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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2 IIT-JEE5-PH- FIITJEE Solutions to IITJEE 5 Mains Paper Time: hours Physics Note: Question number to 8 carries marks each, 9 to 6 carries marks each and 7 to 8 carries 6 marks each. Q. A whistling train approaches a junction. An observer standing at junction observes the frequency to be. KHz and.8 KHz of the approaching and the receding train. Find the speed of the train (speed of sound = m/s) While approaching v f' = f v vs = f v s While receding v f'' = f v+ vs 8 = f + vs On solving velocity of source (train) v s = m/s Q. A 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. q Potential of the bubble (V) = πε a by conservation of volume π a t = π R R = ( a t) a t Hence, potential on the droplet q V' = (as charge is conserved) πε R a V' =.V t Q. The potential energy of a particle of mass m is given by E x V( x) = x> λ and λ are the de Broglie wavelengths of the particle, when x and x > respectively. If the total energy of particle is E, find λ /λ. FIITJEE Ltd. ICES House, 9 A Kalu Saria, Sarvapriya Vihar, New Delhi - 6, Ph : , , 685, Fax : 659

3 IIT-JEE5-PH- K.E. = E E = E (for x ) h λ = me KE = E (for x > ) h λ = me λ λ = Q. A U tube is rotated about one of it s limbs with an angular velocity ω. Find the difference in height H of the liquid (density ρ) level, where diameter of the tube d << L. ω H L L PA = dmxω ω L ρa ρ gha = L ω H = g ω x dx H Q5. A wooden log of mass M and length L is hinged by a frictionless nail at O. A bullet of mass m strikes with velocity v and sticks to it. Find angular velocity of the system immediately after the collision about O. M O L m v Apply conservation of angular momentum about O (mv)l = (ml ML + )ω ω = mv (m + M)L Q6. What will be the minimum angle of incidence such that the total internal reflection occurs on both the surfaces? µ = µ = µ = For first surface sinc = sin9 c = 5 For second surface sinc = sin9 c = 6 θ µ = θ µ = θ θ µ = Minimum angle of incidence = Max {c, c } = 6 FIITJEE Ltd. ICES House, 9 A Kalu Saria, Sarvapriya Vihar, New Delhi - 6, Ph : , , 685, Fax : 659

4 IIT-JEE5-PH- Q7. The side of a cube is measured by vernier callipers ( divisions of a vernier scale coincide with 9 divisions of main scale, where division of main scale is mm). The main scale reads mm and first division of vernier scale coincides with the main scale. Mass of the cube is.76 g. Find the density of the cube in appropriate significant figures. Least count of vernier callipers = ( 9 )mm =.mm Side of the cube = mm +. mm =. mm =. cm.76 Density = =.66g/ cm (.) Q8. An unknown resistance X is to be determined using resistances R, R or R. Their corresponding null points are A, B and C. Find which of the above will give the most accurate reading and why? X R G A B C R = R or R or R rr X = r δx δr δr = + X r r X R δ r = δ r = G r r δx r+ r = A B C X rr δx For to be minimum, rr should be maximum and as r + r is constant. X This is true for r = r. So R gives most accurate value. Q9. A transverse harmonic disturbance is produced in a string. The maximum transverse velocity is m/s and maximum transverse acceleration is 9 m/s. If the wave velocity is m/s then find the waveform. If amplitude of wave is A and angular frequency is ω, ω A = ω = rad/s ω A 9 ω v = k = m k A = cm Considering sinusoidal harmonic function y = ( cm) sin(t ± x ) +φ Q. A cylinder of mass m and radius R rolls down an inclined plane of inclination θ. Calculate the linear acceleration of the axis of cylinder. mgsinθ f = maaxis () fr = I axis α () a axis = Rα () gsinθ aaxis = a axis α θ mg f N FIITJEE Ltd. ICES House, 9 A Kalu Saria, Sarvapriya Vihar, New Delhi - 6, Ph : , , 685, Fax : 659

5 IIT-JEE5-PH- Q. A long solenoid of radius a and number of turns per unit length n is enclosed by cylindrical shell of radius R, thickness d (d<<r) and length L. A variable current i = i sin ωt flows through the coil. If the resistivity of the material of cylindrical shell is ρ, find the induced current in the shell. L a R d φ= ( µ ni sinωt) π a dφ ε= = ( µ niωcosωt) π a dt ρπr Resistance = Ld ( µ niωcosωt) πa (Ld) I = ρπr ` Induced current I R a d Q. Two identical ladders, each of mass M and length L are resting on the rough horizontal surface as shown in the figure. A block of mass m hangs from P. If the system is in equilibrium, find the magnitude and the direction of frictional force at A and B. A L θ P m θ B For equilibrium of whole system, ΣF y = M + m N= g For rotational equilibrium of either ladder Calculating torque about P A N P mg Mg f Mg f N B y x NL cos θ Mg L cosθ fl sinθ = f = (M+m)g cot θ Q. Highly energetic electrons are bombarded on a target of an element containing neutrons. The ratio of radii of nucleus to that of Helium nucleus is () /. Find (a) atomic number of the nucleus. (b) the frequency of K α line of the X ray produced. (R =. 7 m and c = 8 m/s) r = r A / r A = r = He A = 56 and Z = (56 ) = 6 for K α line, Rc v = Z ( ) ν =.56 8 Hz FIITJEE Ltd. ICES House, 9 A Kalu Saria, Sarvapriya Vihar, New Delhi - 6, Ph : , , 685, Fax : 659

6 IIT-JEE5-PH-5 Q. A small body attached to one end of a vertically hanging spring is performing SHM about it s mean position with angular frequency ω and amplitude a. If at a height y* from the mean position the body gets detached from the spring, calculate the value of y* so that the height H attained by the mass is maximum. The body does not interact with the spring during it s subsequent motion after detachment. (aω > g) y m At position B 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 maximum and therefore if the block gets detached at this point, it will rise to maximum height, mg g y* = k = <a ω Position B Natural length Position A Mean Position m Q5. In the given circuit, the switch S is closed at time t =. The charge Q on the capacitor at any instant t is given by Q(t) = Q ( e αt ). Find the value of Q and α in terms of given parameters shown in the circuit. S R + V C R Applying KVL in loop and, R i V i R q C =... () q C i R =... () i i = dq... () dt On solving we get, t(r+ R ) CVR CRR q = e R+ R CVR R+ R Q = and α= R + R CR R + V i i +q C q i R Q6. Two identical prisms of refractive index are kept as shown in the figure. A light ray strikes the first prism at face AB. Find, (a) the angle of incidence, so that the emergent ray from the first prism has minimum deviation. (b) through what angle the prism DCE should be rotated about C so that the final emergent ray also has minimum deviation. (a) For minimum deviation B r = r = sini sin = i = 6 A 6 i r r (b) Prism DCE should be rotated about C in anticlockwise direction through 6 so that the final emergent ray is parallel to the incident ray and angle of deviation is zero (minimum) A B B 6 C C 6 D 6 E FIITJEE Ltd. ICES House, 9 A Kalu Saria, Sarvapriya Vihar, New Delhi - 6, Ph : , , 685, Fax : 659

7 IIT-JEE5-PH-6 Q7. A cylinder of mass kg is given heat of J at atmospheric pressure. If initially temperature of cylinder is C, find (a) final temperature of the cylinder. (b) work done by the cylinder. (c) change in internal energy of the cylinder. (Given that Specific heat of cylinder= J kg C, Coefficient of volume expansion = 9 5 C, Atmospheric pressure = 5 N/m and Density of cylinder = 9 kg/m ) (a) Q = ms T J T = kg (J/kg C) T final = 7 C = 5 C (b) W = P atm V = P atm V γ T = ( 5 N/m ) m 9 (9 5 / C) (5 C) =.5 J (c) U = Q W = J.5 J = J Q8. In a moving coil galvanometer, 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 galvanometer having numbers of turns N, area A and moment of inertia I is placed in magnetic field B. Find (a) k in terms of given parameters N, I, A and B. (b) the torsional constant of the spring, if a current i produces a deflection of π/ in the coil. (c) the maximum angle through which coil is deflected, if charge Q is passed through the coil almost instantaneously. (Ignore the damping in mechanical oscillations) (a) τ = inab sinα For a moving coil galvanometer α = 9 ki = inab k = NAB (b) τ = Cθ i NAB = Cπ/ C = inab π (c) Angular impulse = τ. dt = NABidt = NABQ NABQ = Iω ω = NABQ I Using energy of conservation Iω = Cθ max θ max = ω I NABπ = Q C Ii FIITJEE Ltd. ICES House, 9 A Kalu Saria, Sarvapriya Vihar, New Delhi - 6, Ph : , , 685, Fax : 659

8 IIT-JEE5-M- FIITJEE Solutions to IITJEE 5 Mains Paper Time: hours Mathematics Note: Question number to 8 carries marks each, 9 to 6 carries marks each and 7 to 8 carries 6 marks each. Q. A person goes to office either by car, scooter, bus or train probability of which being,, and 7 respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is,, and 9 respectively. Given that he reached office in time, then what is the probability that he travelled by a car. Let C, S, B, T be the events of the person going by car, scooter, bus or train respectively. Given that P(C) = 7, P(S) = 7, P(B) = 7, P(T) = 7 Let L be the event of the person reaching the office in time. L 7 L 8 L 5 L 8 P =, P =, P =, P = C 9 S 9 B 9 T 9 L P.P( C) 7 C C 7 9 P = = =. L P( L) Q. Find the range of values of t for which sin t = x + 5x x x, t π, π. Let y = sin t x + 5x so, y = x x (y 5)x x(y ) (y + ) = since x R,, so D y y or y + 5 and y 5 or sin t + 5 and sin t 5 Hence range of t is π, π π, π. Q. Circles with radii, and 5 touch each other externally if P is the point of intersection of tangents to these circles at their points of contact. Find the distance of P from the points of contact. FIITJEE Ltd. ICES House, Sarvapriya Vihar (Near Hauz Khas Bus Term.), New Delhi - 6, Ph : , , 685, Fax : 659

9 IIT-JEE5-M- Let A, B, C be the centre of the three circles. Clearly the point P is the in centre of the ABC, and hence s( s a)( s b)( s c ) (s a)(s b)(s c) r = = = s s s Now s = = s =. Hence r = 5.. = 5. A C P 5 5 B Q. Find the equation of the plane containing the line x y + z =, x + y + z = 5 and at a distance of from the point (,, ). 6 Let the equation of plane be (λ + )x + (λ )y + (λ + )z 5λ = 6λ+ +λ λ 5λ = (λ+ ) + ( λ ) + ( λ+ ) 6 6(λ ) = λ + λ + 6 λ =, 5. The planes are x y + z = and 6x + 9y + 9z 5 =. Q5. If f(x ) f(x ) < (x x ), for all x, x R. Find the equation of tangent to the curve y = f(x) at the point (, ). f (x ) f (x ) < (x x ) f(x ) f(x ) lim < lim x x f (x) < δ f (x) =. x x x x x x Hence f (x) is a constant function and P (, ) lies on the curve. f (x) = is the curve. Hence the equation of tangent is y =. n+ Q6. If total number of runs scored in n matches is (n+ n ) where n >, and the runs scored in the k th match are given by k. n+ k, where k n. Find n. Let S n = = [ n+ n] n n k k. + = k= n n+ k k. = k= n+ n. n n+ (sum of the A.G.P.) n + = n = 7. Q7. The area of the triangle formed by the intersection of a line parallel to x-axis and passing through P (h, k) with the lines y = x and x + y = is h. Find the locus of the point P. Area of triangle =. AB. AC = h y = x and AB = k = AC A(,) h =.. (k ) k = ± h. locus is y = x +, y = x +. B(k,k) C( k,k) y =k P(h,k) X O x+y= FIITJEE Ltd. ICES House, Sarvapriya Vihar (Near Hauz Khas Bus Term.), New Delhi - 6, Ph : , , 685, Fax : 659

10 IIT-JEE5-M- Q8. Evaluate π cos x e sin cos x+ cos cos x sin x dx. I = = 6 π cos x e sin cos x+ cos cos x sin x dx π / a cos x e sin x cos cos xdx ( ) Let cos x = t I = 6 t t e cos dt = ecos e sin 5 +., if f(a x) = f(x) a f x dx = f ( x) dx, if f(a x) = f(x) Q9. Incident ray is along the unit vector ˆv and the reflected ray is along the unit vector ŵ. The normal is along unit vector â outwards. Express ŵ in terms of â and ˆv. ˆv â ŵ ˆv is unit vector along the incident ray and ŵ is the unit vector along the reflected ray. â Hence â is a unit vector along the external (9-θ) bisector of ˆv and ŵ. Hence wˆ vˆ = λ aˆ + wˆ vˆ = λ or cos θ = λ or λ = sin θ where θ is the angle between ˆv and ŵ. Hence wˆ vˆ = sin θ aˆ = cos(9 θ )a ˆ = (aˆ v)a ˆ ˆ wˆ = vˆ (aˆ v)a ˆ ˆ. ŵ θ ˆv mirror Q. Tangents are drawn from any point on the hyperbola locus of mid point of the chord of contact. x y = to the circle x + y = 9. Find the 9 x y Any point on the hyperbola = is ( secθ, tanθ). 9 Chord of contact of the circle x + y = 9 with respect to the point ( sec θ, tan θ) is secθ.x + tanθ.y = 9.() Let (x, y ) be the mid point of the chord of contact. equation of chord in mid point form is xx + yy = x + y.() Since () and () represent the same line, secθ tanθ 9 = = x y x + y 9x 9y secθ =, tanθ = x + y x + y Hence ( ) 8y ( + ) ( + ) ( ) 8x = 9 x y x y the required locus is x y x + y =. 9 9 FIITJEE Ltd. ICES House, Sarvapriya Vihar (Near Hauz Khas Bus Term.), New Delhi - 6, Ph : , , 685, Fax : 659

11 IIT JEE QUESTION PAPERS PAPER I 5-5 Publisher : Faculty Notes Author : Panel of Experts Type the URL : 9 Get this ebook

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

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