoelectrons liberated in 2 s is approximately:
(1) $6\times10^{11}$
(2) $9\times10^{11}$
(3) $11\times10^{11}$
(4) $15\times10^{11}$