NARAYANA JUNIOR COLLEGE

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1 NARAYANA JUNIOR COLLEGE SR IIT ALL STREAMS ADV MODEL DPT Date: /4/6 PHYSICS SINGLE ANSWER QUESTIONS. Find the radius of gyration of a cylinder of mass M radius r and length r about an axis passing through its centre and perpendicular to its symmetry axis Axis of Rotation r r A) r B) r C) 7 r D) r. A cylinder of mass m suspended by two strings wrapped around the cylinder one near each ends. The free ends of the strings being attached to hooks on the ceiling, such that the length of the cylinder is horizontal. From the position of rest, the cylinder is allowed to roll down as suspension strings unwind in same sense, calculate the tension in the each string. A) mg B) 6 mg C) mg D) 9 mg. A force F is applied tangential at the topmost point of a sphere of mass M and radius R. If the ground is rough enough to prevent sliding, find the frictional force acting on the sphere. A) 7 F B) 7 F C) 7 F D) 4 7 F

2 4. An equilateral triangle ABC formed from a uniform wire has two small identical beads initially located at A. The triangle is set rotating about the vertical axis AO. Then the beads are released from rest simultaneously and allowed to slide down, one along AB and other along AC as shown, Neglecting frictional effects, the quantities that are conserved as the beads slides down. A) Angular velocity and total energy (kinetic and potential) B) Total angular momentum and total energy C) Angular velocity and moment of inertia about the axis of rotation. D) Total angular momentum and moment of inertia about the axis of rotaion. 5. A cylinder of mass m and radius r rolls down a circular track from point A as shown in the figure. Assume that the friction is just sufficient to support rolling. Velocity of the cylinder at point A was zero. Assume r R. The reaction by the ground on the cylinder at point B is A) 7 mg B) 4 mg C) 5 mg D) mg

3 6. Four identical rods of mass M 6kg each are welded at their ends to form a square and then welded to a massive ring having mass m 4kg having radius R m. If the system is allowed to roll down the incline of inclination, assuming there is no slipping find the minimum value of friction coefficient to prevent slipping. A) 5 7 B) 5 C) 5 7 D) A uniform smooth rod is placed on a smooth horizontal floor is hit by a particle moving on the floor at a distance l /4 from one end. Velocity of particle was perpendicular to length of rod. Then the distance travelled by the centre of the rod after collision when it has completed three revolutions will be [ e and l is the length of the rod] A) l B) l C) l / D) Can t be determined 8. A ball of mass m and radius R rolling without slipping on a rough horizontal surface collides elastically with a smooth wall (inclined as shown) and comes back. It eventually starts pure rolling in back ward direction. The work done by friction in the whole process i.e., starting from pure rolling in forward direction and ultimately pure rolling in backward direction is A) is zero B) is surely positive C) is surely negative D) may be positive, negative or zero 9. A solid sphere is rolling purely on a rough horizontal surface (coefficient of kinetic friction = ) with speed of centre = u. It collides inelastically with a smooth vertical wall at a certain moment, the coefficient of restitution being ½. The sphere will begin pure rolling after a time. A) u/(7 g) B) u/(7 g)

4 C) u/(5 g) D) u/(5 g). A yo-yo arranged as shown, rests on a frictionless surface. When a force F is applied to the string, the yo-yo: A) moves to the left and rotates counterclockwise B) moves to the right and rotates counterclockwise C) moves to the left and rotates clockwise D) moves to the right and rotates clockwise. Velocity of the centre of smaller cylinder is v. If there is no slipping anywhere, then the velocity of the centre of larger cylinder is: A) v B) v C) v / D) v /. A system of uniform cylinders and plates is shown. All the cylinders are identical and there is no slipping at any contact. Velocity of lower & upper plates is v and v respectively as shown. Then the ratio of angular speeds of the upper cylinders to lower cylinders is A) / B) C) D). A thin uniform rod AB of mass M and length L is hinged at one end A to the horizontal floor. Initially it stands vertically. It is allowed to fall freely on the floor in the vertical plane by giving a small disturbance. The angular velocity of the rod when its end B strikes the floor is A) g L B) g L C) g L D) g L 4

5 4. A disc of mass M and radius R is rolling with angular speed on a horizontal plane as shown in Figure. The magnitude of angular momentum of the disc about the origin O is (Friction is sufficient so no slipping takes place) A) MR B) MR C) MR D) MR 5. A thin wire of length L and uniform linear mass density is bent into a circular loop with centre at O as shown in figure. The moment of inertia of the loop about the axis XX is A) L 8 B) L 6 C) 5L 6 D) L 8 SECTION II Multiple Correct Answer Type This section contains 5 multiple correct answer(s) type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct. 6. The moment of inertia of a thin square plate ABCD of uniform thickness about an axis passing through the center O and Perpendicular to the plate is A) I I B) I I4 C) I I D) I I I I4 5

6 where I I I I 4 are respectively the moments of inertia about axcs,, and 4 which are in the plane of the plate. 7. If a rigid body is subjected to two forces f i i 4k acting at (,,4) and f i i 4k acting at (,,) then which of the following is (are) true? A) The body is in translatory equilibrium B) The body is under the influence of a couple only C) The body is under the influence of a single force D) The body is under the influence of a force to gether with a couple. 8. In the figure, the blocks have masses Mand M M has a radius r and some mass the string does not slip on the pulley M and acceleration a. The pulley P A) The two sections of the string have unequal tensions B) The two blocks have accelerations of equal magnitude C) The angular acceleration of P is ar D) Angular acceleration of the pulley is zero. 6

7 9. A particle of mass M is projected from ground with an initial speed u and angle of projection as shown in fig. A) Average torque on a projectile during the time of flight about the point of projection is zero B) Average torque on a projectile during the time of flight about the point of projection is Mu sin C) The torque produced by gravity on the particle at the highest point about the point of projection is not zero D) The torque produced by gravity on the particle at the highest point about the point of projection is zero. A uniform rod of length l and mass m rests on a smooth horizontal table. A point of mass m moving horizontally at right angle to the rod with velocity v collides with one end of the rod and sticks to it, then: A) Angular velocity of the system after collision is v/ l B) Angular velocity of the system after collision is v/l C) The loss in kinetic energy of the system as a whole as a result of the collision is D) The loss in kinetic energy of the system as a whole as a result of the collision is mv /6 7 mv / 4. CHEMISTRY SINGLE ANSWER QUESTIONS 7

8 A) B) C) 4 D) 5. The electrophilic aromatic substitution proceeds through a A) Free radical B) Sigmacomplex C) Benzyne D) Carbene. Heating benzene in a large excess of 8% DSO4 in DOresults in what products A) C6H5SO D B) C6H5OD C) C6H5D D) CD Which of the following organic chlorides will not give a friedel craft alkylation product when heated with Benzene and AlCl? CH CCl B) CH CHCH Cl C) CHCH Cl D) CH A) 5. The compound X in the reaction CHCl A) B) C) D) 6. For the coagulation of ml of arsenous sulphide solution 5 ml of M NaCl is required. The coagulation value of NaCl is A) 48. B) 8. C) 49.5 D) Gold number of gelatin is.. Find the amount of gelatin to be added to ml of a colloidal Sol of gold to prevent its coagulation before adding ml of % NaCl solution. A) mg B) 4 mg C) mg D) 5 mg 8. At 4 K the energy of activation of a reaction is decreased by.8 Kcal in the presence of catalyst hence the rate will be 8

9 A) Increased by.7 times B) Increased by.8 times C) Decreases by.7 times D) Increased by 6.6 times 9. Which of the following is most effective for coagulation of ASSSol A) K FeCN 6 B) K FeCN 4 6 C) CaCl D) NaSO 4. Activators are generally metal ions. The catalytic activity of amylase increases in the presence of A) Na B) Ca C) Cu D). How many moles of Th NO should be dissolved in 5 moles of water so that vapour 4 pressure of water is reduced by 4% A) B) C) D) 4 Mn. H P of. M monobasicacid is measured to be. Its osmotic pressure at a given temperature T K is A). RT B). RT C). RT D). RT. To ml of M BaCl solution 5 ml of.5 M KSO4is added. BaSO4 is precipitated what will happen? A) Freeezing point is increased B) Freezing point is decreased C) Boiling point is lowered D) Freezing point remains same 4. An aqueous solution of methanol in water has vapour pressure A) less than that of water B) Equal to that of water C) More than that of water D) Equal to that of methanol 5. The osmotic pressure of an aqueous solution of a non electrolyte is 8.8 atm at will be the vapour pressure of this solution at 5 C what C (Density of water at C is g / cc) A).958 atm B).6 atm C) 6. atm D) 7. atm SECTION II Multiple Correct Answer Type This section contains 5 multiple correct answer(s) type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct. 9

10 6. The vapour pressure of a solution depends upon A) Surface area B) Temperature C) Mole fraction of solvent D) degree of dissociation of solute

11 7. Which of the following forms an ideal solution? A) Ethyl bromide + Ethyl Iodide B) Ethyl alcohol + water C) Chloroform + Benzene D) Benzene + Toluene 8. For an electrophilic substitution reaction, the presence of a halogen atom in the benzene ring A) Deactivates the ring by inductive effect B) Deactivates the ring by resonance C) Increases the charge density at ortho and para positions relative to metaposition by resonance. D) Directs the incoming electrophile to metaposition by increasing the charge density at metaposition 9. Which of the following acts as negative catalyst? A) Ethanol in oxidation of chloroform B) Tetra ethyl lead used as antiknocking agent C) Glycerol in the decomposition of HO D) Fe in the formation of ammonia by habers process 4. Which of the following is / are correct about Freundlich adsorption isotherm for gases and solutions A) x n KC m B) x m bp a x C) log log k nlog p m D) x log log k log c m n MATHEMATICS 4. Let f x x dx x x SINGLE ANSWER QUESTIONS and f. Then the value of f A) log B) log 4 C) log will be D) None of these 4 4. cos 4x dx is equal to cot x tan x

12 A) n sec x cos x c B) n sec x cos x c 4 4 C) n cos x cos x c D) 4 cos cos 4 n x x c dx a tan x b tan x c, cos x sin x then 4. If A) C) a, b B) 5 5 a, b D) 5 5 a, b 5 5 a, b If I / cos x dx, I x cos / sin x dx, I x sin / cos xsin 4 cos xsin x dx then x A) I I I B) I I I C) I I I D) None of these 45. x t t dt, where. denotes the greatest integer function, and x R, is equal to n n x x A) x B) x n n x x C) x D) x 46. Let f : R R value of be a continuous function and f x f x f f x dx is equal to is true x R. If A) 6 B) C) f D) f f, then 47. The value of / sin x dx, where, is A) cos B) cos C) D) cos 48. The value of n lim tan tan...tan n n n n / n is

13 A) e B) e C) D) e

14 49. If An is the area bounded by y xand y n x, n N, then A, A... An A) n n B) n n n C) n n n D) n n n 5. The area of the region bounded by and y x is x y x A) sq. units B) sq. units C) 4 sq. units D) sq. units 5. The area bounded by the loop of the curve 4y x 4 x is A) 7 / sq. units B) 8 / sq. units C) / sq. units D) 6 / sq. units 5. The form of the differential equation of the central conics ax by is A) dy dy x y B) x y dx dx C) dy d y dy x xy y dx dx dx D) None of these 5. The equation of the curve which is such that the portion of the axis of x cut off between the origin and tangent at any point is proportional to the ordinate of that point is A) x ya blog x B) C) x ya blog y log x by a D) None of these 54. An object falling from rest in air is subject not only to the gravitational force but also to air resistance. Assume that the air resistance is proportional to the velocity with constant of proportionality as velocity cannot exceed. k, and acts in a direction opposite to motion g 9.8 m / s. Then k A) 9.8 / k m / s B) 98 / k m / s C) m / s 9.8 D) None of these 4

15 SECTION II Multiple Correct Answer Type This section contains 5 multiple correct answer(s) type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct. 55. The population of a country increases at a rate proportional to the number of inhabitants. f is the population which doubles in years, then the population will triple in approximately A) years B) 45 years C) 48 years D) 54 years 56. If g x A) g x x t dt, then x x B) gx is monotonic C) gx is differentiable at x = D) ' 57. If f x is integral over,, then f x dx is equal to g x is differentiable at x = A) lim n r f n n r n B) lim n n n r f n C) rn lim n r n f n n r n D) lim n n n r r f n 58. If At is the area bounded by x ai y bi, i N, where a i a i bi and bi b i, a, b, then A) A 8 B) A 56 n C) lim A i D) lim A 6 i n i 8 cos 4x dx Af x B, then cot x tan x 59. If A) A B) 8 n n i B 4 C) f x has fundamental period D) f x is an odd function 5

16 6. The graph of the function y f x passing through the point, and satisfying the dy differential equation y cos x cos x is such that dx A) it is a constant function B) it is periodic C) It is neither an even nor an odd function D) It is continuous and differentiable for all x NARAYANA JUNIOR COLLEGE SR IIT ALL STREAMS ADV MODEL DPT Date: /4/6 PHYSICS C B C B A B A C A B B B C C D A, B, C A A, B B, C A, C CHEMISTRY C B D D B D C A C A 4 B B B A A B, C, D A, D A, C A, B, C A, D MATHEMATICS 4 5 B C B C A A C C D A 5 6 D C A A C A, B, C B, C A, C A, C A, B, D 6

17 HINTS PHYSICS. 4 4 I mr ml mr m r 7mr 7 mk K r. mg T ma &TR I HenceT Ma mg ma 4 a g T mg 6. F f Ma FR fr MR 5 9 R F so a Hence f F 7M 7 f F f a 4. No torque acts on the system so the angular momentum will remain conserved. Also if no external force does any work, hence the total energy remain conserved. 7

18 5. Acceleration = a n V and N mg man ; mg R R ( ) mv K r 6. M ( R ) R I M 4 mr 8 I MR mr kg m. (i) (4 M m) g sin f (4 M m) a a f. R I. (ii) R Solving (i) and (ii) a 7g 4 5 f a 8g cos l Ml 7. Fi dt MV and Fi dt 4 So, V l l 6 V t l 8. Conceptual 9. Conceptual. Conceptual. In the figure C and C are IC of two cylinders. In the absence of slipping between plank and cylinders, point A and A have same velocity. 8

19 Angular velocity of larger cylinder is v v 4R R v vcm ( R) v R. In the absence of slipping, velocities of contact points of upper cylinders and lower cylinders are respectively. v vab up R AB, vcd v lower CD R up lower. Loss is PE = gain in rotational KE. As the centre of mass of the rod falls through a distance mgl L/, the loss in PE = Gain in ml KE = Iω = ω Equating the two, we have mgl ml ω 6 Or g L, which is choice c 4. Let OC = Rc and let v c be the velocity of the centre of mass of the disc. The linear momentum of the centre of mass is P c = Mv c. If L c is the angular momentum of the disc about C, then the angular momentum about origin O is 9

20 MagnitudeL = I ω +R Mv sinθ c c c MR ω + MR c v c sin θ I c = MR MR ω + MR Rω R sin θ = R and V = Rω c c = MR ω 5. Let m be the mass of the loop and r its radius. The moment of inertia of the loop about ax axis passing through the centre O is I = mr From the parallel axes theorem, the moment of inertiabout XX is I = I + mr = mr + mr = mr The mass of the loop, m = pl and radius r = L/ π Hence L ρl I = ρl = 8 SECTION II Multiple Correct Answer Type This section contains 5 multiple correct answer(s) type questions. Each question has 4 choices (A), (B), (C) and (D), out of which ONE OR MORE is/are correct. 6. Conceptual 7. Conceptual 8. Conceptual 9. Conceptual

21 . W.r.t centre of mass frame. From law of conservation of momentum mv mv v v ( v - velocity of C.M) From law of conservation of angular momentum w.r.t to centre of mass of the system m m 6 L mvl ( m ) L L L v Loss in K.E = K f Ki mv ( m) v I Where ( m ) L L I m m L 6

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