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Physics Part 1 Straight Objective Type This section contains 8 multiple choice questions numbered 1 to 8. Each question has 4 choices (A), (B), (C) and (D), outt of which ONLY ONE is correct. 1. A disc of radius R is rolling without slipping on a fixed circular track of radius 2R. Line joining the centre of the disc and centre (O) of circular track is rotating with constant angular velocity in clockwise sense. A rod AB whose end A is attached to a small ring which can move only along a fixed horizontal rod and other end of rod B is pivoted at B, R/2 distance above centre of disc as shown. Find the angular velocity of rod AB at the instant when rod AB is making an angle 30 with horizontal and centre of the disc is directly above the centre (O) of the track as shown in figure.(Assume rod can rotate freely about hinge point B) (A) Zero (B) (C) (D)
2. Two blocks A and B of masses m and 2m respectively are connected together by a light spring of stiffness fness k and then placed on a smooth horizontal surface. The blocks are pushed towards each other the block A by the spring, by the time the spring acquires its natural length, is (A) (B) (C)
(D)
3. A cylinder of mass m rests in a supporting block as shown. If β = 60 and θ = 30 , calculate the maximum acceleration a which the block may be given up the incline so that the cylinder does not lose contact at B. (neglect friction anywhere) (A) g/2 (B) g (C) 2g (D) g/4 4. A block of mass m is attached to the frame by a light spring of stiffness k. The frame and block are initially at rest with x = ,the uncompressed length of the spring. If the frame is given a constant horizontal acceleration towards left, determine the maximum velocity of the block relative to the frame (block is free to move inside frame). Ignore any friction.
(A) (B) (C)
(D)
5. A heavy disc with radius R is rolling down hanging on two non-stretched stretched string wound around the disc very tightly. The free ends of the string are attached to a fixed horizontal support. The strings are always tensed during the motion. At some instant, the angular velocity of the disc is ω,, and the angle between the strings is α. Find the velocity of
centre of mass of the disc at this moment (A) (B) (C) (D)
!
"# !
!
"# !
6. A small particle of mass m is attached at B to a hoop as mass m and radius r, whole system is placed on the rough horizontal ground. The system is released from rest when B is directly above A and rolls without slipping. Find the angular acceleration of the system at the instant when AB becomes horizontal as shown in the ffigure. (A) (B) (C) (D)
$
$ % $ & &$
7. A uniform rod of mass m and length L is placed on the fixed cylindrical surface of radius R at a small angular position ' from the vertical (vertical means line joining centre and vertex of the cylindrical path) as sho shown in the figure and released from rest. Find the angular velocity ω of the rod at the instant when it crosses the horizontal position (Assume that when it crosses the horizontal position its midpoint and vertex of the circular surface coincide). Friction is sufficient to prevent any slipping. (A) (B) (C) (D)
)3*+',-.' / 01,' 2 1 ( )*+',-.' / 01,' 2 1 ( )*+',-.' / 01,' 2 1 ( )6*+',-.' / 01,' 2 1 (
8. A small ball of mass 100g is attached to a light and inextensible string of length 50cm. The string is tied to a support O and the mass m released from point A which is at a horizontal distance of 30 cm from the support. Calculate the speed of the ball as its lowest point of the trajectory. (A) 2.2 m/s (B) 2.5 m/s (C) 3.2 m/s (D) 2.5 m/s Comprehensive Type This section contains 3 groups of questions. Each group has 2 multiple choice question based on a paragraph. Each question has 4 choices (A), (B), (C) and (D), for its answer, out of which ONLY ONE is correct. Paragraph for question 9 to 10 A block of mass 10 kg is placed at the centre of a rough disc which is at rest on a horizontal surface with its plane horizontal. The disc starts moving with a constant acceleration : =
5̂ /
7̂ m/, as shown in figure. The coefficient of friction
between the block and the disc is μ; = μ = 0.1
Answer the following question on the basis of given data. 9. Now if an elongated spring of the spring constant k = 50 N/m is attached so that one end of spring is fixed to periphery of disc and other end to the block as shown in figure, the block just starts moving in +y (positive y axis) direction with negligible acceleration with respect to disc. The friction force on block immediately af after ter block starts moving with respect to disc is 9 (A) 2N(-7 9 (B) 5N(-7 9 (C) 10N(-7
9 (D) 15N(-7 10. In the previous problem initial elongation in the spring is (A) (B) (C) (D)
m
√ m √
m
√= √
m
Paragraph for questions 11 and 12 Three small identical spheres A, B and C each of mass m, are connected to a small ring D of negligible mass by means of three identical light inextensible strings of length l each, which are equally spaced as shown. The spheres may slide freely on a fric frictionless horizontal surface. All three spheres have given same speed > perpendicular to string, such that, all are moving n a circle about ring D which is at rest. Suddenly string CD breaks. After the other two string becomes taut again, determine 11. Speed of ring D (A) > ? (B) (C)
(D)
? ?
12. The angular speed of A with respect to D is (when string become taut) (A) (B) (C) (D)
?
? ? & ?
Paragraph for questions 13 and 14 Two identical discs A and B of mass m and radius R, each are placed on the rough horizontal surface. Their centres are connected with the light spring of spring constant k. Initially spring is in its natural length and discs are at rest. Now centre of disc A has given velocity > in the horizontal direction as shown in the figure. There is sufficient friction between discs and ground to prevent the slipping at all instant. 13. Find the maximum compression of the spring (A) (B) (C) (D)
? ? ? ?
14. Find the angular velocity of disc A at the instant of maximum compression in the spring (A) (B) (C) (D)
? ? ? ?
Multiple Correct Choice Type This section contains 6 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONE OR MORE is/are correct 15. A uniform semicircular wire of mass M =
kg and radius R = 1m is free to rotate about a fixed
√
horizontal zontal axis coinciding with the diameter passing through open ends. First the wire is taken aside such that its plane becomes horizontal and then it is released from rest. Choose the
correct option(s), when vertical component of velocity of the centre of m mass ass is maximum (take g= 10m/, , @ = 10)
(A) Angular displacement of wire is 01B C √2
(B) Vertical component of force exerted by the axis on the wire is 55√2 N
(C) Vertical component of force exerted by the axis on the wire is 10 10√2 N (D) Horizontal component of force exerted by the axis on the wire is 8 N 16. A subway train travels between two of its station stops with the acceleration schedule shown in the acceleration verses time graph. Then (A) The time interval ∆tt during which the train brakes to a stop with a deceleration of 2 m/, is 8 sec. (B) The distance between stations is 350 m. (C) The time interval ∆tt during which the train brakes to a stop with a deceleration of 2 m/, is 10 sec. (D) The distance between stations is 416 m.
17. A small sphere of mass m is suspended by a ligh light and inextensible string of length l from a point O fixed on a smooth inclined plane of inclination ' with the horizontal. The sphere is moving in a circle on the incline plane as shown. If the sphere has a velocity u at the top most position A. then, (A) The tension in the string as the sphere passes the 90 F
position B equals to m G 2 2*,-.'!.
F
(B) The tension in the string at the bottom most position equals to m G / 5*,-.' *,-.'!. F
(C) The tension in the string as the sphere passes the 90 position B equals to m G / 2*,-.'!.
F
(D) The tension in the string at the bottom most position equals to m G 2 5*,-.' *,-.'!. 18. In the adjacent figure, a uniform disc of mass 2m and radius l/2 and radius l/2 /2 is lying at rest on a smooth horizontal surface. A particle ‘A’ of mass m is connected to a light string of length l , whose other end is attached to the circumference of the disc. Initially string is just taut and tangential to the disc, particle A is at rest. In the same horizontal plane another particle B of same mass m moving with velocity > perpendicular to string collides elastically with A. just after impact which of the following statements will be true (A) Tension in the string is
? =G
(B) Acceleration leration of the centre of the disc is (C) Tension in the string is
? =G
(D) Acceleration of the centre of the disc is
? =G ? =G
19. In the adjacent figure a block A of mass m is hanging vertically with the help of a light inextensible string which is passi passing ng over massless and frictionless pulley its other end is attached to block C of mass 2M as shown in the figure figure. Pulley is fixed to the block B of mass 2M as shown. Then, (ignore any friction) (A) Tension in the string immediately after system is I$
released from rest is J JI (B) Tension in the string immediately after system is released from rest is
I$ I$ JI I
$
(C) The acceleration of block of mass m immediately after system is released from rest is JI
$
(D) The acceleration of block of mass m immediately after system is released from rest is J I
20. A rod CD of length L and mass M is placed horizontally on a frictionless horizontal surface as shown. A second identical rod AB which is also placed horizontally (perpendicular to CD) on the same horizontal surface is moving along the surface with a velocity ty v in a direction perpendicular to rod CD and its end B strikes the rod CD at end C and sticks to it rigidly. Then,
?
(A) Velocity of centre of mass of the system just after impact is
?
(B) The ω(angular speed) of system just after collision is =(
?
(C) Velocity of centre of mass of the system just after impact is =?
(D) The ω(angular speed) of system just after collision is (
Chemistry Part 2 Straight Objective Type This section contains 8 multiple choice questions numbered 1 to 8. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. 1. Which observation did not contribute to the development of Bohr’s model of the atom? (A) Photons have specific frequency and wavelength (B) Electrons have dual nature (wave (wave-particle duality) (C) Light is emitted by K gas at low pressure when electricity is passed through it (D) The nuclear model proposed by Rutherford 2. What wavelength of light is required to convert Given bond enthalpies: N=N (711 kj/mole) N-N N (163 kj/mole) (A) 218 nm
(B) 310 nm (C) 425 nm (D) 612 nm 3. Boron nitride is reffered to as white graphite because it is lubricious material with the same platy hexagonal structure as black graphite (Carbon graphite). Which of the following properties of hBN is untrue? (A) Hcp unit cell (B) High thermal conductivity (C) High electrical resistance (D) Chemically inert 4. Lanthanum (III) chloride is a white hygroscopic powder which attains the following equilibrium in a sealed vessel: LaCL (s)+ K M (g) ⇌ LaCIO (s)+ 2HCI(g) More water vapour is added and the equilibrium is allowed to re-establish. If at new equilibrium, OP Q has doubled, OPRS increases: (A) 2 times (B) √2 times (C) 3 times
(D) √3 times 5. Electron capture accomplishes the same end result for the nucleus as: (A) α emission (B) T C emission (C) T J emission (D) U emission 6. If the reaction A + B → P is exothermic to the extent of 30 kcal/mole and V for the forward reaction is 294 kj/mole, V for the backward reaction in kcal/mole is: (A) 324 (B) 264 (C) 100 (D) 40 7. Which physical constant for K M has higher magnitude than W M: (A) Boiling point
(B) Temperature of maximum density (C) Dielectric constant (D) Bond dissociation energy 8. Ratio of frequency of revolution of electron in second excited state of HX J and second state of H Y
is Z . x+y is
(A) (B) (C) (D)
6 7 8 9
Comprehensive Type This section contains 3 groups of questions. Each group has 2 multiple choice question based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which ONLY ONE is correct.
Paragraph for question Nos. 9 to 10 Orthophosphoric acid K OM is a weak tribasic mineral acid K OM ⇌ K J + K OM C
[\ = 10C
K OM C ⇌ K J + K OM C
[ = 10C]
K OM C ⇌ K J + K OM C
[^ = 10C
9. If 1,2,3,4 moles of K OM , NaK OM , N KOM and N OM respectively are mixed together to form an aqueous solution, resulting pH is: (A) 8 (B) 9 (C) 10
(D) 12 10. If 2.5 mole each of K OM , NaK OM , N KOM and N OM are mixed together to form an aqueous solution, resulting pH is (A) 5 (B) 7 (C) 9 (D) 11 Paragraph for questions Nos. 11 to 12 The isomerisation of cyclopropane K% to propene is believed to occur by the following mechanism: Step-1: K% + K% → K% + K%∗ Step-1: K%∗ → CK - CH = CK 11. At low pressure, step-1 unindirectional and slow whereas step-2 is bidirectional and fast. Order of the reaction is: (A) 0 (B) 1 (C) 2 (D) 3 12. At high pressure, step-1 is bidirectional and fast whereas step-2 is unidirectional and slow. Order of the reaction is: (A) 0 (B) 1 (C) 2 (D) 3 Paragraph for Question Nos. 13 to 14 Redox reaction involves transfer of electrons between 2 chemical species. An unbalanced and incomplete example is shown below: ` + K OM → H` + O M%
13. If the above reaction were possible in either direction, maximum equivalent weight would belong to: (A) ` (B) K OM (C) K` (D) O M% 14. If the above reaction were unidirectional towards right, what fraction of O M% would be left unreacted if reaction were started with stoichiometric amount of reactants. (A) (B) (C) (D)
Multiple Correct Choice Type This section contains 6 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for its answer, out which ONE OR MORE is/are correct. 15. Bubbling a K% through a solution of alkali metal (M) in liquid NK produces which of the following chemical species: (A) MBK
(B) `K aK (C) K (D) baK 16. If equatorial plane in PCL= molecule is the X-Y plane, the orbitals hybridizing to produce axial bonds will be: (A) cd (B) e Cf (C) cf (D) ed
17. Which of the following anions is/are bigger than hydride ion(K C ) (A) g C (B) L C (C) ah C (D) L C 18. Which of the following changes can be experimentally realized in the laboratory: (A) Combustion of Mg ribbon in CM (B) Reduction of FeCL (aq.) on adding Zn granules (C) Dry heating of hydrated MgCL to get anhydrous MgCL (D) Evolution of brown gas on heating LiNM 19. Which digits would appear as stoichiometric coefficient of either reactant or product when the following skeletal redox reaction is balanced: ib.M + K j + K jM → iKjM + b.jM + b.jM + j + K M (A) 2 (B) 4 (C) 6 (D) 8 20. Identify the correct sequencing of hydrides based on the parameter specified in bracket: (A) NK > OK > l,K > jmK (bond angle) (B) SbK > `K > l,K > OK (boiling point) (C) NK > OK > l,K > jmK (dipole moment) (D) NK > OK > l,K > jmK (basic nature)
Mathematics Part 3 Straight Objective Type This section contains 8 multiple choice questions numbered 1 to 8. Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.
1. The value of lim →r s t.01B C u is (A) −
(B) (C)
(D) −
x
2. The value of v t.|2,-. + 1| e, is (A) Is equal to -2πln2 x
(B) Is equal to − t.2
(C) Is equal to 0 (D) Does not exist 3. Let f be a differentiable function such that |y| ≤ 1∀ ∈ }−1,1 and g(x) = |y′|, g(0) = 4, then choose the correct statement (A) There is no point x in the interval (-1,0) at which g(x) ≤ 2 (B) g(x) > 2 ∀ ∈ (0,1) (C) there is a point of local maxima of g(x) in (-1,1) (D) x=0 is a point of local maxima of g(x) 4. v
"# J√
J J !J ; J ! ^ ^
;
(A)
(B)
J ; J ! ^ ;
(C)
;
^
J ; J ! ^
J ; J !
dx, is
+0 +0 +0
x
(D) B.C 1 + 2,-. + ! + 0
f
= ^ C f C , is fCf ^ f C C B.C C ! = 0 f C C B.C C ! = 0 f C C 2B.C C ! = 0
5. Solution of the differential equation = (A) In| − 4 + 5 + 2 + 2| −
(B) In| − 4 + 5 + 2 + 2| + (C) In| − 4 + 5 + 2 + 2| +
(D) In| − 4 + 5 + 2 + 2| − 2B.C
6. The value of v (A) )2√2
J
J ^/
e, is
f C C !
C
=0
(B) )3√2 (C) )2√3
(D) )3√3
7. Let = 0, m = 8, J = + m .e mJ = m ∀ - = 1, 2, 3, … ….. let be area of the loop formed by | − | + || = m . If all the loops are plotted on the same X – Y plane, then the value of ⋃ l is (A) (B) (C) (D)
= ,. .-B,.
,. .-B,. & ,. .-B,. & ,. .-B,.
8. The number of real roots of the equation 54 − 36 + 18 − 6 + 1 = 0, is (A) 0 (B) 2 (C) 3 (D) 4
Comprehension type This section contains three groups of questions. Each group has two multiple choice question based on a paragraph. Each question has 4 choices (A), (B), (C) and (D) for its answer, out of which only one is correct. Paragraph for question Nos. 9 and 10 Read the following write up carefully and answer the following questions:
Let y =
,
∈ 0, @.e *be inverse of y
9. If ‘A’ is the area of the region bounded by = *, x − axis, the line = x and the line = x, then the value of ‘A’ lies in the interval
(A) 0, %!
(B) % , !
(C) , !
(D) , 1!
10. The slope of tangent to the curve = *, ∈ 0, @ at the point where it meets the curve = 1, is x x −
− x − x
(A) − (B) (C) (D)
Paragraph for question Nos. 11 to 12 Let f: R → R; fx = 2x − 3k + 2x + 12kx − 7, −4 ≤ k ≤ 6, k ∈ l, then 11. The number of values of k for f(x) to be invertible, is (A) 0 (B) 1 (C) 2 (D) 3 12. The number of values of k for f(x) to have three real and distinct roots, is (A) 2 (B) 4 (C) 5
(D) 6
Paragraph for question Nos. 13 to 14 Let a tangent be drawn at a point on the locus f(x, y) = 0 and it meets the positive X and Y – axis at points F and Q. let A and G be respectively the arithmetic and geometric means of the segments OP and OQ. Now, 13. If A = 1 then the locus f(x, y) = 0, is f
(A) = + C , (where k is a parameter) (B)
f
+ C = 2, (where k is a parameter)
+ , (where k is a parameter) C
f + C = 1, (where k is a parameter)
(C) = (D)
14. If G = 1, then the locus of f(x, y) = 0, is (A) [ − 2 + [ + 1 = 0, (where k is a parameter) (B) 1 − [ − 2 + [ + 1 = 0, (where k is a parameter)
(C) [ − 2 + + 1 = 0, (where k is a parameter)
(D) 1 − [ − 2 + + 1 = 0, (where k is a parameter)
Multiple correct choice type This section contains 6 multiple choice questions. Each question has four choices (A), (B), (C) and (D) for its answer, out of which one or more is/are correct. 15. Let y =
√ Y C . √ Y J
The graphs of y = f(x) and y = f(4x) are symmetrical about the points (a1, 0)
and (a2, 0) respectively, than (A) a1 = 1, a2 = 4
(B) a1 = 2, a2 =
(C) v y2e = 1
(D) v y2e = 0
16. Let g(x) = f(x) + f(1-x) where f(x) = sin@ , © 0,1 then
(A) g(x) = y ! y
C !
(B) g(x) > π ∀ x∈ (0,1) (C) g(x) ≤ 4 ∀ x ∈ (0,1)
(D)
x
/ ;x
< v
C
e < 2
Cx
17. Let f(x) = 5x tan x + 8sin(tan x) + 5ln(cos x), then in the interval , 0! (A) (B) (C) (D)
F(x) is increasing F(x) has a point of local maxima F(x) has a root F(x) is always negative Cx x
18. Let f(x) = 01, . X , x ∈ , ! then
x
(A) F’(x) has a point of local maxima at x=
Cx
(B) F’(x) has a point of local maxima in the interval , 0!
Cx x
(C) F’(x) has exactly two points of local maxima/minima in , ! Cx x
(D) F’(x) has no root in , !
19. Let f be any twice differentiable function ∀ x∈ + and x = b be its points of local maxima and local minima respectively then (A) y = [f(x)], ([.] denotes the G.I.F) must be continuous at x = a (B) y = [f(x)], ([.] denotes the G.I.F) must be continuous at x = b (C) f”(a) must be negative (D) f”(b) must be negative
/ 2 ! dx, then / v 1 − e
v 1 − e
20. let | = v (A) | =
(B) | =
x/
(C) | = v
(D) | = %
,-. ' 01, ' e'
ANSWER KEY Q. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Physics A D B C B C D A C A B A C A A,B,D C,D B,C A,B A,C B,C
Chemistry B A A B C C C C C B C B C D A,B,C A,D B,C,D A,B,D A,B,D A,B,C,D
Mathematics A C C C D B A A B B B B A C B,D A,B,C,D A,D A,B,C B,D A,B,C,D
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