Paper - 1

PHYSICS (PART I)

SECTION I

This section contains 9 multiple-choice questions numbered 1 to 9. Each question has 4 choices (A), (B), (C) and (D),

out of which Only One is correct

1. A and B are two concentric spheres with their centers

at O and of radius R and 2R respectively. C and D are

two other concentric spheres of radius R and 2R and

their center is at O. O and O are at a separated by a large distance. Initially the charge on sphere A is q and on B is q. When the switch is connected the

charge on the sphere D and C is

2R

A

B

S

R

2R

C

D

R

O O

(A) 2

q, zero (B)

2

q, zero (C) ,

2 2

q q (D) Zero, Zero.

2. The pitch of a screw gauge is 1 mm and there are 100 divisions on its circular scale. When

nothing is put in between its jaws, the zero of the circular scale lies 4 divisions below the

reference line. When a steel wire is placed between the jaws, two main scale divisions are

clearly visible and 67 divisions on the circular scale are observed. The diameter of the wire is

(a) 2.71 mm (b) 2.67 mm (c) 2.63 mm (d) 2.65 mm

3. If the ground state of hydrogen atom is chosen as zero potential energy level, the value of the

total energy in the second excited state (in eV) is

(A) 25.69 (B) 12.09 (C) 23.8 (D) 20.4

4. In a sample of rock, the ratio of 206Pb to 238U nuclei is found to be 0.5. The age of the

rock is. (Assume that all the Pb nuclides in the rock was produced due to the decay of

Urarium nuclides and 238 9

1/ 2T ( U) 4.5 10 year).

(A) 2.25 109 year (B) 4.5 109 ln 3 year

(C) 4.5 109 32ln

ln 2 year (D) 2.25 109 ln

3

2

year

5. Interference pattern is observed at P due to superposition of two rays coming out from a

source S as shown in the figure. The value of l for which maxima is obtained at P is

2l P

l/3 30

o

S

(A) 13

2

nl (B)

132

12

nl

(C)

324312

nl (D)

13

12

nl

6. A thin plane convex glass lens ( = 1.5) has its plane surface silvered and R is the radius of curvature of the curved part. Then which of the following ray diagram is the correct

representation for an object placed at O.

O 2R O 2R

(A) (B)

O 3R O 2R

(C) (D)

7. A rod of length l is standing vertically on a frictionless surface. It is disturbed slightly from

this position. Let and be the angular speed and angular acceleration of the rod when the

rod turns through an angle with the vertical, the value of acceleration of center of mass of the rod is

(A) 2

sin cos2 2

l l (B)

2

sin cos2 2

l l

(C) cos2

l (D)

2

sin2

l

8. Two blocks A (5kg) and B(2kg) attached to the ends of a spring

constant 1120N/m are placed on a smooth horizontal plane with the

spring undeformed. A and B are imparted simultaneously velocities of

3m/s and 10m/s along the line of the spring in the same direction

shown. Consider the following statements. Choose the correct answer.

(1) when the extension of the spring is maximum the velocities of A and B are zero.

(2) the maximum extension of the spring is 25cm.

(3) maximum extension and maximum compression occur alternately.

(4) the maximum compression occur for the first time after 56

sec.

(A) Statements (2), (3), (4) are correct (B) Statements (1), (2), (3) are correct

(C) Statements (2), (3) are correct. (D) All the statements are correct

9. Initially spring is compressed by distance x0 from equilibrium

position. At this compression block is given velocity

0

3K

mx , so that compression in the spring increases and

block start S.H.M. (Spring constant is K). Equation of motion

of the block is :

EquilibriumPosition

x0

K

(A) 0

ky 3 sin t

m 3

x (B) 0

Ky 2 sin t

m 6

x

(C) 0

Ky 3 sin t

m 6

x (D) 0

Ky 2 sin t

12 3

x

SECTION II

Assertion Reason Type

This section contains 4 questions numbered 10 to 13. Each question contains STATEMENT1 (Assertion) and STATEMENT2 (Reason). Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

NOTE : (A) Statement 1 is True, Statement 2 is True; Statement 2 is a correct explanation for Statement 1.

(B) Statement 1 is True, Statement 2 is True; Statement 2 is NOT a correct explanation for Statement 1.

(C) Statement 1 is True, Statement 2 is False. (D) Statement 1 is False, Statement 2 is True.

10. STATEMENT 1: Good reflectors are poor emitters of thermal radiation. STATEMENT 2 : The ratio of the emissive power (e) and absorptive power (a) is constant

for all substances at any given temperature and for radiation of the same wavelength.

11. Statement-1: A horizontal rim of a wheel is free to rotate about a vertical axis. A line charge

is glued onto the rim of the wheel. The shaded central region has a uniform magnetic field

pointing in vertical direction. If the field is switched off, the wheel starts rotating with certain

angular speed which does not depend on how fast or slow the field is switched off.

+ + + + + +

+ + +

+ + +

+ + + + + + +

+ +

+

Statement-2: Induced electric field due to a time varying magnetic field does not depend on

time taken to change magnetic field.

12. STATEMENT 1

Mirrors are free from chromatic aberration.

STATEMENT 2

The refractive index of the material of the mirror is independent of wavelength (color) of

light.

13. STATEMENT 1

The kinetic energies of all particles, emitted by a sample containing active radiocobalt nuclei, are identical.

STATEMENT 2

particles get their energy from the energy differences between the mother and daughter nuclei, which is fixed for a particular decay scheme.

SECTION III

Linked Comprehension Type

This section contains 2 paragraphs P14-16, and P17-19 . Based upon each paragraph, 3 multiple choice questions have to

be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

P14-16 : Paragraph for Questions Nos. 14 to 16

A uniform rod is lying at rest on a frictionless horizontal surface. A

particle, of same mass as the rod, collides with the rod at its end

with an angle from the normal as shown in the figure. Assume that there is no friction between the rod and particle and that the

collision is elastic.

Onormal

vm

m,

2

14. The magnitude of the velocity of the centre of mass of the rod after collision is

(a) v cos (b) v sin (c) 2

v cos5

(d) 2

v sin5

15. The magnitude of change in angular momentum of the rod about the point O after collision is

(a) zero (b) m v cos

5

(c)

m vsin

5

(d)

6m vsin

5

16. The magnitude of the change in velocity of the particle after collision is

(a) 8

v cos5

(b) 2

v cos5

(c) 8

vsin5

(d) 2

v sin5

P17-19 : Paragraph for Questions Nos. 17 to 19

For the two given circuits at t = 0, a constant force F acts at the middle points of the rigid

conducting wires ab and cd. At t = 0 both wire are at rest. The electric resistance of the circuit

(a) is zero, while for the circuit (b) electrical resistance is R. The electrical resistance of the

horizontal rails is zero. There is no friction between rails and rigid wires ab and cd. Both the

circuits are placed in a vertical constant magnetic field B (as shown in the figure). The mass and

length of the each wire ab and cd is m, respectively

R

FCC F

m, m,

Circuit (X) Circuit (Y)

BB

a

b c

d

17. At t =m

F

, electric current for circuit (X) is

(A) zero (B) 2 2

B CF

(m B C)

(C)

B CF

m

(D)

F

B

18. At t = 0, acceleration of wire cd for circuit Y is

(B) F

m (B)

2 2

F

(m B C) (C)

2 2

F

B C (D)

2 2

2F

B C

19. At t = 0, acceleration of wire ab for circuit X is

(A) F

m (B)

2 2

F

(m B C) (C)

2 2

F

B C (D)

2 2

2F

B C

SECTION IV Matrix Match Type

This section contains 3 questions. Each question contains statements given in two columns

which have to be matched. Statements (A, B, C, D) in Column I have to be matched with

statements (p,q,r,s) in Column II. The answers to these questions have to be appropriately

bubbled as illustrated in the following example. If the correct matches are A-p, A-s, B-q, B-r, C-

p, C-q and D-s, then the correctly bubbled 4 4 matrix should be follows :

20. The entries in Column I depict certain current distributions, while the entries in Column II

depict the variation of the magnetic field (B) as one moves along the xaxis for each of these distributions, but in a different order. Match the entries in Column I with the proper entries in

Column II.

Column I Column II

(A)

O x(perpendicular

to wire)

Straight current carrying wire

(P) B

x

(B)

Ox

(axis of wire)

Circular current carrying wire

(Q) B

x

(C)

Ox

(perpendicular to the plane of the wires;

O being equidistant)

Parallel current carrying wires in the same plane

i

i

(R) B

x

p q r s

A

B

C

D

(D)

x(parallel to one of the wires)

Two perpendicular current carrying wires in the same plane

i

i

(S) B

x

21. Charge Q is distributed on two identical capacitors in

parallel. Separation of the plates in each capacitor is d0.

If the first plate of the capacitor C1 and the second plate

of the capacitor C2 start moving to the left with constant

speed v, then match the options in Column I with those

in Column II.

+ -

+ -

Column I Column II (A) Charge on capacitor C1 as a

function of time (P) Reduces

(B) Charge on capacitor C2 as a

function of time (Q)

0

0

Q(d vt)

2d

(C) Current in the circuit (R)

0

0

Q(d vt)

2d

(D) Energy of the system (S)

V

0

Q

2d

22.

Column I Column II

(A) Mass of products formed is less than

that of reactants.

(P) decay.

(B) Binding energy per nucleon

increases.

(Q) decay.

(C) Emitted particles have variable

kinetic energy.

(R) +decay.

(D) Two body decay process. (S) decay.

CHEMISTRY PART - II Useful Data: Gas Constant R = 8.314 J K1 mol1

= 0.0821 Lit atm K1 mol1

= 1.987 2 Cal K1 mol1

Avogadro's Number Na = 6.023 1023

Plancks constant h = 6.625 1034 J s

= 6.625 1027 erg s 1 Faraday = 96500 Coulomb

1 calorie = 4.2 Joule

1 amu = 1.66 1027 kg

1 eV = 1.6 1019 J Atomic No: H = 1, D = 1, Li = 3, Na = 11, K = 19, Rb = 37, Cs = 55, F = 9, Ca = 20, He = 2, O = 8, Au = 79, Ni = 28,

Zn = 30, Cu = 29, Cl = 17, Br = 35, Cr = 24, Mn = 25, Fe = 26, S = 16, P = 15, C = 6, N = 7, Ag = 47.

Atomic Masses: He = 4, Mg = 24, C = 12, O = 16, N = 14, P = 31, Br = 80, Cu = 63.5, Fe = 56, Mn = 55, Pb = 207, Au = 197,

Ag = 108, F = 19, H = 1, Cl = 35.5, Sn = 118.6, Na = 23, D = 2, Cr = 52, K = 39, Ca = 40, Li = 7, Be = 4,

Al = 27, S = 32.

SECTION - I

This section contains 9 multiple choice questions numbered 23 to 31. Each question has 4 choices (A), (B), (C) and

(D), out of which ONLY ONE is correct

23.

3

(i)t BuO

(ii)H OD.

CH3

N R

O

D is

(a)

CH3

N R

O

t-BuO

(b)

N R

(c)

N

R

(d)

N

R

24. How many grams of sucrose (Molecular weight = 342) should be dissolved in 100 gm water

in order to have a solution with sum of lowering of freezing point and elevation of boiling

point equal to 5oC (Kb = 0.51, Kf = 1.86)?

(a) 34.2 gm (b) 72 gm

(c) 342 gm (d) 460 gm

25.

H

BrCl

H H

BrH

Cl Br

HH

Cl Br

HCl

H

(A) (B) (C) (D)

(I) (A) and (B) are diastereomers

(II) (B) and (D) are enantiomers

(III) (A) and (D) are geometric isomer

(IV) (A) and (C) are optical isomer

(a) only I is true (b) I, II & III are true

(c) I & II are true (d) all are true

26. 23

(Reaction similar to acidic -Halogenation)

HNO &EtONO/ HCl

H O(A).

Et C CH3

O

(A) is

(a) Et C C H

OO

(b) Et C CH N

O

OH

(c) No reaction (d) Et C CH2NO

O

27. The IUPAC name of complex K3[Al(C2O4)3] is

(a) potassium aluminoxalate (b) potassium trioxalato aluminate (III)

(c) potassium aluminium (III) oxalate (d) potassium trioxalatoaluminate (VI)

28. CH3 C Cl / AlCl 3

O

Me C Cl

O2I / NaOH(i)LDA

(Acid)(Acid)

(ii)

A B C D E(ppt.)

(C) & D are :

(I) C O

O

(II) CH3 C O

O

(III) C CH2

O

C O

O

(a) I only (b) I and II (either)

(c) III only (d) none of these

29. A solution of 0.1 M Na2CO3 is titrated against HCl using (i) phenolpthalene as an indicator

and (ii) Methyl orange as an indicator. If 1 2 3 2

6 10

a (H CO ) aK 2 10 ,K 4 10 , then

[pH(1) pH(2)] equals to? (Where pH1 is at end point with phenalpthalene and pH2 is at end point with methyl orange) (gases are evolved off if any).

(a) 1 (b) 3

1 log 22

(c) 0 (d) 3

1 log 22

30. 2

3 2Cu (aq) NO NH OH Cu

NH2OH + Fe2+

Fe3+ + NH3

NH3 + HCl 4NH + Cl

If 0.72 M of 100 ml Cu+ was required to completely reduce 3NO

to NH2OH; if Fe

2+ was in

excess then volume of 0.1 M HCl needed to react with NH3 formed in ml will be?

(a)240 (b)360 (c)120 (d)480

31. The molar conductivity at infinity dilution of AgNO3, NaCl and NaNO3 are 116, 110, 106

s-cm2-mol

1 respectively. The conductivity of AgCl in water is 2.4 106 s/cm and that of

water is 1.2 106 s/cm. Find the value of x. Where x 106 is molarity of AgCl = (a) 10 (b) 20 (c) 30 (d) 40

SECTION - II

Assertion - Reason Type

This section contains 4 questions numbered 32 to 35. Each q ues t io n contains STATEMENT-1 (Assertion) and

STATEMENT-2 (Reason). Each q u e s t i o n has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Note: (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for

Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for

Statement-1 (C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True

32. STATEMENT-1 : [Fe(CN)6]4

is an outer orbital complex.

STATEMENT-2 : Electronic configuration of Fe2+

ion is 1s2, 2s

2, 2p

6, 3s

2, 3p

6, 3d

6,4s

0

33. STATEMENT-1 :CH3CH2OCH2Cl reacts faster when treated with water than

CH3CH2OCH2CH2CH2Cl.

STATEMENT-2 : Carbonium ion formed by the ionization of CH3CH2OCH2Cl is stabilized

by resonance

34. STATEMENT 1 Specific activity of the same radioactive substance is same for 10g radioactive substance as

well as 50 g radioactive substance.

STATEMENT 2 Since specific activity of a radioactive substance is its activity per g.

35. STATEMENT 1 The value of equilibrium constant may have lower or higher value at different temperatures.

STATEMENT 2 The values of equilibrium constant depends on enthalpy of reaction.

SECTION - III

Linked Comprehension Type

This section contains 2 paragraphs C36-38 and C39- 41. Based upon each paragraph, 3 mul t iple choice questions have

to he answered. Each question has-4 choices (A), (B), (C) and (D), o ut of which ONLY ONE is correct.

C36 - 38: Paragraph for Question Nos. 36 to 38

When a liquid is completely miscible with another liquid, a homogeneous solution consisting

of a single phase is formed. If such a solution is placed in a closed evacuated vessel, the total

pressure exerted by the vapour, after the system attained equilibrium will be equal to the sum

of partial pressures of the constituents. A solution is said to be ideal if its constituents follow

Raoults law under all conditions of concentrations. i.e., the partial pressures of each and every constituents is given by pi = xipi

0

Where pi is the partial pressures of the constituent i, whose mole fraction in the solution is xi

and pi0 is the corresponding vapour pressure of the pure constituent.

Since both the components of an idel binary liquid system follow Raoults law of the entire range of the composition, the partial pressure exerted by the vapours of these constituents

over the solution will be given by

pA = xApA0 (i)

pB = xB pB0

(ii) where xA and xB are the amount fractions of the two constituents in the liquid phase and pA

0

and pB0 are the respective vapour pressures of the pure constituents. The total pressure (p)

over the solution will be the sum of the partial pressures. The composition of the vapour

phase (yA) can be determined with the help of Daltons law of partial pressures.

36. For an ideal solution in which pA0 > pB

0, the plot of total pressure (p) verses the mole fraction

of A at constant temperature in the vapour phase is :

(a)

0 1 1 0

yA

yB

pB0

pA0

p

(b)

0 1 1 0

yA

yB

pB0

pA0

p

(c)

0 1 1 0

yA

yB

pB0

pA0

p

(d)

0 1 1 0

yA

yB

pB0

pA0

p

37. A plot of reciprocal of total pressure 1

p

verses yA gives

(a) an linear plot with slope = 0 0

B A

1 1

p p

(b) a linear plot with slope =

0 0A B

1 1

p p

(c) a linear plot with slope = 0

B

1

p (d) a linear plot with slope = pA

0pB

0

38. Two liquids A and B form an ideal solution at temperature T. When the total vapour pressure

above the solution is 600 torr, the amount fraction of A in the vapour phase is 0.35 and in the

liquid phase is 0.70. The vapour pressures of pure B and A are

(a) 800 torr; 1300 torr (b) 1300 torr; 300 torr

(c) 300 torr; 1300 torr (d) 300 torr; 800 torr

C39 - 41: Paragraph for Question Nos. 39 to 41

When CH3CH=CH2 is treated with Br2 it forms an optically active product, further the same

compound when treated with HBr forms optically inactive product.

CH3CH=CH2 + Br2 Product

CH3CH=CH2 + HBr Product When HBr adds to 2-methyl-1-butene in the presence of benzoyl peroxide, the product is

racemic mixture.

CH3CH2 C CH2

CH3

+ HBr Peroxide Racemic mixture

However, when HBr adds on 3-chloro-1-butene, the product is found to be optically active.

+ HBr Optically active

CH3

HCl

CH CH2

39. Formation of optically inactive product by the additions HBr on in presence and absence of

peroxide to CH3CH=CH2 is (a) of same type

(b) of different type

(c) due to formation of carbocation in I reaction and free radical in II reaction

(d) due to formation of carbocation in both reactions.

40. The type of intermediate formed in the above mentioned reaction of 2-methyl-1-butene is

(a) carbocation (b) free radical

(c) carbanion (d) carbene

41. Optically activity in the product of the reaction of (+)-3-chloro-1-butene is due to

(a) formation of (+)enantiomer (b) formation of the ()enantiomer

p q r s

A

B

C

D

(c) formation of non-50 : 50 racemic mixture

(d) formation of diastereomeric pair

SECTION IV Matrix Match Type

This section contains 3 questions. Each question contains statements given in two columns

which have to be matched. Statements (A, B, C, D) in Column I have to be matched with

statements (p, q, r, s) in Column II. The answers to these questions have to be appropriately

bubbled as illustrated in the following example

If the correct matches are A p, A s, B q, B r, C p, C q and D s, then the

correctly bubbled 4 4 matrix should be as follows:

42. Reaction sequence I

O2 2 2(i) I / NaOH SO Cl EtOH / H EtMgBr

1 1 1 1(ii)HA B C D

Reaction sequence II

H

O2 2SO Cl[O] EtOH EtMgBr

2 2 2 2A B C D

Reaction sequence III

NH2

O2 5 2P O (i)I / NaOHEtMgBr MeOH

3 3 3 3(ii)H HA B C D

Reaction sequence IV Cl

3H OKCN MeOH / H MeMgBr

4 4 4 4A B C D

Match the compounds given in Column I with respect to the sequences in which there are found in Column II Column I Column II

(A) CH3 C Cl

O

(P) Sequence I

(B) CH3 C Et

O

(Q) Sequence II

(C) Et C OH

O

(R) Sequence III

(D) Et C O

O

Me

(S) Sequence IV

43. Match the following :

Column I Column II

(A) Monomer of natural rubber (P) RI (B) Most reactive alkyl halide for SN

2 reactions (Q) CHCl3

(C) Gives alcohol with NaOH (aq) (R) Isoprene

(D) Give carboxylic acid with aq. NaOH (S) 2-methyl buta-1, 3-diene

44. Match the following :

Column I Column II

(A) Spontaneous process (P) H = ve (B) Heat flow from high temperature

of system towards low

temperature of surroundings

(Q) G = +ve

(C) Non spontaneous process (R) Stotal = +ve (D) Increase in randomness of system

by heating

(S) G = ve

MATHEMATICS PART - III

SECTION I

This section contains 9 multiple-choice questions numbered 45 to 53. Each question has 4 choices (A), (B), (C) and

(D), out of which Only One is correct

45.

2sec x

2

2x 2

4

f t dt

lim

x16

equals

(a) 8

f 2

(b) 2

f 2

(c) 2 1

f2

(d) 4f 2

46. Let f(x) = x + sin x. The area bounded by 1y f x ,y x, x 0, is (a) 1 (b) 2

(c) 3 (d) canot be found because f-1

(x) cannot be determined

47. If x = a cos t, y = a sin t, then 2

2

d y

dx at t =

4

is

(a) a

2 2 (b)

a

2 2 (c)

2 2

a (d)

2 2

a

48. The reflection of the point P(1, 0, 0) in the line x 1 y 1 z 10

2 3 8

is

(a) (3, 4, 2) (b) (5, 8, 4) (c) (1, 1, 10) (d) (2, 3, 8)

49. The line x + y = 6 is a normal to the parabola y2 = 8x at the point

(a) 18, 12 (b) (4, 2) (c) (2, 4) (d) (3, 3)

50. The line x + y = 1 meets x-axis at A and y-axis at B. P is the mid-point of AB. P1 is the foot

of the perpendicular from P to OA; M1 is that from P1 to OP; P2 is that from M1 to OA and so

on. If Pn denotes the nth foot of the perpendicular on OA from Mn-1, then OPn =

(a) 1/2 (b) 1/2n (c) 1/2

n/2 (d) 1/ 2

51. If 0 < x < 1, the number of solutions of the equation

1 1 1 1tan x 1 tan x tan x 1 tan 3x is (a) 0 (b) 1 (c) 2 (d) 3

52. If a1 > 0 for i = 1, 2, , n and a1a2.an = 1, then (2 + a1)(2 + a2).(2 + an) is greater than (a) 2

n/2 (b) 2

3n/2 (c) 2

2n (d) none of these

53. If the ratio 1 z

1 z

is purely imaginary, then

(a) 0 < |z| < 1 (b) | z | 1

(c) | z | 1 (d) bounds for |z| can not be decided

SECTION II

Assertion Reason Type

This section contains 4 questions numbered 54 to 57. Each question contains STATEMENT1 (Assertion) and STATEMENT2 (Reason). Each question has 4 choices (A), (B), (C) and (D), out

of which ONLY ONE is correct

Note: (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for

Statement-1

(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct

explanation for Statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True

54. Statement-1: Curve satisfying the differential equation y

y '2x

passing through (2, 1) is a

parabola with focus 1

,04

.

Statement-2: The differential equation y

y '2x

is of variable separable.

55. Let

x

x

x

2 1, 1 x 0

f x 2 , x 0

2 1, 0 x 1

Statemnet-1: f is bounded but never reaches its maximum and minimum.

Statement-2: f has a discontinuity at 0.

56. Statement-1: P is a point (a, b, c). Let A, B, C be the images of P in yz, zx ad xy planes

respectively, then equation of the plane passing through the points A, B and C is

x y z1

a b c .

Statement-2: The image of a point P in a plane is the foot of the perpendicular drawn from P

on the plane.

57. Statement-1: A chord y = mx + c of the curve 3x2 y2 2x + 4y = 0, which passes through

the point (1, 2), subtend a right angle at the origin. Statement-2: Lines represented by the equation

(3c + 2m)x2 2(1 + 2m)xy + (4 c)y2 = 0 are perpendicular if c + m + 2 = 0.

SECTION III

Linked Comprehension Type

This section contains 2 paragraphs P58-60 and P61-63. Based upon each paragraph, 3 multiple choice questions have to

be answered. Each question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct.

M58-60 : Paragraph for Questions Nos. 58 to 60

If f, g and h are functions having a common domain D and h x f x g x , x D and if

x a x alimh x limg x l

then x alimf x = l

. This is known as Sandwich Theorem.

58. 4

x 0

1lim x sin

3 x

is

(a) 0 (b) 1

(c) 1

3 (d) does not exist

59. Let 1/ x 1/ x

2

1/ x 1/ x

e ef x x , x 0

e e

and f(0) = 1 then

(a) x 0lim f x

does not exist (b)

x 0lim f x

does not exist

(c) x 0lim f x

exists (d) f is continuous at x = 0.

60. Let 5 31

f x x , x 0x

and f(0) = 0 ([.] denotes the greatest integer function)

(a) x 0lim f x

does not exist (b) f is not continuous at x = 0

(c) x 0lim f x

= 1 (d) x 0lim f x

= 0

M61-63 : Paragraph for Questions Nos. 61 to 63

P(a, 5a) and Q(4a, a) are two points. Two circles are drawn through these points touching the

axis of y.

61. Centre of these circles are at

(a) (a, a), (2a, 3a) (b) 205a 29a

,18 3

, 5a

,3a2

(c) 29a

3a,3

, 205a 29a

,9 18

(d) none of these

62. Angle of intersection of these circles is

(a) tan-1

(4/3) (b) 1tan 40/9 (c) tan

-1(84/187) (d) /4

63. If C1, C2 are the centres of these circles then area of OC1C2, where O is the origin, is (a) a

2 (b) 5a

2 (c) 10a

2 (d) 20a

2

SECTION IV

Matrix Match Type

This section contains 3 questions. Each question contains statements given in two columns which

have to be matched. Statements (A, B, C, D) in Column I have to be matched with statements

(p,q,r,s) in Column II. The answers to these questions have to be appropriately bubbled as

illustrated in the following example. If the correct matches are A-p, A-s, B-q, B-r, C-p, C-q and

D-s, then the correctly bubbled 4 4 matrix should be follows:

64. The anti-derivative of

COLUMN I COLUMN II

(A)

2

secx

secx tan x (P)

sin x 2log C

sin x 1

(B)

cos x

sin x 1 sin x 2 (Q)

2

2

cos xC

2 1 sin x

(C)

1

2

2xsin

1 x

,|x| < 1 (R) 1 22x tan x log 1 x C

p q r s

A

B

C

D

(D) tan x cot x (S)

1 tan x 12 tan C2 tan x

65. Letters of the word INDIANOIL are arranged at random. Probability that the word formed

COLUMN I COLUMN II

(A) Contains the word INDIAN (P)

9

5

1

C

(B) Contains the word OIL

(Q)

5 72 21

C C 9!

(C) Begins with I and ends with L

(R) 1

24

(D) Has vowels at the odd places

(S)

7 93 21

C C

66. Let

2 sin cos

p 1 cos sin

1 sin cos

,

sin 2 1 1

q cos 2 4 3

2 7 5

,

cos sin cos

r sin cos sin

cos sin cos

and

2

2 2 2

2 2

sec 1 1

s cos cos cosec

1 cos cot

Match the functions on the left with their range on the right.

COLUMN I COLUMN II (A) p (P) [0, 1]

(B) q (Q)

0,2 2

(C) r (R) [2, 2]

(D) s (S) 5 2, 5 2