Paper - i (Question Paper)

(JEE-Advanced) P-I

SECTION I: PHYSICS PART I: Single Correct Answer Type This section contains 10 multiple choice questions. Each question has four choices (a), (b), (c) and (d) out of which ONLY ONE is correct.

1.

A bomb of mass 5 kg explodes at rest into three pieces of masses in the ratio 1 : 1 : 3. The pieces with equal masses fly in mutually normal direction with speed of 21 ms 1 . Velocity of the third piece is

(a) 9.12 ms 1 2.

(b) 7.98 ms 1

(c) 9.87 ms 1

(d) 8.97 ms 1

A uniform solid cylinder of radius R is spinned with angular velocity  about its axis and then placed into a corner. The coefficient of friction between the cylinder and planes is  . The number of turns taken by the cylinder before stopping is given by

(a)

2 r  1    8g

(b)



2 R 1   2



8g  1   

(c)



2 R 1   2



4g  1   

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(d)



2 R 1   2 g  1   



(JEE-Advanced) P-I

3.

A block is placed on a incline plane at angle . The coefficient of friction between the block and the plane is  = tan . The block is given a hit so that it gets initial velocity v horizontally along the plane. The speed of the block after a very long time is v v (a) v (b) (c) 2v (d) 2 4

4.

Two beads of mass m each are positioned at the top of a frictionless hoop of mass M and radius R, which stands vertically on the ground. The beads are given tiny kicks, and they slide down the hoop, one to the right and other-one to the left, as shown. The smallest value of m/M for which the hoop will rise up off the ground at some time during the motion, is 2 3 3 (a) (b) (c) (d) 1 3 4 2

m

m M

5.

A wedge is lying on a rough horizontal surface. A small sphere collides with wedge as shown in figure. Then before and after collision conservation of momentum is applicable. (There is no friction between sphere and wedge ) (a) for (wedge + sphere) along x-axis (b) for (wedge + sphere) along y-axis (c) for sphere only along y ' (d) for sphere only along x '

6.

A uniform solid square plate ABCD of mass m and side a is moving in x – y horizontal smooth plane. The velocity of centre of mass is v0 (2iˆ  4 ˆj ) m/s. The end A of square plate is suddenly fixed by a pin, find the new velocity of centre of mass of square 3v0 ˆ 3v0 ˆ (i )  j (a) v0 (2iˆ  4 ˆj ) (b) 4 4 3v0 ˆ 3v0 ˆ i j (c) (d) 3v0iˆ  3v0 ˆj 2 2 SPACE FOR ROUGH WORK

(JEE-Advanced) P-I

7.

Two right angled prisms having the same refracting angle 60º, are placed in contact as shown in the figure. A ray of light incident on the first prism is finally deviated by an angle 90º. The refractive index of second prism is 2 . The refractive index of the first prism is equal to

(a) (c) 8.

9.

sin 30º sin 60º sin 15º 2 sin 60º 2

(b) (d)

sin 45º sin 60º sin 60º 2 sin 30º 2

The diagram shows four identical capacitors each of capacitance C. Now a +2Q and +4Q charges are given simultaneously at the positions shown in the figure. The value of V A  VB will be (a)

Q 4C

(b) 10

(c)

5Q 2C

(d)

Q 3C

Q C

A plane mirror is moving vertically downward at 2 m/s. There exists a magnetic field of value

1 Tesla inward (as shown). A 90 

C 





charged particle of specific change     53  is rotating in kg the field with a constant speed of 10 m/s. Initially the particle is located at point A along the horizontal. The magnitude of velocity of the image of the particle at t = 0.5s is (a) 10.2 m/s (b) 12.8 m/s (c) 5 m/s (d) 15 m/s SPACE FOR ROUGH WORK

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10.

A disc is under pure rolling on a horizontal surface as shown. The kinetic energy of the disc about point P is

(a)

1 mv 2 4

(b)

3 mv 2 8

(c)

3 mv 2 4

(d)

5 mv 2 4

PART II: Multiple Correct Answer(s) Type This section contains 5 multiple choice questions. Each question has four choices (a), (b), (c) or (d) out of which ONE or MORE are correct. 11.

A small ring of mass m can slide on a smooth rod. The ring is attached to a particle of mass m by a string of length l. A horizontal velocity v0  2 gl is given to the ring (a) the maximum angle that the string will make in subsequent motion is 60º with vertical (b) the maximum angle that the string will make in subsequent motion is 30º with vertical (c) velocity of ring when string makes maximum angle with vertical is

gl 2

(d) velocity of particle when string makes maximum angle with vertical is

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gl 2

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12.

There are two concentric metallic spherical shells of radii a and b such that a < b. An ideal cell of emf  is connected across the two shells. The medium between the shells is filled with a dielectric of dielectric constant K and resistivity . For a point P at a distance r from the common centre C where (a < r < b) (a) rate of fall of potential is (b) electric field is

 ab

 ba

2

r (b  a )  ab

(c) current density is

 r 2 (b  a )

(d) net current supplied by the cell is 13.

4 ab  (b  a )

The diagram shows a circuit with zero resistance. At time t = 0 the connector (mass = M and length l) is given a velocity v m/sec. At this time spring is in relaxed state. Choose the correct statement(s) (a) the charge which will appear in the capacitor immediately after the connector is given a velocity is BlvC (b) the maximum compression in the spring will be v

C v

m k

(c) the time period of oscillations of the connector is 2 14.

k

B

m  BlC k

(d) all the above statements are incorrect Four identical charged particles each of mass 0.1kg and charge 2C connected to each other via massless non-conducting rods of equal length. The whole arrangement is placed in a cylindrical region carrying a uniform magnetic field as shown in the figure ( B 0  4T , a  1m) . Suddenly the magnetic field is switched off. Then choose the correct statement(s) (a) the angular momentum of the system is 8 Nm/s (b) the angular momentum of the system is 8 2 Nm/s (c) Angular velocity of the system is 40 rad/s (d) the magnetic field at point O immediately after the magnetic field is switched off is non zero. SPACE FOR ROUGH WORK

(JEE-Advanced) P-I

15.

Two identical point particles A and B are placed in front of a concave mirror of focal length 20cm, at distances 10 cm and 30 cm respectively. The particles oscillate perpendicular to the principal axis, such that the displacement equation for both the particles is given by YA  YB  0.1sin( t ) . The correct statement(s) is/are (a) displacement equation of the image of particle A is YA '  0.2 sin( t   / 2) (b) displacement equation of the image of particle B is YB '  0.3 sin( t  ) (c) time period of oscillation of both the images is 2s. (d) relative vertical displacement of image of A with respect to image of B is YAB '  0.4 sin( t ) PART III: Integer Answer Type

This section contains 5 questions. The answer to each question is a single digit integer, ranging from 0 to 9 (both inclusive) 16.

A finite chain of resistors is shown in the figure. All the resistors in the chain are of the same value. A current of 1A flows through the last resistor. The current supplied by the battery is found to be 7n. Find the value of n.

17.

Two similar and coherent point sources S1 and S 2 vibrating in phase emit light of wavelength . The separation between the sources is 2. A person initially at S 2 starts walking along a straight line which is passing through S 2 and perpendicular to S1 S 2 . During his journey he finds variation in intensity of light which is obviously because of interference of light waves. At some place during his walk the person finds resultant intensity of light is equal to intensity due to one of the sources. The path difference  between the waves arriving at that place is . Find value of n. n

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18.

A string of length l cm sustains a standing wave such that all the points on the string having displacement amplitude 1mm are separated by d cm. The string is oscillating in its third overtone. What is the ratio

l . d

19.

An electric bulb of 100 W – 300 V is connected with an AC supply of 500 V and ( 150 /  )Hz. The required inductance to save the electric bulb is : x Henry. Find x?

20.

A solid sphere of radius R = 0.5m is placed on a rough surface. A force F  7 N is applied at point A. Sphere starts to perform pure rolling. Height of point A varies as h  h0  kt where h0  0.1 m k = 0.4 m/sec. A t

3 sec find frictional force acting on sphere. 2

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