fulltest iv main paper question paper aits 2013 ft iv jeem

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FULL TEST IV

Time Allotted: 3 Hours Maximum Marks: 432 P lease read the inst ruct ions care fu l ly. You are a l lo t ted 5 minutes

spec i f ica l ly for th is purpose. You are not a l lowed to leave the Examinat ion Hal l before the end of

the test .

INSTRUCTIONS

A. General Instructions 1. Attempt ALL the questions. Answers have to be marked on the OMR sheets. 2. This question paper contains Three Parts. 3. Part-I is Physics, Part-II is Chemistry and Part-III is Mathematics. 4. Each part has only one section: Section-A. 5. Rough spaces are provided for rough work inside the question paper. No additional sheets will be

provided for rough work. 6. Blank Papers, clip boards, log tables, slide rule, calculator, cellular phones, pagers and electronic

devices, in any form, are not allowed. B. Filling of OMR Sheet 1. Ensure matching of OMR sheet with the Question paper before you start marking your answers

on OMR sheet. 2. On the OMR sheet, darken the appropriate bubble with black pen for each character of your

Enrolment No. and write your Name, Test Centre and other details at the designated places. 3. OMR sheet contains alphabets, numerals & special characters for marking answers. C. Marking Scheme For All Three Parts.

(i) Section-A (01 to 02 and 09 to 30) contains 24 multiple choice questions which have only one correct answer. Each question carries +4 marks for correct answer and 1 mark for wrong answer.

Section-A (03 to 08) contains 6 multiple choice questions which have only one correct answer.

Each question carries +8 marks for correct answer and 2 mark for wrong answer.

Name of the Candidate

Enrolment No.

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AITS-FT-IV-PCM-JEE(Main)/13


2

Useful Data

PHYSICS

Acceleration due to gravity g = 10 m/s2

Planck constant h = 6.6 1034 J-s

Charge of electron e = 1.6 1019 C

Mass of electron me = 9.1 1031 kg

Permittivity of free space 0 = 8.85 1012 C2/N-m2

Density of water water = 103 kg/m3

Atmospheric pressure Pa = 105 N/m2

Gas constant R = 8.314 J K1 mol1

CHEMISTRY

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 Js = 6.625 1027 ergs 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, He = 2, Li=3, Be=4, B=5, C=6, N=7, O=8,

N=9, Na=11, Mg=12, Si=14, Al=13, P=15, S=16, Cl=17, Ar=18, K =19, Ca=20, Cr=24, Mn=25, Fe=26, Co=27, Ni=28, Cu = 29, Zn=30, As=33, Br=35, Ag=47, Sn=50, I=53, Xe=54, Ba=56, Pb=82, U=92.

Atomic masses: H=1, He=4, Li=7, Be=9, B=11, C=12, N=14, O=16, F=19, Na=23, Mg=24, Al = 27, Si=28, P=31, S=32, Cl=35.5, K=39, Ca=40, Cr=52, Mn=55, Fe=56, Co=59, Ni=58.7, Cu=63.5, Zn=65.4, As=75, Br=80, Ag=108, Sn=118.7, I=127, Xe=131, Ba=137, Pb=207, U=238.



3

PPhhyyssiiccss PART I

SECTION A

Single Correct Choice Type This section contains 30 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 1. An alternating voltage 220V and 50Hz is applied across a circuit containing pure resistance R,

inductance L and capacitance C in series. Across which of the following can the voltage be greater than 220V ?

(A) R, L & C (B) either L or C (C) L and C taken together (D) both (B) and (C) 2. A small ball strikes at one end of a stationary uniform frictionless rod of mass m and length l

which is free to rotate, in gravity-free space. The impact is elastic. Instantaneous axis of rotation of the rod will pass through

(A) its center of mass. (B) the center of mass of rod plus ball. (C) the point of impact of the ball on the rod. (D) the point which is at a distance 2l / 3 from the striking end. 3. The value of R for which 20 % of the main current passes

through galvanometer of resistance 80 is (A) 10 (B) 20 (C) 30 (D) 40

G

R

I

(main current)

4. A sphere of radius R is uniformly charged so that total charge on sphere is Q. Potential difference

between center of sphere and surface of sphere is (A) 0Q/4 R (B) 0Q/8 R (C) 0Q/16 R (D) 0Q/32 R 5. A large metallic plate is facing a charged sheet having charge

density placed parallel to plate at a distance A from the plate. Potential at point P at a distance x from the sheet is

(A) 0

( x)2 + A (B) 0

( x)2 A

(C) 0

( x) A (D) 0( x) + A

l

xP

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4

6. The circuit shown in the figure consists of a battery of emf E,

resistances R1 and R2, inductance L and a two way switch ABC. First A is connected to B for a long time and then A is connected to C. The total heat produced in R2 is

(A) 2

21

ELR

(B) 2

22

EL2R

(C) 2

21

EL2R

(D) 2

1 2

EL2R R

L

A

R2

R1E

C

B

7. As shown in figure, sphere of mass m is constrained to rotate about in

a vertical plane by connecting it to O by means of a rigid light rod of length A . What is the velocity of the sphere when it hits the wall of container ? It is given that specific gravity of ball is 2 and the container is filled with water.

(A) 2gA (B) 0.5gA

(C) gA (D) 3 g2A

ml

o

8. Consider three quantities x = EB

, 1/2

0 0

1y =

and RCz = A . Here A is the length of a wire, C is a capacitance and R is a resistance. All other symbols have standard meanings

(A) y, z have the same dimensions (B) z, x have the same dimensions (C) x, y have the same dimensions (D) all of the three have the same dimensions 9. A projectile is projected in the earths gravitational with initial kinetic energy E. The horizontal of

the projectile range is R. If the mass of the projectile is 1 kg then the angle of projection of the projectile will be equal to

(A) 1 gRsin2E

(B) 1 gR2sin

2E

(C) 1 gR0.5sin2E

(D) 1 gR4sin

2E

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5

10. Three rods of identical cross-sectional area are made from the same metal from the sides of an isosceles right angled triangle ABC, right angled at B. The points A and B are maintained at temperatures T and 2T respectively. In the steady state, the temperature of point C is TC. There

is a point D on BA, whose temperature is same as TC. Then the ratio BDDA

is

(A) 1 (B) 2

(C) 12

(D) 2

11. Neglect the effect of rotation of the earth. Suppose the earth suddenly stops attracting objects

placed near its surface. A person standing on the surface of the earth will (A) fly up. (B) slip along the surface. (C) fly along a tangent to the earths surface. (D) remain standing. 12. Variation of internal energy with density of one mole of monatomic gas is

depicted in the adjoining figure, corresponding variation of pressure with volume can be depicted as (Assuming the curve is rectangular hyperbola)

U

(A) P

v

(B) P

v (C) P

v

(D) P

v 13. The acceleration time graph of a particle is shown in the figure.

What is the velocity of particle at t = 8s, if initial velocity of particle is 3 m/s?

(A) 4 m/s (B) 5 m/s (C) 6 m/s (D) 7 m/s

t841O

4

a

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6

14. Let B and denote induction to magnetic field and energy density at mid point of a long solenoid carrying a current i. The graph between and B will be (A)

B X O

Y

(B)

B X O

Y

(C)

B

Y

X

(D)

B X O

Y

15. In the given figure, the angle of inclination of the inclined plane is 30.

Find the horizontal velocity V0 so that the particle hits the inclined plane perpendicularly.

(A) 02gHV

5= (B) 0 2gHV 7=

(C) 0gHV5

= (D) 0 gHV 7=

90

H

3090

V0

16. A chain of mass 10 kg and length 10 m is resting on a rough horizontal surface ( = 0.2). A

constant force of 20 newton is applied at one end. The tension in the mid point of the chain is (A) 20 N (B) 15 N (C) 10 N (D) 5 N 17. Find the speed of the block when it covers a horizontal

distance A. It is given that the block never loses contact with the smooth horizontal surface, and the force always acts at an angle with the horizontal.

F0

u = 0M

smooth

(A) 0F cos

mA

(B) 02 F cos

mA

(C) 02 F cos

mA

(D) 0F cos

mA

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7

18. Two fixed long conducting wires cross each other perpendicularly so

that they do not actually touch but are close to each other as shown in the figure. Equal current i exist in each wire in the directions indicated. In what region(s) will there be some points with zero net magnetic field?

(A) IV, III (B) III, II (C) II, IV (D) I, IV

I II

III IV

19. A sample of hydrogen gas is excited by means of a monochromatic radiation. In the subsequent

emission spectrum, 10 different wavelengths are obtained, all of which have energies greater than or equal to the energy of the absorbed radiation. It follows that the initial quantum number of the state (before absorbing radiation) was

(A) 5 (B) 4 (C) 3 (D) 2.

20. The ratio of the min in a Coolidge tube to deBroglie of the electrons striking the target depends on

accelerating potential V as

(A) mindeBroglie

V (B)

min

deBroglieV

(C) mindeBroglie

1V

(D) min

deBroglie

1V

. 21. The minimum distance between an object and its real image formed by a thin convex lens of focal

length 30 cm is (A) 45 cm (B) 50 cm (C) 75 cm (D) 120 cm 22. A thin concave and a convex lens have the same focal length of 40 cm, and are put into contact

to form a lens combination. The combination is used to view an object of 5 mm length kept at 20 cm form the lens combination. As compared to the object, the image will be

(A) Magnified and inverted (B) Reduced and erect (C) Of the same size as the object and erect (D) Of the same size as the object but inverted 23. A uniform plank of Youngs modulus Y is moved over a smooth horizontal surface by a constant

horizontal force F0. The area of cross-section of the plank is A. The compressive strain on the plank in the direction of the force is

(A) 0FAY

(B) 02FAY

(C) 0F

2AY (D) 0

3FAY

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8

24. A solid ball of radius 0.2m and mass 1kg lying at rest on a smooth

horizontal surface is given an instantaneous impulse of 50 N-s at point P as shown. The number of rotations made by the ball about its diameter before hitting the ground is

(A) 625 32 (B)

2500 32

50N-sP

30

(C) 3125 32 (D)

1250 32

25. A thin uniform spherical shell and a uniform solid cylinder of

the same mass and radius are allowed to roll down a fixed incline, without slipping, starting from rest. The times taken by them to roll down the same distance are in the ratio, tsph : tcyl

(A) 1415

(B) 1514

(C) 310

(D) 103

.

26. A particle A is attached to a spring and the time period for small oscillations is observed to be T;

when an additional mass dm is added on, the time period becomes T + dT. The mass of particle A is

(A) 12dT dTdm 2

T T

+ (B)

12dT dTdm 2 2T T

+ +

(C) 1dTdm 2

T

(D) none of these

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9

27. The equation of a vibrating string, fixed at both ends, is given by

xy (3 mm)sin sin(400 t)15 =

where x is the distance (in cm) measured from one end of the string, t is the time (in seconds), and y gives the displacement. The string vibrates in 4 loops.

The speed of transverse waves along the string equals, for the fundamental mode, (A) 60 m/s (B) 120 m/s (C) 30 m/s (D) 15 m/s. 28. The vibrations produced in a sitar wire are (A) progressive transverse (B) progressive longitudinal (C) stationary transverse (D) stationary longitudinal. 29. Given 2 2 F (xy )i (x y) j= +G newton. Find the work done by FG when a particle is taken along the

semicircular path OAB where the co-ordinates of B are (4, 0).

(A) 65 J3

(B) 75 J2

(C) 73 J4

(D) 0 J

30. A small cube of mass m slides down a circular path of radius R cut into a

larger block of mass M, as shown in the figure. M rests on a table, and both blocks move without friction. The blocks are initially at rest, and m starts from the top of the path. The velocity v of the cube as it leaves the block is

mR

M

(A) 2mgRM

(B) 2gR

(C) 2mgRm M+ (D)

2MgRm M+

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10

CChheemmiissttrryy PART II

SECTION A Single Correct Choice Type

This section contains 30 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 1. One of the reaction coordinate energy diagrams of two nucleophilic substitution reactions are

given below:

Ene

rgy

R - XNuc

R - Nuc X-

Reaction coordinates(A)

R - XNuc

R - Nuc X-

Reaction coordinates(B) Which of the following statement(s) is/are true? (A) (A) corresponds to SN1 and (B) corresponds to SN2. (B) (A) corresponds to SN2 and (B) corresponds to SN1. (C) Both (A) and (B) correspond to SN2. (D) Both (A) and (B) correspond to SN1. 2. A liquid (A) is treated with Na2CO3 solution. A mixture of two salts (B) and (C) are produced in the

solution. The mixture on acidification with sulphuric acid produces the liquid A again (A) is (A) H2O2 (B) Br2 (C) CS2 (D) CCl4 3. The 22NO Br 2NOBr+ follows the mechanism below (i) 2 2NO Br NOBr+ fast (ii) 2NOBr NO 2NOBr+ slow If the concentration of both NO and Br2 are increased two times the rate of reaction would

become (A) 4 times (B) 6 times (C) 16 times (D) 8 times

Rough work



11

4. Which of the following statement is/are correct? (i) The ligand thiosulphato 22 3S O

can give rise to linkage isomers. (ii) In metal carbonyls the ligand CO molecule acts both as donor and acceptor. (iii) The complex K[Cu(CN)2] has coordination number 2. (iv) The complex ferricyanide does not follow effective atomic number (EAN) rule. (A) (i), (ii) and (iii) (B) (i), (ii) and (iv) (C) (i), (iii) and (iv) (D) (ii), (iii) and (iv) 5. In the following reaction sequence:

O

N

H

+ [ ]6 63

3

(i) p TSA, C H Heat(ii) CHCl , KOH(iii) H O

X+

What is the major product [X]?

(A) O O

OH

(B) O

Cl

Cl

(C) O O

H

(D) OCl

6. The 2s-orbital of H-atom has a radial node at 2a0 because 2s is proportional to (A) (1/2 + r/a0) (B) (2 + r/a0) (C) (2 r/2a0) (D) (2 r/a0) 7. The standard emf E0 for the reaction 2 2fumarate lactate succinate pyruvate + + is on the basis of the following information 2 2 0fumarate 2H 2e succinate E 0.031 V + + + = 0pyruvate 2H 2e lactate E 0.185 V + + + = (A) 0.154 V (B) 0.216 V (C) 0.216 V (D) 0.154 V

Rough work



12

8.

O

OLi

( )

( )

2R CuLi A

HydrolysisC

( )

( )

RMgXB

HydrolysisD

Identify the compounds (C) and (D) respectively. (A) O

R

OH

O

OH

R

and

(B) O

OH

R

O

R

OH

and

(C) O

OH

R

O

OH

R

and

(D) OOHR

O

OH

R

and

9. Consider the following statement(s): (i) Hypophosphorous acid shows reducing properties. (ii) Pyrophophoric acid can show basicities from one to four because it has 4-OH groups

attached to the phosphorous. (iii) Orthophosphoric acid is obtained during disproportionation of hypophosphorous acid. (iv) Metaphosphoric acid on heating gives P4O10. Which of the following is/are correct: (A) (i), (ii) and (iii) are correct (B) (i), (ii) and (iv) are correct (C) (i), (iii) and (iv) are correct (D) All are correct 10. What is the correct order of bond angle in the following? (A) CH4 > NF3 > NH3 > H2O (B) CH4 > NH3 > NF3 > H2O (C) CH4 > NH3 > H2O > NF3 (D) NF3 > CH4 > NH3 > H2O

Rough work



13

11. How many chiral centre(s) (excluding N centre) is/are there in morphine?

O

OH

OH

N CH3H

H

(A) 4 (B) 5 (C) 6 (D) more than 6 12. The products formed when a mixture of the following two amides (I and II) are treated with

bromine and aqueous KOH are

CONH2 CONH2

NO2

15

(A) NH2 NH2

NO2

and

(B) NH2 NH2

NO2

and

15

(C)

NH2 NH2

NO2

and

15

(D) A mixture of all of there.

Rough work



14

13. Which of the following cannot be taken as unit cell of any lattice?

P

S

Q

R

(A) P (B) Q (C) R (D) S 14. Which of the following graphs is/are inconsistent with ideal gas behaviours?

(A)

V

P

T2

T1T2 >

T1

(B)

P

PV

T2

T1

(C)

T (K)

V

P1P2 > P1

P2

(D)

N

P

T1T2 >

T1

T2

Rough work



15

15. Which statement is true if the following reaction is spontaneous? 2 2Zn Cu Zn Cu+ ++ +ZZZXYZZZ (A) Kc is larger than 1; oG is negative, and oE is positive. (B) Kc is larger than 1; oG is positive, and oE is negative. (C) Kc is larger than 1; oG is negative, and oE is negative. (D) Kc is smaller than 1; oG is negative, and oE is positive. 16. ( ) ( )2 2SnCl HCl I A B+ + + The compounds (A) and (B) are: (A) SnI2, Cl2 (B) H2SnCl4, HI (C) SnCl4, HI (D) HSnCl3, HI 17. In an O2 molecule how many sets of degenerate molecular orbitals are present? (A) 1 (B) 2 (C) 3 (D) 4 18. In the following graph

0 5 10 15 20

0.1

0.2

Rad

ialp

roba

bilit

y di

strib

utio

n (P

n,l)

Pn3, 3A Pn2, 2A Pn1, 1A

Radius (a0) Then, what is correct set of quantum nos. for the given graphs? (A) n1 = 3 l1 = 1, n2 = 2 l2 = 2, n3 = 2, l3 = 2 (B) n1 = 3 l1 = 0, n2 = 3 l2 = 1, n3 = 3, l3 = 2 (C) n1 = 3 l1 = 2, n2 = 3 l2 = 1, n3 = 3, l3 = 0 (D) n1 = 3 l1 = 0, n2 = 3 l2 = 1, n3 = 2, l3 = 2

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16

19. A colourless gas X forms a brown coloured gas when mixed with air. The gas X is (A) NH3 (B) NO (C) N2O (D) NO2 20. Among the following the optically inactive compound is

(A) NCH2

CH3

CH3

(B) PCH3

(C)

CCH3

C C C

CH3

(D)

H

OHCOOH

21. Consider the ionization 3 2 3 2 2 2H H N CH COO H N CH COOH H NCH COOH H

+ + + ++ +ZZZX ZZZXYZZZ YZZZ Assuming that the following acid dissociation constants apply to the ionizations: 103 2 aNH NH H ; K 1.6 10 M

+ + + = ZZZXYZZZ 3aCOOH COO H ; K 4.0 10 M

+ + = ZZZXYZZZ Estimate a value for the equilibrium constant for the process 3 2 2 2H N CH COO H NCH COOH

+ ZZZXYZZZ is (A) 4.0 108 (B) 0.25 10+7 (C) 6.4 1013 (D) 1 108 22. The gas phase reaction between nitric oxide and oxygen is third order. The following rate

constant have been measured: T/K : 80.0 143.0 228.0 300.0 413.0 564.0 K109/cm6mol7s1: 41.8 20.2 10.1 7.1 4.0 2.8 The behaviour is interpreted in terms of a temperature dependent pre exponential factor; the rate

constant is of the form n E/RTK aT e= Where a and n are constants. Assume the activation energy to be zero. What is the value of n to the nearest half integer. (A) 1.5 (B) 1.5 (C) 0.5 (D) 0.5

Rough work



17

23. T

HP

for an ideal gas is equal to

(A) RTVP

(B) RT VP

(C) Zero (D) ( )V

T

24. In the freezing point depression experiment, it can be noted that: (A) Vapour pressure of the solution is less than that of pure solvent. (B) Only solute molecules solidify at freezing point. (C) Solvent molecules solidify at a temperature above the freezing point of pure solvent. (D) If weight of solute is much less than the weight of solvent and Hfus is heat of fusion per mole

of solvent than the gradient (slope) of the following plot is found to be 0f

fus

RTH , where y = mass

fraction of solute in the solution.

Tf

y

25. Calcium phosphide (Ca3P2) formed by reacting calcium orthophosphate [Ca3(PO4)2] with

magnesium was hydrolysed by water. The evolved PH3 was burnt in air to yield P2O5. How many moles of Mg(PO3)2 would be obtained, if 384 g of Mg were used for reducing calcium phosphide.

3 4 2 3 2Ca (PO ) Mg Ca P MgO+ + ( )3 2 2 32Ca P H O Ca OH PH+ + 3 2 2 5 2PH O P O H O+ + ( )2 5 3 2MgO P O Mg PO+ (A) 2 moles (B) 16 moles (C) 8 moles (D) 4 moles

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18

26. In the following reactions:

CH3

CH3 CH3

Cl( )NaOEt A

CH3

CH3 CH3

Cl( )NaOEt B

What are the structures of (A) and (B) respectively

(A) CH3

CH3 CH3

CH3

CH3 CH3

and

(B) CH3

CH3 CH3

CH3

CH3 CH3

and

(C) CH3

CH3 CH3

Only

(D) CH3

CH3 CH3

Only

27. Ester hydrolysis in acidic condition can follow (A) SN1 and SN2 both (B) Only SN1 (C) Only SN2 (D) SN1, SN2 and tetrahedral mechanism

Rough work



19

28. NO2

NaOEt2NH OH ?

+

(A) NO2

NH2

(B) NO2

NH2(C) NH2

NH2

(D) NH2NH2

29.

Cl

OMe

CN( )

2NaNHX major

What is the structure of (X)? (A) CN

OMe

(B) CN

OMe

(C)

OMe

CN

(D)

OMe

CN

30. Which of the following statements are in correct regarding Co2+ and Cr3+ ions? (i) The spin magnetic moment of both the ions is same (ii) The magnetic moment of both the ions in different (iii) The magnetic moment of both the ions is not different (iv) They both have same numbers of unpaired electrons (A) (i) only (B) (iii) only (C) (i) and (iii) both (D) All

Rough work



20

MMaatthheemmaattiiccss PART III

SECTION A Single Correct Choice Type

This section contains 30 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 1. The roots of 1 + z + z3 + z4 = 0 are represented by the vertices of (A) a square (B) an equilateral triangle (C) a rhombus (D) a rectangle 2. The equation of plane containing the lines r a b= + GG G and r b a= + GG G is (A) [r ab] 0=GG G (B) [r ab] a.b=G GG G G (C) [r ba] a.b=G GG G G (D) [r ab] 2a.b=G GG G G

3. If sin4 + cos4+ 2 = 4 sin cos , 0 , ,2 then sin + cos is equal to

(A) 2 (B) 0 (C) 2 (D) 1 4. Number of points with integral coordinates lying on the circle x2 + y2 = 2008 is (A) 0 (B) 4 (C) 8 (D) 16 5. If the equation ax2 bx + 12 = 0 where a and b are +ve integers not exceeding 10, has both roots

greater than 2 then the number of ordered pair (a, b) is (A) 0 (B) 1 (C) 3 (D) 5

Rough work



21

6. The number of different ordered triplets (a, b, c), a, b, cI such that these can represent sides of a triangle whose perimeter is 21, is

(A) 12 (B) 31 (C) 55 (D) 91 7. If the cube of a natural number ends with a prime digit then the probability of its fourth power

ending not with a prime digit, is (A) 3/10 (B) 9/10 (C) 4/9 (D) 3/4 8. The number of different permutations of all the letters of the word 'PERMUTATION' such that any

two consecutive letters in the arrangement are neither both vowels nor both identical is (A) 63 6! 5! (B) 57 5! 5! (C) 33 6! 5! (D) 7 7! 5! 9. When 3233 is divided by 34, it leaves the remainder (A) 2 (B) 4 (C) 8 (D) 32

10. The value of 1 1111 11

2 3 431 2 CC C2 2 2 ...2 3 3 4 4 5

+ + upto 12 terms is (A) 1/10 (B) 1/11 (C) 1/12 (D) 1/13

11. The maximum value of f(x) = 2| x | 2 x

| x | 1

+ , (xR), is (A) 4 (B) 3 (C) 2 (D) 1 12. Maximum number of common normals of y2 = 4ax and x2 = 4by can be equal to (A) 3 (B) 4 (C) 6 (D) 5

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22

13. The equation 2x3 15x2 + 36x + k = 0 has three real and distinct roots. The total number of integral values of k is

(A) 0 (B) 1 (C) 2 (D) none of these 14. A cubic f(x) vanishes at x = 2 and has relative minimum/maximum at x = 1 and 1/3 such that

( )11

14f x dx3

= . Then f(x) is equal to (A) 3 2x x x+ (B) 3 2x x x 1+ + (C) 3 2x x x 2+ + (D) 3 2x x x 2+

15. The function ( ) ( )( ) ( ) ( ){ }x 3 2 21f x 2 t 1 t 2 3 t 1 t 2 dt= + attains its maximum at x is equal to (A) 1 (B) 2 (C) 3 (D) 4

16. The value of 2

x 0

xlimsinx tanx

, where [] denotes greatest integer function, is equal to (A) 0 (B) 1 (C) limit does not exist (D) none of these 17. Let A be a square Matrix all of whose entries are integers. Then the true statement is (A) If det A = 1, then A1 exists but all its entries are not necessarily integers (B) If det A 1, then 1A exists and all its entries are non integers (C) If det A 1= , then 1A exists and all its entries are integers (D) If det A 1= , 1A need not exist

18. If A is a square Matrix such that ( )4 0 0

A. AdjA 0 4 00 0 4

= then

(A) A 5= (B) adjA 18=

(C) ( )adj adjA

16adjA

= (D) adj2A 128=

Rough work



23

19. 1 1sin cos x cos sinx a = has at least one solution iff a (A) { }0 (B) 0,

2

(C) 3,2 2 (D) ( )0,

20. Area bounded by the minimum of { }x 1, y 1 1 = and the maximum of { }x 1, y 1 4 = in

square units is (A) 16 (B) 36 (C) 25 (D) 45

21. The range of function f(x) defined by ( ) ( )2

2e

x 1f xlog x 1

+= + , { }x R 0 is (A) ( )0,1 (B) ( )0,2 (C) [ )e, (D) ( ), 22. The area bounded by the parabola x2 + 8x +12y + 4 = 0 and its latus rectum is 6k. Then the value

of k is (A) 1 (B) 2 (C) 3 (D) 4 23. Let ( ) 3 2f x x ax bx c= + + + and x1, x2 be the roots of ( )f x 0 = , if 1 2x x< then ( )f x 0= will have (A) No real root if ( ) ( )1 2f x 0 or f x 0< > (B) Only one real root if ( ) ( )1 2f x 0 or f x 0< > (C) Three real roots if ( ) ( )1 2f x 0 or f x 0< > (D) cannot say any thing 24. If a, b, c are unit vectors then 2 2 2| a b | | b c | | c a |+ + + + + can not be less than (A) 3 (B) 6 (C) 9 (D) 12

Rough work



24

25. If f(x) is a twice differentiable function such that f(0) f(2) 3;f (0) f (2) 1 = = = = then 2

0

x f (x)dx is

(A) 3 (B) 2 (C) 1 (D) 0

26. The value of x 0

100x 99sinxlimsinx x

+ where [ ]. denotes the greatest integer function is (A) 197 (B) 196 (C) 198 (D) does not exist 27. General solution of (xy sin xy + cos xy)y dx + (xy sin xy cos xy) x dy = 0 is (A) y.sin (xy) = cx (B) x.sec (x y) = cy (C) y tan (xy) = cx (D) y cot (xy) = cx 28. The general solution of the differential equation (3x + 2y2) y dx + 2x (2x + 3y2) dy = 0, is (A) x6y2 + x4y3 = c (B) x6y2 + x3y4 = c (C) x2y6 + x4y3 = c (D) x2y6 + x3y4 = c

29. Let x 1, x 0

f(x)x 1, x 0+

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