Question number 1 to 8 carries 2 marks each, 9 to 16 carries 4 marks each and 17 to 18 carries 6 marks each.

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1 IIT-JEE5-PH-1 FIITJEE Solutions to IITJEE 5 Mains Paper Tie: hours Physics Note: Question nuber 1 to 8 carries arks each, 9 to 16 carries 4 arks each and 17 to 18 carries 6 arks each. Q1. whistling train approaches a junction. n observer standing at junction observes the frequency to be. KHz and 1.8 KHz of the approaching and the receding train. Find the speed of the train (speed of sound = /s) While approaching v f' = f v vs = f v s While receding v f'' = f v+ vs 18 = f + vs On solving velocity of source (train) v s = /s Q. conducting liquid bubble of radius a and thickness t (t <<a) is charged to potential V. If the bubble collapses to a droplet, find the potential on the droplet. 1 q Potential of the bubble (V) = 4 πε a by conservation of volue 4 4π a t = π = ( a t) 1 a t Hence, potential on the droplet 1 q V' = (as charge is conserved) 4 πε 1 a V' =.V t Q. The potential energy of a particle of ass is given by E x 1 V( x) = x> 1 λ 1 and λ are the de roglie wavelengths of the particle, when x 1 and x > 1 respectively. If the total energy of particle is E, find λ 1 /λ. FIITJEE td. IES House, 9 Kalu Saria, Sarvapriya Vihar, New Delhi - 16, Ph : , , , Fax : 65194

2 IIT-JEE5-PH- K.E. = E E = E (for x 1) h λ 1 = E KE = E (for x > 1) h λ = 4E λ1 λ = Q4. U tube is rotated about one of it s libs with an angular velocity. Find the difference in height H of the liquid (density ρ) level, where diaeter of the tube d <<. H P = dx ρ ρ gh = H = g x dx H Q5. wooden log of ass M and length is hinged by a frictionless nail at O. bullet of ass strikes with velocity v and sticks to it. Find angular velocity of the syste iediately after the collision about O. M O v pply conservation of angular oentu about O (v) = ( M + ) = v ( + M) Q6. What will be the iniu angle of incidence such that the total internal reflection occurs on both the surfaces? µ 1 = µ = µ = For first surface sinc 1 = sin9 c 1 = 45 For second surface sinc = sin9 c = 6 µ 1= µ = µ = Miniu angle of incidence = Max {c 1, c } = 6 FIITJEE td. IES House, 9 Kalu Saria, Sarvapriya Vihar, New Delhi - 16, Ph : , , , Fax : 65194

3 IIT-JEE5-PH- Q7. The side of a cube is easured by vernier callipers (1 divisions of a vernier scale coincide with 9 divisions of ain scale, where 1 division of ain scale is 1 ). The ain scale reads 1 and first division of vernier scale coincides with the ain scale. Mass of the cube is.76 g. Find the density of the cube in appropriate significant figures. east count of vernier callipers = (1 9 ) =.1 1 Side of the cube = = 1.1 = 1.1 c.76 Density = =.66g/ c (1.1) Q8. n unknown resistance X is to be deterined using resistances 1, or. Their corresponding null points are, and. Find which of the above will give the ost accurate reading and why? X G = 1 or or r 1 X = r δx δr1 δr = + X r1 r X δ r1 = δ r = G r 1 r δx r1+ r = X rr 1 δx For to be iniu, rr 1 should be axiu and as r 1 + r is constant. X This is true for r 1 = r. So gives ost accurate value. Q9. transverse haronic disturbance is produced in a string. The axiu transverse velocity is /s and axiu transverse acceleration is 9 /s. If the wave velocity is /s then find the wavefor. If aplitude of wave is and angular frequency is, = = rad/s 9 v = k = 1 k = 1 c onsidering sinusoidal haronic function y = (1 c) sin(t ± x ) +φ Q1. cylinder of ass and radius rolls down an inclined plane of inclination. alculate the linear acceleration of the axis of cylinder. gsin f = aaxis (1) f = I axis α () a axis = α () gsin aaxis = a axis α g f N FIITJEE td. IES House, 9 Kalu Saria, Sarvapriya Vihar, New Delhi - 16, Ph : , , , Fax : 65194

4 IIT-JEE5-PH-4 Q11. long solenoid of radius a and nuber of turns per unit length n is enclosed by cylindrical shell of radius, thickness d (d<<) and length. variable current i = i sin t flows through the coil. If the resistivity of the aterial of cylindrical shell is ρ, find the induced current in the shell. a d φ= ( µ ni sint) π a dφ ε= = ( µ nicost) π a dt ρπ esistance = d ( µ nicost) πa (d) I = ρπ ` Induced current I a d Q1. Two identical ladders, each of ass M and length are resting on the rough horizontal surface as shown in the figure. block of ass hangs fro P. If the syste is in equilibriu, find the agnitude and the direction of frictional force at and. P For equilibriu of whole syste, ΣF y = M + N= g For rotational equilibriu of either ladder alculating torque about P N P g Mg f Mg f N y x N cos Mg cos f sin = f = (M+)g cot Q1. Highly energetic electrons are bobarded on a target of an eleent containing neutrons. The ratio of radii of nucleus to that of Heliu nucleus is (14) 1/. Find (a) atoic nuber of the nucleus. (b) the frequency of K α line of the X ray produced. ( = and c = 1 8 /s) r = r 1/ 1 r = 14 r = 4 He 1 = 56 and Z = (56 ) = 6 for K α line, c v = Z 1 4 ( ) ν = Hz FIITJEE td. IES House, 9 Kalu Saria, Sarvapriya Vihar, New Delhi - 16, Ph : , , , Fax : 65194

5 IIT-JEE5-PH-5 Q14. sall body attached to one end of a vertically hanging spring is perforing SHM about it s ean position with angular frequency and aplitude a. If at a height y* fro the ean position the body gets detached fro the spring, calculate the value of y* so that the height H attained by the ass is axiu. The body does not interact with the spring during it s subsequent otion after detachent. (a > g) y t position as the potential energy of the spring will be zero, the total energy (Gravitational potential energy + Kinetic energy) of the block at this point will be axiu and therefore if the block gets detached at this point, it will rise to axiu height, g g y* = k = <a Position Natural length Position Mean Position Q15. In the given circuit, the switch S is closed at tie t =. The charge Q on the capacitor at any instant t is given by Q(t) = Q (1 e αt ). Find the value of Q and α in ters of given paraeters shown in the circuit. S 1 + V pplying KV in loop 1 and, 1 i 1 V i 1 1 q =... (1) q i =... () i 1 i = dq... () dt On solving we get, t(1+ ) V 1 q = 1 e 1+ V 1+ Q = and α= V 1 i 1 i +q q i Q16. Two identical priss of refractive index are kept as shown in the figure. light ray strikes the first pris at face. Find, (a) the angle of incidence, so that the eergent ray fro the first pris has iniu deviation. (b) through what angle the pris DE should be rotated about so that the final eergent ray also has iniu deviation. (a) For iniu deviation r 1 = r = sini sin = i = 6 6 i r 1 r (b) Pris DE should be rotated about in anticlockwise direction through 6 so that the final eergent ray is parallel to the incident ray and angle of deviation is zero (iniu) 6 6 D 6 E FIITJEE td. IES House, 9 Kalu Saria, Sarvapriya Vihar, New Delhi - 16, Ph : , , , Fax : 65194

6 IIT-JEE5-PH-6 Q17. cylinder of ass 1 kg is given heat of J at atospheric pressure. If initially teperature of cylinder is, find (a) final teperature of the cylinder. (b) work done by the cylinder. (c) change in internal energy of the cylinder. (Given that Specific heat of cylinder= 4 J kg 1 1, oefficient of volue expansion = , tospheric pressure = 1 5 N/ and Density of cylinder = 9 kg/ ) (a) Q = s T J T = 1kg (4J/kg ) T final = 7 = 5 (b) W = P at V = P at V γ T = (1 5 N/ 1 ) 9 1 (9 1 5 / ) (5 ) =.5 J (c) U = Q W = J.5 J = J Q18. In a oving coil galvanoeter, torque on the coil can be expressed as τ = ki, where i is current through the wire and k is constant. The rectangular coil of the galvanoeter having nubers of turns N, area and oent of inertia I is placed in agnetic field. Find (a) k in ters of given paraeters N, I, and. (b) the torsional constant of the spring, if a current i produces a deflection of π/ in the coil. (c) the axiu angle through which coil is deflected, if charge Q is passed through the coil alost instantaneously. (Ignore the daping in echanical oscillations) (a) τ = in sinα For a oving coil galvanoeter α = 9 ki = in k = N (b) τ = i N = π/ = in π (c) ngular ipulse = τ. dt = Nidt = NQ NQ = I = NQ I Using energy of conservation 1 1 I = ax ax = I Nπ = Q Ii FIITJEE td. IES House, 9 Kalu Saria, Sarvapriya Vihar, New Delhi - 16, Ph : , , , Fax : 65194