fiitjee main sample paper

7/27/2019 fiitjee main sample paper

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FIITJEE Ltd., FIITJEE House, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -110016, Ph 46106000, 26569493, Fax 26513942website: www.fiitjee.com

CONCEPT RECAPITULATION TEST(Set I)

Time Allotted: 3 Hours Maximum Marks: 432

Please read the ins t ruct ions careful ly . You are al lot ted 5 minutesspec i f ical ly for th is purpose.

You are not al 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 beprovided for rough work.

6. Blank Papers, clip boards, log tables, slide rule, calculator, cellular phones, pagers and electronicdevices, 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 answerson OMR sheet.

2. On the OMR sheet, darken the appropriate bubble with black pen for each character of yourEnrolment 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 onecorrect answer. Each question carries +4 marks for correct answer and 1 mark for wronganswer.

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.

ALLINDIA

TESTSE

RIES

FIITJEE JEE (Main), 2013

FromLongTermC

lassroomProgramsandMedium/

ShortClassroomProgram4

inTop10,10inTop

20,43inTop100,

75in

Top200,159inTop500Ranks&3542

totalselec

tions

in

IIT-JEE

2012


2/22

AITS-CRT(Set-I)-PCM-JEE (Mains)/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 C

2/N-m

2

Density of water water= 103

kg/m3

Atmospheric pressure Pa = 10

5

N/m

2

Gas constant R = 8.314 J K1

mol1

CHEMISTRY

Gas Constant R = 8.314 J K1

mol1

= 0.0821 Lit atm K1mol

1

= 1.987 2 Cal K1 mol1Avogadro's Number Na = 6.023 10

23

Plancks constant h = 6.625 1034 Js= 6.625 1027 ergs

1 Faraday = 96500 coulomb

1 calorie = 4.2 joule1 amu = 1.66 1027 kg1 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.


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3

PPhhyyssiiccss PART I

SECTION ASingle 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. A small block of mass m slides along a frictionless loop inside looptrack as shown in figure. The minimum value of the ratio R/r so thatthe block may not lose contact at the highest point the inner loop is(A) 7/2 (B) 2(C) 5/2 (D) 3

R

r

B

A

2. Two blocks A and B each of mass m are connected by a

massless spring of natural length A and spring constant k. Theblocks are initially resting on a smooth horizontal floor with thespring at its natural length as shown in figure. A third identical

block C also of mass m moves on the floor with a speed v alongthe line joining A and B collides with A. Then (assuming thecollision is elastic),

mC A B

m m

(A) the KE of the A and B block as a system is zero, when the spring is maximum compressed.

(B) the KE the A and B block as a system is 21

mv4

and that of C is zero, when the spring is

under maximum compression(C) the total KE of A+B+C as a system is (1/2) mv

2when spring is at maximum compression

(D) spring1

(PE ) KE2

= of (A+B) as a system when the spring is at maximum compression

3. One end of a spring of force constant k is fixed to a vertical wall and the other to a body of mass

m resting on a smooth horizontal surface. There is another wall at a distance x0 from the body.The spring is then compressed by 2x0 and released. The time taken to strike the wall is

(A)m

6 k

(B)

m

k

(C)2 m

3 k

(D)

m

4 k

Rough work


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4

4. A force ( ) F yi xj N= +G

acts on a particle as it undergoes counterclockwise circular motion in

x-y plane in a circle of radius 4 m and with the centre at origin. The work done by the force whenthe particle undergoes one complete revolution is (assume x, yare in m)

(A) zero (B) 32 J (C) 8 J (D) 16 J

5. A uniform rod AB of mass m and length A is at rest on a smooth horizontal surface. An impulse Jis applied to the end B perpendicular to the rod in horizontal direction. Speed of the point A of therod after giving impulse is

(A)J

2m

(B)J

2m

(C)J

m(D)

J2

m

6. In figure there are two sliders and they can slide on twofrictionless parallel wires in uniform magnetic field B, which ispresent everywhere. The mass of each slider is m, resistance Rand initially these are at rest. Now if one slider is given a velocity

v0, the velocity of other slider after considerably long time will be(neglect the self induction)

B

v0

(A) 0v

4(B) 0

v

2

(C) v0 (D) Zero.

7. Charge on the capacitor having capacitance C2 in steady state is(A) Zero (B) (C1 + C2)V(C) C2V (D) C1V

R

C1

V

C2

R

Rough work


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5

8. A conductor of variable cross-section is connected across a battery. Let usconsider two cross-sections A and a. Let V and v be drift speeds ofelectrons at those cross-sections respectively then(A) AV > av (B) AV = av(C) Av= aV (D) AV < av

a

v

A

V

9. The displacement of a particle varies according to the relation:4sin 3 3cos3x t t= + where x is in metre and t is in second.The maximum acceleration of the particle is given by(A) 12 m/s

2(B) 36 m/s

2

(C) 15 m/s2

(D) 45 m/s2

10. A charged plate of charge Q is kept in between two plates. Both theplates have an initial charge of Q and 3Q respectively. The finalcharges on q3 & q5 surfaces of the plate are

(A) ,2 2

Q 3Q (B) ,

2 2

Q 5Q

(C) ,2 2

3Q 5Q

(D) ,2 23Q 3Q

q1

q2

q3

q4

q5

q6

3Q

Q

Q

11. A particle of mass 0.002 kg and a charge 1 C is held at rest on a frictionless horizontal surface

at a distance of 1 m from a fixed charge of 1 mC. If the particle is released, it will be repelled. Thespeed of the particle when it is at a distance of 10 m from the fixed charge is(A) 60 ms

1 (B) 75 ms

1

(C) 90 ms1

(D) 100 ms1

12. The switch is closed at time t = 0. The potential difference acrossinductor and resistor (at t = 0), respectively, is(A) 5V, 5V (B) 8V, 6V(C) 10V, 0V (D) 0V, 10V

10V

5H 5

Rough work


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6

13. Acceleration vs time graph is shown in the figure for aparticle moving along a straight line. The particle is initiallyat rest. Find the time instant(s) when the particle comes torest?(A) t = 0, 1, 2, 3, 4(B) t = 0, 2, 4

(C) t = 1, 3(D) None of these

+2

O

2

1 2 3 4t(sec)

a(m/s )2

14. A block of mass 10 kg connected to another hollow block of same

size and negligible mass, by a spring of spring constant 500 N/m,floats in water as shown in the figure. The compression in the

spring is (water= 1 103

kg/m3, g = 10 m/s

2)

(A) 10 cm (B) 20 cm(C) 50 cm (D) 100 cm

m=10 kg

15. The minimum horizontal acceleration of the container so thatthe pressure at the point A of the container becomesatmospheric is

(A) g23

(B) g

34

(C) g (D) g4

3

a2m

3m

A

16. A satellite in an equatorial orbit has a time period of 6 hrs. At a certain instant, it is directly

overhead an observer on the equator of the earth. It is directly overhead the observer again aftera time T. The possible value(s) of T is/are(A) 8 hr (B) 4.8 hr(C) both (A) and (B) (D) none of these

17. The internal energy of a gas is given by U = 2PV. It expands from V0 to 2V0 against a constantpressure P0. The heat absorbed by the gas in the process is(A) 2P0V0 (B) 4P0V0(C) 3P0V0 (D) P0V0

Rough work


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7

18. Two identical conducting rods AB and CD are connected to acircular conducting ring at two diametrically opposite points Band C. The radius of the ring is equal to the length of rods ABand CD. The area of cross-section, and thermal conductivityof the rod and ring are equal. Points A and D are maintainedat temperatures of 100C and 0C. Temperature of point C

will be

A B C D100C 0C

(A) 62C (B) 37C(C) 28C (D) 45C

19. Two balls of same material and finish have their diameters in the ratio 2 : 1. Both are heated tothe same temperature and allowed to cool by radiation. Rate of cooling of big ball as compared tosmaller one will be in the ratio(A) 1 : 1 (B) 1 : 2(C) 2 : 1 (D) 4 : 1

20. Three bars each of area of cross section A and length L areconnected in series as shown in the figure. Thermalconductivities of their materials are K, 2K and 1.5K. If thetemperatures of free end of first and the last bar are 200C and

18C. The value of1 and 2 are (in steady state)

K 2K 1.5K

200C 18C1 2

(A) 120C, 80C (B) 116C, 80C(C) 116C, 74C (D) 120C, 74C

21. The acceleration time graph of a particle is shown in thefigure. What is the velocity of particle at t = 8s, if initial velocityof particle is 3 m/s? (Assume motion is 1 dimension)(A) 4 m/s(B) 5 m/s(C) 6 m/s(D) 7 m/s

t84

1

O

4

a

Rough work


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8

22. 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 inclinedplane perpendicularly.

(A)2gH

5(B)

2gH

7

(C)gH

5 (D)gH

7

90

H

3090

V0

23. A uniform bar of square cross-section is lying along africtionless horizontal surface. A horizontal force isapplied to pull it from one of its ends then(A) The bar is under same stress throughout its length

F

(B) The bar is not under any stress because force has been applied only at one end(C) The bar simply moves without any stress in it(D) The stress developed reduces to zero at the end of the bar where no force is applied

24. A block Q of mass 2m is placed on a horizontal frictionless plane.A second block of mass m is placed on it and is connected to a

spring of spring constant K, the two block are pulled by distance A.Block Q oscillates without slipping. The work done by the frictionforce on block Q when the spring regains its natural length is

Q

P s

K

(A)1

3KA

2(B)

2

3KA

2

(C)1

2KA

2(D)

1

4KA

2

25. A long straight wire, carrying current I, is bent at its mid

point to form an angle of. Magnetic field at point p, (asshown in figure)

(A) 0I

(1 sin )

4 d

(B) 0 0 I (1 sin )

4 d

+

(C) 0I

(1 cos )4 d

(D) 0I

(1 cos )4 d

+

I

p

dI

Rough work


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9

26. An electron moving with a velocity 1V 1i=

Gm/s at a point in a magnetic field experiences a force

1F 2 jN=

G. If the electron is moving with a velocity 2

V 1j=G

m/s at the same point, it experiences a

force 2F 2i=

GN. The force that an electron would experience if it were moving with a velocity

3V 2k=

Gm/s at the same point is

(A) zero (B)

2kN (C) 2k N (D) information is insufficient

27. Xray from a tube with a target A of atomic number z shows strong K lines for target A and weak

K lines for impurities. The wavelength of K lines is z for target A and 1 and 2 for two impurities

z z

1 2

14 &

4

= =

Screening constant of K lines to be unity. Select the correct statement.(A) the atomic number of first impurity is (2z 1)(B) the atomic number of first impurity is (2z + 1)(C) the atomic number of second impurity is (z + 1)

(D) the atomic number of second impurity isz

1

2

+

.

28. All electrons ejected from the surface by incident light of wavelength 200 nm can be stoppedbefore travelling 1 m in direction of uniform electric field of 4 N/C. The work function of the surfaceis (hc = 1240 (nmeV)(A) 4eV (B) 6.2eV(C) 2.2eV (D) 2eV.

29. The plane surface of a plano-convex lens of focal length 20 cm is silvered. It will behave as(A) Plane mirror(B) Convex mirror of focal length 40(C) Concave mirror of focal length 10 cm(D) None of the above

30. Figure shows three lenses of equal radii of curvature of the curvedsurfaces. The ratio of focal lengths of P, Q and R is

(A) 1 : 1 : 1 (B) 1 : 1 : 1(C) 1 : 2 : 1 (D) 1 : 2 : 1

P Q R

Rough work


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10

CChheemmiissttrryy PART II

SECTION AStraight Objective Type

This section contains 30 multiple choice questions numbered 1 to 30. Each question has 4 choices (A),(B), (C) and (D), out of which ONLY ONE is correct.

1. In Bohrs model of hydrogen atom the ratio between period of revolution of an electron in the orbitn = 1 to the period of revolution of electron in orbit n = 2 is(A) 1 : 12 (B) 2 : 1(C) 1 : 8 (D) 1 : 4

2. Magnetic moment of V (Z = 23), Cr (Z = 24) and Mn (Z = 25) are x, y, z hence(A) z < y < x (B) x = y = z(C) x < z < y (D) x < y < z

3. Which one is the correct expression below for the solution containing n number of weak acids?

(A)n i

ii 1

kHc

+

=

= (B)n

i ii 1

H k c+

=

=

(C)n

i ii 1

H k c+

=

= (D) None of these

4.

O O

CH3NO2

Cl

NaOHX

Product X is(A) NO2 O

O

(B) NO2

OCH3

O

(C)

O O

CH3NO2

OH

(D)

C

O

CH2 C CH3

O

O2N

Rough Work


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11

5.CH3

H OH

D

MeS

O

O

Cl

( )3CH COONa

2 ( )

( )2OH /H O Esterhydrolysis

3

( ) ( )Acidification

34CH COOH P+ +

( )0X+

The product P has optical rotation(A) (+ X

0) (B) ( X

0)

(C) Zero (D) some other value, but NOT (X0)

6. 5.0 gram of mixture containing NaHCO3, NaCl and Na2CO3 is dissolved in 500 ml water and its10 ml portion required 12.4 mL 0.1 M HCl solution to reach the equivalence point. In an anotherexperiment, 10 mL portion of the same stock solution is mixed with 10 mL 0.15 M NaOH solution.Excess NaOH required 12.6 mL 0.1 M HCl solution for back titration. Determine the NaHCO3 % inthe original mixture.(A) 3% (B) 20.16%(C) 26.04% (D) 40 %

7. How long a current 1 amp be passed through 150 ml of 0.2 M NaCl solution in order to change its

pH by 4 units assuming no volume change(A) 193 sec (B) 29 sec(C) 96.5 sec (D) 14.5 sec

8. The radioactivity of a sample is R1 at time t1 and R2 at time t2. If the half life of specimen is t the

number of atoms that have disintegrated in time (t2 t1) is proportional to

(A) (R1t1 R2t2) (B) (R1 R2)

(C) 1 2R R

t

(D) t (R2 R1)

Rough Work


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12

9. On the basis of the following information determine x and y.

( ) ( ) ( )3AgNO Excess4

2 5 y 2 6x0.10 g 1.435 g0.612 g

C H TiCl C H Ti Cl AlCl s+ + +

(A) 1, 3 (B) 3, 1(C) 2, 6 (D) 4, 8

10. 25 ml of p-hydroxy benzoic acid titrated with 0.02 M NaOH solution.Volume of NaOH added pH8.12 mL 4.5716.24 ml (equivalence point) 7.02

OH

COOH

2H O+ ZZZXYZZZ

OH

COO

3 a2H O : K++

OH

COO

2H O+ ZZZXYZZZ

O

COO

3 a2H O : K++

from the above data, derive value of PKa1 and PKa2

(A) PKa1 = 4.5%, PKa2 = 9.4% (B) PKa1 = 5.9%, PKa2 = 8.34%(C) PKa1 = 6.23%, PKa2 = 8.43% (D) PKa1 = 9.4%, PKa2 = 4.5%

11. Consider the following energy diagram for a chemical reaction

E1

E2PE

Reaction co-ordianate

Which of the following statement is (are) true for the reaction(A) The reaction is endothermic(B) Enthalpy of reaction = E1 E2(C) the reaction is spontaneous under all conditions(D) If the peak value is lowered, the enthalpy of reaction will also be lowered.

Rough Work


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13

12. The equation for the reaction in the figure below is

( ) ( ) ( )5 3 2AB g heat AB g B g+ +ZZZXYZZZ

0.8

0.6

0

0.2

0.4

1 2 3

AB2

B2

AB5

Time (min)

Conc.

(Mm

ol/t)

At time 3 min, what change was imposed into the equilibrium(A) Pressure was increased (B) Temperature was increased(C) B2 was added to the system (D) AB3 was added to the system

13. Which one of the following about SF4, SOF4, CH2SF4 and OCF2 molecules is correct?(A) Equitorial FSF bond angle in SOF4 will be less than in SF4 molecule(B) The two hydrogens, carbon, sulphur and two flourines of equatorial positions in molecule

CH2SF4 will be lying in the same plane(C) The bond angle FCO will be


14/22



14

15.Cl

Cl

( )2Mg Et O1 eq

Product is

(A) Anti aromatic (B) Non aromatic(C) Aromatic (D) None of these

16.

C Cl

O

CHO

2

2

(i) CdMe

(ii) H O?

Product is(A)

OH

OH

(B)

OH

O

(C)

O

O

(D)

OH

O

17. Which of the following will produce chiral centre after reaction is completed?

(A) O

COOH

(B) CHOOHC

CHO CHO

Conc. NaOH

(C) O

Dil KOH

(D)3 3CHCl CH COCl

KOHPhOH

Rough Work


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15

18. The linkage between the two monosaccaride unit in lactose is

(A) C1 ofD glucose and C4 is D galactose(B) C1 ofD galactose and C4 ofD glucose(C) C1 ofD galactose and C4 ofD glucose(D) C1 ofD glucose and C4 ofD galactose

19. The product P isOH

CH3I

H

D

I

*( ) ( )

18

aq K OH/ 2 eq / H acidificationP

+

(A) OH

CH3OH

H

D

I*

18

(B) OH

CH3I

H

D

OH

*18

(C)OH

CH3I

H

D

I

*

18 (D) OH

CH3I

H

OH

D

*

18

20. I

( )2 2Me CuLi Br / h / Mg Zn(Hg) / HCl /ether

P

O

Me

The major product P is(A)

CH2

Me

HO

(B)

CH2

MeOH

(C)

CH2 Me

(D)

CH2

Me

Rough Work


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16

21. Which of the following is correct?(A) In C2H4 the bond angle < HCl < bond angle < HCH(B) N2F2 can exhibit geometrical isomerism(C) In solid state nitryle chloridericle (O2NCl) will be consisting of NO

+and OCl

ions

(D) the correct order of CO bond length is CO < CO2 < CO2

22. Which of the following is false?(A) NaF is more than NaBF4 in water

(B) EAB < [ ]A A B B1

E E2

+ (E is representing bond energy

(C) The two bonds in CO2 molecules are coplanar(D) The largest ion among Se

2O

2, F

and Rb

+is Se

2

23. The IUPAC nomenclature oxidation state and hybridisation of the given complex

K2 [Cr(NO)(NH3)(CN)4] with = 1.73 BM respectively are(A) potassiumanninetetracyano (C) nitrosyl chromium (I), +1, d

2sp

3

(B) Potassium ammine tetracyano (c) nitrosylchromium (I), +2, sp3d

2

(C) Potassiumamminetracyano (c) nitrosonium chromium (I), +2, sp3d

2

(D) Potassium amminetetra cyano (c) nitrosonium chromium (I), +1, d2sp

3

24.( )

4 4Y

X NH OH Gelatinous ppt. NH Cl+ +

Y NaOH Soluble.+ Identify x

(A) FeCl3 (B) CrCl3(C) AlCl3 (D) SbCl3

25. CHO

NO2

( ) ( ) ( )2 43

Ac O,AcONa FeSO HCl/

NHreduction

B C D

( )A

What is (D)(A)

O CH2

(B)CH

CH

NO2

C

OHO

(C)

O O

NO2

(D)

CHC

C

OHO

NO2

Rough Work


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17

26. The correct order of decreasing order of ionic product of lead dihalide is(A) PbF2 > PbCl2 > PbBr2 > PbI2 (B) PbF2 > PbBr2 > PbCl2 > PbF2(C) PbF2 > PbI2 > PbCl2 > PbBr2 (D) PbCl2 > PbBr2 > PbF2 > PbI2

27. 2Fe 2e Fe+ +

3 2

Fe e Fe

+ + +

+ The standard potential (in volt) corresponding to the reaction (i) and (ii) are E1 and E2respectively. The value (in volt) of standard potential corresponding to the reaction

3Fe 3e Fe is+ +

(A) E1 + E2 (B)1 22E E

3

+

(C) 1 2E 2E

2

+(D) 1 2

E E

3

+

28. Dehydration of the following in increasing order is(i)

OH

(ii)OH

(iii)

OH

(iv)

OH

(A) I < II < III < IV (B) II < III < IV < I(C) I < III < IV < II (D) None of these

29. A deliquescent white crystalline hydroxide X reacts with a nitrate Y to form another hydroxidewhich decomposes to give an insoluble brown layer of its oxide. X is a powerful cantery andbreaks down the proteins of skin flesh to a pasty mass. X and Y are(A) NaOH, AgNO3 (B) NaOH, Zn (NO3)2(C) NaOH, Al(NO3)2 (D) Ca(OH)2, Hg(NO3)

30.

( )( ) ( ) ( ) ( )

i NBS

ii Mg/ether ii HA B+

OCH3

(A)

OH

(B)

OH

(C) OH

(D)

OH

Rough Work


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18

MMaatthheemmaattiiccss PART III

SECTION A

Straight Objective Type

This section contains 30 multiple choice questions numbered 1 to 30. Each question has 4 choices (A),(B), (C) and (D), out of which ONLY ONE is correct.

1.cot x

2

e

sin x [2 ln cosec x + sin 2x] dx is equal to(A) 2ecotx ln (cosec x) + c (B) ecotx ln x + c(C) e

cotx(ln (cosec x) + c (D) none of these

2. A closed right circular cylinder has volume 2156 cubic units. The radius of its base so that its totalsurface area may be minimum is

(A) 7 (B)22

7

(C) 5 (D) 11

3. If f is continuous function satisfying f(f(x)) = 1 + x x R. then f(1) is equal to(A) 1 (B 0(C) 1 (D) none of these

4. The value of { }100

0

x dx (where {x} is the fractional part of x is(A) 50 (B) 1(C) 100 (D) none of these

Rough work


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19

5. Let N be any four digit number say x1 x2 x3 x4. Then maximum value of1 2 3 4

N

x x x x+ + +is equal

to

(A) 1000 (B)1111

4

(C) 800 (D) None of these

6. ABCD is a quadrilateral with side lengths AB = 4, BC = 10, CD = 6 and AD = 6, and diagonalBD = 8 units. If the incircles of triangles ABD and BCD touch BD at P and Q respectively, thenarea of quadrilateral C1PC2Q (where C1 and C2 are incentres of triangle ABD and BCDrespectively), is

(A)15

32

+ sq. units (B) 3 sq. units

(C)15

6sq. units (D) none of these

7. If ( )f x dx 1

=

then

1f x dx

x

is equal to

(A) 0 (B) 1

(C) 1 (D) 2

8. In a ABC, if base BC is unity. If sinA

2= x1, sin

B

2= x2, cos

A

2= x3, cos

B

2= x4 such that

20062007

31

2 4

xx0

x x

=

then side AC is equal to

(A) 1 (B) 2(C) 3 (D) 5

9. Let f(r) be the number of integral points inside a circle of radius r and centre at origin (integralpoint is a point both of whose coordinates are integers), then

( )

2r

f rlim

ris equal to

(A) 1 (B) (C) 2 (D) /2

Rough work


20/22



20

10. f(x) +1

f 1x

= 1 + x for x R {0, 1}. The value of f(2) is equal to

(A) 1 (B)1

4

(C)3

4

(D)1

2

11. If (h, k) is a point on the axis of the parabola 2(x 1)2 + 2(y 1)2 = (x + y + 2)2 from where threedistinct normals may be drawn then(A) h > 2 (B) h < 4(C) h > 8 (D) h < 8

12. Let P be a point on x2

+ y2

= 5, Q be a point on 7x + y + 3 = 0. Line x y + 1 = 0 is theperpendicular bisector of PQ, if P is(A) (2, 1) (B) (1, 2)

(C) (1, 2) (D) none of these

13. If |z 1| + |z+ 3| 8, then the range of values of |z 4| is

(A) (0, 8) (B) [0, 8](C) [1, 9] (D) [5, 9]

14. The value(s) of 'a' for which exactly one root of the equation eax

2 e

2ax + e

a 1 = 0 lies between

1 and 2 are given by

(A) ln(5 17) 5 17

a ln4 4

+< < (B) 0 < a < 100

(C) ln5

4< a < ln

10

3(D) none of these

15. If a, b, cGG G

, are three nonzero vectors, no two of which are collinear, a 2b+G G

is collinear with cG

and b 3c+G G

is collinear with aG

, then | a 2b 6c+ +GG G

| will be equal to

(A) Zero (B) 1(C) 9 (D) none of these

16. The number of subsets of {1, 2, 3, , n} having least element m and greatest element k,

1 m < k n, is(A) 2

n (k m)(B) 2

k m 2

(C) 2k m 1 (D) 2

k m + 1

Rough work


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21

17. If A and B are two independent events, such that P(A B) = 8/25 and P(B) = 13/75, then P(A) is(A) 11/25 (B) 7/25(C) 12/31 (D) 9/11

18. If f : R R is a function satisfying the property f(2x + 3) + f(2x + 7) = 2, x R, then the periodof f(x) is

(A) 2 (B) 4(C) 8 (D) 12

19. The value of3 5

2 4

cos x cos x

sin x sin x

+

+ dx is(A) sin x 6 tan

1(sin x) + c (B) sin x 2 (sin x)

1+ c

(C) sin x 2(sin x)1

6 tan1

(sin x) + c (D) none of these

20. If A =

sin

2

1

t dt

1 t

+ and B =cosec

2

1

dt

t(1 t )+ , then the value of2

A B 2

2 2

A A B

e e B 1

1 A B 1

+

is

(A) sin (B) cosec (C) 0 (D) 1

21. A tangent to the ellipse 4x2

+ 9y2

= 36 is cut by the tangent at the extremities of the major axis at

T and T. The circle on TT as diameter passes through the point(A) (5, 0) (B) (5, 1)(C) (0, 0) (D) (3, 2)

22. The point of intersection of two tangents to the hyperbola x2/a

2 y

2/b

2= 1, the product of whose

slopes is c2, lies on the curve

(A) y2

b2

= c2

(x2

+ a2) (B) y

2+ a

2= c

2(x

2 b

2)

(C) y2

+ b2

= c2

(x2

a2) (D) y

2 a

2= c

2(9x

2+ b

2)

23. If sin1

x + cos1

x + tan1

x , then(A) =

4

, =

3

4

(B) = , = 2

(C) = 0, = (D) none of these

Rough work


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24. If I be the centre of the circle inscribed in the ABC whose sides BC, CA and AB are of lengths , and respectively, then IA

JJG+ IB

JJG+ IC

JJGis

(A) unit vector (B) null vector

(C) vector of magnitude + + (D) none of these

25. If f (x + y) = f (xy) x, y R and f (2000) = 1999 then f (2001) is equal to

(A) 2000 (B) 2001(C) 2002 (D) none of these

26. If the parabola y = ax2

+ bx + c has vertex at (4, 2) and a [1, 3], then difference between theextreme values of abc is equal to,(A) 3600 (B) 144(C) 3456 (D) None of these

27. If I(n) =

/ 2

n

0

.sin d

, n N, n > 3, then the value of I(n) ( )n 1

I n 2n

is

(A)2

1

n(B)

1

n 1

(C)2

1

n 1(D)

1

n 1+

28. If z is a complex number satisfying |z|2 |z| 2 < 0, then the value of |z2 + zsin |, for all values

of, is(A) equal to 4 (B) equal to 6(C) more than 6 (D) less than 6

29. If x, y, r and s are positive real numbers, such that x2

+ y2

= r2

+ s2

= 1, then maximum value ofxr + ys is(A) 2 (B) 0

(C) 1 (D)1

2

30. If a line segment AM = a moves in the xy plane remaining parallel to OX, so that left end point Aslides along the circle x

2+ y

2= a

2, then the locus of M is

(A) x2

+ y2

= 4a2

(B) x2

+ y2

= 2ax

(C) x2

+ y2

= 2ay (D) x2

+ y2 2ax 2ay = 0

Rough work

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