A parallel plate capacitor having a separation between the plates d, plate area A and material with dielectric constant K has capacitance $\mathrm { C } _ { 0 }$. Now one-third of the material is replaced by another material with dielectric constant 2K, so that effectively there are two capacitors one with area $\frac { 1 } { 3 } \mathrm {~A}$, dielectric constant 2K and another with area $\frac { 2 } { 3 } \mathrm {~A}$ and dielectric constant K. If the capacitance of this new capacitor is C then $\frac { \mathrm { C } } { \mathrm { C } _ { 0 } }$ is
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$

In a metre bridge experiment null point is obtained at 40 cm from one end of the wire when resistance $\mathbf { X }$ is balanced against another resistance Y . If $\mathrm { X } < \mathrm { Y }$, then the new position of the null point from the same end, if one decides to balance a resistance of 3 X against Y , will be close to :
(1) 80 cm
(2) 75 cm
(3) 67 cm
(4) 50 cm

A letter ' $\mathrm { A } ^ { \prime }$ is constructed of a uniform wire with resistance $1.0 \Omega$ per cm , The sides of the letter are 20 cm and the cross piece in the middle is 10 cm long. The apex angle is 60 . The resistance between the ends of the legs is close to:
(1) $50.0 \Omega$
(2) $10 \Omega$
(3) $36.7 \Omega$
(4) $26.7 \Omega$

A shunt of resistance $1 \Omega$ is connected across a galvanometer of $120 \Omega$ resistance. A current of 5.5 ampere gives full scale deflection in the galvanometer. The current that will give full scale deflection in the absence of the shunt is nearly :
(1) 5.5 ampere
(2) 0.5 ampere
(3) 0.004 ampere
(4) 0.045 ampere

An electric current is flowing through a circular coil of radius $R$. The ratio of the magnetic field at the centre of the coil and that at a distance $2 \sqrt { 2 } R$ from the centre of the coil and on its axis is :
(1) $2 \sqrt { 2 }$
(2) 27
(3) 36
(4) 8

A current $i$ is flowing in a straight conductor of length $L$. The magnetic induction at a point on its axis at a distance $\frac{L}{4}$ from its centre will be:
(1) Zero
(2) $\frac{\mu_0 i}{2\pi L}$
(3) $\frac{\mu_0 i}{\sqrt{2}L}$
(4) $\frac{4\mu_0 i}{\sqrt{5}\pi L}$

A series LR circuit is connected to an ac source of frequency $\omega$ and the inductive reactance is equal to $2R$. A capacitance of capacitive reactance equal to $R$ is added in series with $L$ and $R$. The ratio of the new power factor to the old one is:
(1) $\sqrt { \frac { 2 } { 3 } }$
(2) $\sqrt { \frac { 2 } { 5 } }$
(3) $\sqrt { \frac { 3 } { 2 } }$
(4) $\sqrt { \frac { 5 } { 2 } }$

In a series L-C-R circuit, $C = 10^{-11}$ Farad, $L = 10^{-5}$ Henry and $R = 100$ Ohm, when a constant D.C. voltage E is applied to the circuit, the capacitor acquires a charge $10^{-9}\mathrm{~C}$. The D.C. source is replaced by a sinusoidal voltage source in which the peak voltage $E_0$ is equal to the constant D.C. voltage E. At resonance the peak value of the charge acquired by the capacitor will be:
(1) $10^{-15}\mathrm{~C}$
(2) $10^{-6}\mathrm{~C}$
(3) $10^{-10}\mathrm{~C}$
(4) $10^{-8}\mathrm{~C}$

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) $$\mathrm{v}_{1}=\mathrm{m}_{2}\sqrt{\frac{2\mathrm{G}}{\mathrm{m}_{1}}}$$ $$\mathrm{v}_{2}=\mathrm{m}_{2}\sqrt{\frac{2\mathrm{G}}{\mathrm{m}_{2}}}$$