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Q 1. The centre of mass of three particles of masses {tex} 1 \mathrm { kg } , 2 \mathrm { kg } {/tex} and {tex} 3 \mathrm { kg } {/tex} is at {tex}\mathrm {( 3,3,3 )}{/tex} with reference to a fixed coordinate system. Where should a fourth particle of mass {tex} 4 \mathrm { kg } {/tex} be placed so that the centre of mass of the system of all particles shifts to a point {tex}\mathrm {( 1,1,1 )} ? {/tex}

A

{tex}\mathrm {( - 1 , - 1 , - 1 )}{/tex}

{tex}\mathrm {( - 2 , - 2 , - 2 )}{/tex}

C

{tex}\mathrm {( 2,2,2 )}{/tex}

D

{tex}\mathrm {( 1,1,1 )}{/tex}

Explanation



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Q 2. One end of a uniform wire of length {tex} L {/tex} and of weight {tex} W {/tex} is attached rigidly to a point in the roof and a weight {tex} W _ { 1 } {/tex} is suspended from its lower end. If {tex} S {/tex} is the area of cross- section of the wire, the stress in the wire at a height {tex} \frac {3 L}{ 4} {/tex} from its lower end is

A

{tex} \frac { W _ { 1 } } { S } {/tex}

B

{tex} \frac { W _ { 1 } + (\frac {W} { 4 }) } { S } {/tex}

{tex} \frac { W _ { 1 } + (\frac {3 W}{ 4} ) } { S } {/tex}

D

{tex} \frac { W _ { 1 } + W } { S } {/tex}

Explanation

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Q 3. A block {tex}\mathrm {C}{/tex} of mass {tex} \mathrm { m } {/tex} is moving with velocity {tex} \mathrm { v } _ { 0 } {/tex} and collides elastically with block {tex}\mathrm {A}{/tex} of mass {tex} \mathrm { m } {/tex} and connected to another block {tex}\mathrm {B}{/tex} of mass {tex} 2 \mathrm { m } {/tex} through spring constant {tex} \mathrm { k } {/tex}. What is {tex}\mathrm {k}{/tex} if {tex} \mathrm { x } _ { 0 } {/tex} is compression of spring when velocity of {tex} \mathrm { A } {/tex} and {tex} \mathrm { B } {/tex} is same?

A

{tex}\mathrm { \frac { m v _ { 0 } ^ { 2 } } { x _ { 0 } ^ { 2 } }} {/tex}

B

{tex} \frac { \mathrm { mv } _ { 0 } ^ { 2 } } { 2 \mathrm { x } _ { 0 } ^ { 2 } } {/tex}

C

{tex}\mathrm { \frac { 3 } { 2 } \frac { m v _ { 0 } ^ { 2 } } { x _ { 0 } ^ { 2 } }} {/tex}

{tex}\mathrm { \frac { 2 } { 3 } \frac { m v _ { 0 } ^ { 2 } } { x _ { 0 } ^ { 2 } } }{/tex}

Explanation



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Q 4. Let {tex} P ( r ) = \frac { Q } { \pi R ^ { 4 } } r {/tex} be the charge density distribution for a solid sphere of radius {tex} R {/tex} and total charge {tex} Q {/tex}. For a point ' {tex} p {/tex} ' inside the sphere at distance {tex} r _ { 1 } {/tex} from the centre of the sphere, the magnitude of electric field is

A

{tex} \frac { Q } { 4 \pi \epsilon _ { 0 } r _ { 1 } ^ { 2 } } {/tex}

{tex} \frac { Q r _ { 1 } ^ { 2 } } { 4 \pi \epsilon _ { 0 } R ^ { 4 } } {/tex}

C

{tex} \frac { Q r _ { 1 } ^ { 2 } } { 3 \pi \epsilon _ { 0 } R ^ { 4 } } {/tex}

D

Zero

Explanation





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Q 5. The potential at a point {tex} x {/tex} (measured in {tex} \mu \mathrm { m } {/tex} ) due to some charges situated on the {tex} x {/tex} -axis is given by {tex} V ( x ) = \frac {20} { \left( x ^ { 2 } - 4 \right)} {/tex} Volt. The electric field {tex} E {/tex} at {tex} x = 4 \mu \mathrm { m } {/tex} is given by

{tex} ( 10 / 9 ) {/tex} Volt {tex}/ \mu \mathrm { m } {/tex} and in the {tex} + {/tex}ve {tex} x {/tex}- direction

B

{tex} ( 5 / 3 ) {/tex} Volt {tex} / \mu \mathrm { m } {/tex} and in the {tex} - {/tex} ve {tex} x {/tex}- direction

C

{tex} ( 5 / 3 ) {/tex} Volt {tex} / \mu \mathrm { m } {/tex} and in the {tex} + \mathrm { }{/tex}ve {tex}x {/tex}- direction

D

{tex} ( 10 / 9 ) {/tex} Volt {tex} / \mathrm{\mu m} {/tex} and in the {tex}-{/tex}ve {tex}x{/tex}- direction

Explanation

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Q 6. A material '{tex} B{/tex}' has twice the specific resistance of '{tex} A {/tex}'. A circular wire made of '{tex} B {/tex}' has twice the diameter of a wire made of '{tex}A{/tex}'. Then for the two wires to have the same resistance, the ratio {tex} \ell _ { B } / \ell _ { A } {/tex} of their respective lengths must be

A

{tex}1{/tex}

B

{tex} \frac { 1 } { 2 } {/tex}

C

{tex} \frac { 1 } { 4 } {/tex}

{tex}2{/tex}

Explanation

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Q 7. Figure shows an arrangement of a rod of length {tex} \ell {/tex} and mass {tex} \mathrm { M } {/tex} and a bead of mass {tex} \mathrm { m } {/tex} attached to a weightless string passing over a frictionless pulley. At {tex} \mathrm { t } = 0 , {/tex} the bead is in level with the upper end of the rod. The bead slides down the string with considerable friction, and is opposite to the other end of the rod after {tex} \mathrm {t}{/tex} second. Assuming friction between the bead and the string to be constant all through, the frictional force is

{tex} \frac { 2 \mathrm { Mm } \ell } { ( \mathrm { M } - \mathrm { m } ) \mathrm { t } ^ { 2 } } {/tex}

B

{tex} \frac { 2 \mathrm { Mm } \ell } { ( \mathrm { M } + \mathrm { m } ) \mathrm { t } ^ { 2 } } {/tex}

C

{tex} \frac { \mathrm { Mm } \ell } { ( \mathrm { M } - \mathrm { m } ) \mathrm { t } ^ { 2 } } {/tex}

D

{tex} \frac { 4 \mathrm { Mm } \ell } { ( \mathrm { M } - \mathrm { m } ) \mathrm { t } ^ { 2 } } {/tex}

Explanation

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Q 8. Initial velocity with which a body is projected is {tex} 10 \mathrm { m } / \mathrm { sec } {/tex} and angle of projection is {tex} 60 ^ { \circ } , {/tex} find the range {tex} \mathrm { R } {/tex}

A

{tex} \frac { 15 \sqrt { 3 } \mathrm { m } } { 2 } {/tex}

B

{tex} \frac { 40 } { 3 } \mathrm { m } {/tex}

C

{tex} 5 \sqrt { 3 } \mathrm { m } {/tex}

{tex} \frac { 20 } { 3 } \mathrm { m } {/tex}

Explanation

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Q 9. From a tower of height {tex} \mathrm { H } , {/tex} a particle is thrown vertically upwards with a speed {tex} \mathrm { u } {/tex}. The time taken by the particle, to hit the ground, is {tex} \mathrm n {/tex} times that taken by it to reach the highest point of its path. The relation between {tex} \mathrm { H } , \mathrm { u } {/tex} and {tex} \mathrm { n } {/tex} is:

A

{tex} \mathrm { 2 g H = n ^ { 2 } u ^ { 2 }} {/tex}

B

{tex} \mathrm { g H = ( n - 2 ) ^ { 2 } u ^ { 2 } d} {/tex}

{tex} 2 \mathrm { gH } = \mathrm { nu } ^ { 2 } ( \mathrm { n } - 2 ) {/tex}

D

{tex} \mathrm { gH } = ( \mathrm { n } - 2 ) \mathrm { u } ^ { 2 } {/tex}

Explanation



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Q 10. An oscillator is producing FM waves of frequency {tex} 2 \mathrm { kHz } {/tex} with a variation of {tex} 10 \mathrm { kHz } {/tex}. What is the modulating index?

A

{tex} \mathrm {0.20}{/tex}

{tex} \mathrm {5.0}{/tex}

C

{tex} \mathrm {0.67}{/tex}

D

{tex} \mathrm {1.5}{/tex}

Explanation

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Q 11. In a {tex} \mathrm { p - n} {/tex} junction diode, a square input signal of {tex} 10 \mathrm { V } {/tex} is applied as shown in fig.

The output signal across {tex} \mathrm { R } _ { \mathrm { L } } {/tex} will be

A

C

D

Explanation

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Q 12. The activity of a freshly prepared radioactive sample is {tex} 10 ^ { 10 } {/tex} disintegrations per second, whose mean life is {tex} 10 ^ { 9 } \mathrm { s } {/tex}. The mass of an atom of this radioisotope is {tex} 10 ^ { - 25 } \mathrm { kg } {/tex}. The mass (in {tex} \mathrm { mg } {/tex} ) of the radioactive sample is

{tex} \mathrm {1}{/tex}

B

{tex} \mathrm {3}{/tex}

C

{tex}5{/tex}

D

{tex}6{/tex}

Explanation



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Q 13. A stationary object at {tex} 4 ^ { \circ } \mathrm { C } {/tex} and weighing {tex} 3.5 \mathrm { kg } {/tex} falls from a height of {tex} 2000 \mathrm { m } {/tex} on a snow mountain at {tex} 0 ^ { \circ } \mathrm { C } {/tex}. If the temperature of the object just before hitting the snow is {tex} 0 ^ { \circ } \mathrm { C } {/tex} and the object comes to rest immediately {tex} \left( \mathrm { g } = 10 \mathrm { m } / \mathrm { s } ^ { 2 } \right) {/tex} and latent heat of ice {tex} = 3.5 \times 10 ^ { 5 } {/tex} Joule , then the mass of ice that will melt is

A

{tex} 2 \mathrm { kg } {/tex}

{tex} 200 \mathrm { g } {/tex}

C

{tex} 20 \mathrm { g } {/tex}

D

{tex} 2 \mathrm { g } {/tex}

Explanation

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Q 14. A source of frequency {tex} f {/tex} is stationary and an observer starts moving towards it at {tex} t = 0 {/tex} with constant small acceleration. Then the variation of observed frequency {tex} f {/tex} registered by the observer with time is best represented by

B

C

D

Explanation

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Q 15. A body executes simple harmonic motion. The potential energy(P.E.), the kinetic energy(K.E.) and total energy(T.E.) are measured as a function of displacement {tex} x {/tex}. Which of the following statements is true

A

P.E. is maximum when {tex} x = 0 {/tex}

K.E. is maximum when {tex} x = 0 {/tex}

C

T.E. is zero when {tex} x = 0 {/tex}

D

K.E. is maximum when {tex} x {/tex} is maximum

Explanation

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Q 16. For a gas at a temperature {tex} T {/tex} the root-mean-square velocity {tex} v _ { r m s } , {/tex} the most probable speed {tex} v _ { m p } , {/tex} and the average speed {tex} v _ { a v } {/tex} obey the relationship

A

{tex} v _ { a v } > v _ { r m s } > v _ { m p } {/tex}

{tex} v _ { r m s } > v _ { a v } > v _ { m p } {/tex}

C

{tex} v _ { m p } > v _ { a v } > v _ { r m s } {/tex}

D

{tex} v _ { m p } > v _ { r m s } > v _ { a v } {/tex}

Explanation

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Q 17. A series R-C combination is connected to an AC voltage of angular frequency {tex} \omega = 500 {/tex} radian/s. If the impedance of the R-C circuit is {tex} R \sqrt { 1.25 } {/tex}, the time constant (in millisecond) of the circuit is

A

{tex}5{/tex}

{tex}4{/tex}

C

{tex}6{/tex}

D

{tex}8{/tex}

Explanation

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Q 18. If {tex} c {/tex} is the speed of electromagnetic waves in vacuum, its speed in a medium of dielectric constant {tex} \mathrm { K } {/tex} and relative permeability {tex} \mu _ { \mathrm { r } } {/tex} is

A

{tex} \mathrm { v } = \frac { 1 } { \sqrt { \mu _ { \mathrm { r } } \mathrm { K } } } {/tex}

B

{tex} \mathrm { v } = \mathrm { c } \sqrt { \mu _ { \mathrm { r } } \mathrm { K } } {/tex}

{tex} \mathrm { v } = \frac { \mathrm { c } } { \sqrt { \mu _ { \mathrm { r } } \mathrm { K } } } {/tex}

D

{tex} \mathrm { v } = \frac { \mathrm { K } } { \sqrt { \mu _ { \mathrm { r } } \mathrm { C } } } {/tex}

Explanation

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Q 19. For the angle of minimum deviation of a prism to be equal to its refracting angle, the prism must be made of a material whose refractive index

A

lies between {tex} \sqrt { 2 } {/tex} and {tex} \mathrm {1}{/tex}

lies between {tex} \mathrm {2}{/tex} and {tex} \sqrt { 2 } {/tex}

C

is less than {tex} \mathrm {1}{/tex}

D

is greater than {tex} \mathrm {2}{/tex}

Explanation





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Q 20. Two polaroids are placed in the path of unpolarized beam of intensity {tex} \mathrm { I } _ { 0 } {/tex} such that no light is emitted from the second polaroid. If a third polaroid whose polarization axis makes an angle {tex} \theta {/tex} with the polarization axis of first polaroids, is placed between these polaroids then the intensity of light emerging from the last polaroid will be

{tex} \left( \frac { \mathrm { I } _ { 0 } } { 8 } \right) \sin ^ { 2 } 2 \theta {/tex}

B

{tex} \mathrm { \left( \frac { I _ { 0 } } { 4 } \right) \sin ^ { 2 } 2 \theta} {/tex}

C

{tex} \left( \frac { \mathrm { I } _ { 0 } } { 2 } \right) \cos ^ { 4 } \theta {/tex}

D

{tex} \mathrm { I } _ { 0 } \cos ^ { 4 } \theta {/tex}

Explanation

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Q 21. A uniform thin rod {tex} \mathrm { AB } {/tex} of length L has linear mass density {tex} \mu {/tex} {tex} ( x ) = a + \frac { b x } { L } , {/tex} where {tex} \mathrm { x } {/tex} is measured from A. If the CM of the rod lies at a distance of {tex} \left( \frac { 7 } { 12 } \right) \mathrm { L } {/tex} from {tex} \mathrm { A } , {/tex} then {tex} a {/tex} and {tex} b {/tex} are

A

{tex} a = 2 b {/tex}

{tex} 2 a = b {/tex}

C

{tex} a = b {/tex}

D

{tex} 3 a = 2 b {/tex}

Explanation

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Q 22. The radii of the two columns in U tube are {tex} r _ { 1 } {/tex} and {tex} r _ { 2 } {/tex}. When a liquid of density {tex} \rho {/tex} (angle of contact is {tex} 0 ^ { \circ } {/tex} ) is filled in it the level difference of liquid in two arms is {tex} h {/tex}. The surface tension of liquid is {tex} ( g = {/tex} acceleration due to gravity)

{tex} \frac { \rho g h r _ { 1 } r _ { 2 } } { 2 \left( r _ { 2 } - r _ { 1 } \right) } {/tex}

B

{tex} \frac { \rho g h \left( r _ { 1 } - r _ { 2 } \right) } { 2 r _ { 1 } r _ { 2 } } {/tex}

C

{tex} \frac { 2 \left( r _ { 2 } - r _ { 1 } \right) } { \rho g h r _ { 1 } r _ { 2 } } {/tex}

D

{tex} \frac { \rho g h } { 2 \left( r _ { 2 } - r _ { 1 } \right) } {/tex}

Explanation

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Q 23. Three objects {tex} A , B {/tex} and {tex} C {/tex} are kept in a straight line on a frictionless horizontal surface. These have masses {tex} m , 2 m {/tex} and {tex} m , {/tex} respectively. The object {tex} A {/tex} moves towards {tex} B {/tex} with a speed {tex} 9 \mathrm { m/s } {/tex} and makes an elastic collision with it. Thereafter, {tex} B {/tex} makes completely inelastic collision with {tex} C . {/tex} All motions occur on the same straight line. Find the final speed (in {tex} m / s ) {/tex} of the object {tex} C . {/tex}

4

B

6

C

8

D

10

Explanation

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Q 24. A square surface of side {tex} L {/tex} meter in the plane of the paper is placed in a uniform electric field {tex} E ( \text { volt } / \mathrm { m } ) {/tex} acting along the same plane at an angle {tex} \theta {/tex} with the horizontal side of the square as shown in Figure. The electric flux linked to the surface, in units of volt. {tex} m {/tex}, is

A

{tex} \mathrm { EL } ^ { 2 } {/tex}

B

{tex} \mathrm { EL } ^ { 2 } \cos \theta {/tex}

C

{tex} \mathrm { EL } ^ { 2 } \sin \theta {/tex}

zero

Explanation

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Q 25. In the figure below, what is the potential difference between the point {tex} A {/tex} and {tex} B {/tex} and between {tex} B {/tex} and {tex} C {/tex} respectively in steady state

A

{tex} \mathrm { V } _ { \mathrm { AB } } = \mathrm { V } _ { \mathrm { BC } } = 100 \mathrm { V } {/tex}

B

{tex} \mathrm { V } _ { \mathrm { AB } } = 75 , \mathrm { V } _ { \mathrm { BC } } = 25 \mathrm { V } {/tex}

{tex} \mathrm { V } _ { \mathrm { AB } } = 25 \mathrm { V } , \mathrm { V } _ { \mathrm { BC } } = 75 \mathrm { V } {/tex}

D

{tex} \mathrm { V } _ { \mathrm { AB } } = \mathrm { V } _ { \mathrm { BC } } = 50 \mathrm { V } {/tex}

Explanation