Concept Recap Test Mains 2

7/28/2019 Concept Recap Test Mains 2

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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 II)

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 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 03 and 10 to 12) contains 6 multiple choice questions which have only onecorrect answer. Each question carries +8 marks for correct answer and 2 mark for wronganswer.

Section-A (04 to 09 and 13 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.

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


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AITS-CRT(Set-II)-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 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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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. An electric bulb is designed to work at 100 volt D.C. If it is connected to an A.C. source to givethe same brightness; then the average A.C. voltage is :

(A) 100 Volt (B)100 2

Volt

(C)200 2

Volt (D)200

Volt

2. In the figure a wedge of mass M is placed at rest on a smooth

horizontal surface. A small sphere of mass m hits the inclined face ofthe wedge with horizontal speed v0. The impact is perfectly inelastic.There is no friction anywhere and bodies are rigid. M

mv0

Following four statements are given regarding this problem.(i) After impact both bodies will be moving with same velocity along horizontal.

(ii) After impact the sphere will be moving only parallel to the inclined plane with speed v0cosas observed from the ground.

(iii) Just after collision component of the velocity of the sphere along the plane is v0cos.

(iv) After collision sphere will come to stop and wedge will move with the speed of 0mv

M.

(A) only (i) is true (B) (i) and (ii) both are correct(C) only (iii) is correct (D) all four are wrong

3. The capacitor shown in the figure is in steady state. Theenergy stored in the capacitor is(A) CI

2R

2

(B) 2CI2R

2

(C) 4CI2R

2

(D) None of the above

I

R

CR

R

Rough work


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4

4. Potential within a hypothetical charged sphere varies with the distance of a point from centre as

.V a r= G G

where aG

is a vector of constant magnitude parallel to rG

and rG

is position vector of

point under consideration taking centre of sphere as origin. Then the total charge stored within asphere of radius R is.

(A) R20a (B) 2R20a

(C) 3R20a (D) 4R20a

5. A small square loop of wire of side l is placed inside a large square loop of wire of sideL(L >>l). The loops are co-planar and their centers coincide. The mutual inductance of the systemis proportional to(A) l/L (B) l

2/L

(C) L/l (D) L2/l

6. A metallic rod hinged at one end is rotated in a horizontal plane. If the angular velocity of rotationis doubled(A) there will be no elongation in rod in any case(B) elongation in rod will be same(C) elongation in rod becomes two times of initial value(D) elongation in rod becomes four times of initial value

7. A satellite S is moving in an elliptical orbit around the earth. The mass of the satellite is very smallcompared to the mass to the earth(A) Angular momentum ofSabout the centre of the earth remains constant.(B) The linear momentum ofS remains constant in magnitude.(C)The total mechanical energy of S varies periodically with time.(D) both (A) & (C).

8. A gas is found to obey the law P2V = constant. The initial temperature and volume are T0 and V0.

If the gas expands to a volume 3V0, then the final temperature becomes

(A) 03 T (B) 02 T

(C) 0T

3

(D) 0T

2

Rough work


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5

9. Find the value of u so that the ball reaches at point B.

(Take g = 10 m/s2)

(A) 20 m/s (B) 40 m/s

(C) 15 2 m / s (D) 50 m/s

10m

45

B

A

u

20m

15m

10. A person wants to drive on the vertical surface of a large cylindrical wooden well commonly

known as deathwell in a circus. The radius of the well is 2 meter, and the coefficient of frictionbetween the tyres of the motorcycle and the wall of the well is 0.2 The minimum speed themotorcyclist must have in order to prevent slipping should be (take g = 10 m/s

2)

(A) 10 m/s (B) 15 m/s(C) 20 m/s (D) 25 m/s

11. A metal rod PQ (carrying current from P to Q) is placed perpendicular to the

infinite long wire carrying current i0. If this arrangement lies in a horizontalplane; then the rod PQ will rotate:(A) clockwise (B) anticlockwise(C) along PQ (D) does not rotate

i0

P Q

12. A large number of infinite long thin wires each carrying current i0 are

kept along z-axis. Then the value of magnetic field B, at any point Pat a distance x from O varies as

(A) B x (B)1

Bx

(C) B is independent of x (D) 2

1

B x

y-axis

x-axis

O

P

x

Rough work


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6

13. 50% of the X-ray coming from a Coolidge tube is able to pass through a 0.1 mm thick aluminiumfoil. If the potential difference between the target and the filament is increased, the fraction of theX-ray passing through the same foil will be(A) 0% (B) < 50%(C) 50% (D) > 50%

14. For a right angled prism PQR immersed in water, the incident andemergent rays are parallel as shown in figure. The minimum valueof refractive index of the prism is

(A)3

22

(B)4

23

(C)3

4(D) 2

Emerg

entray

Incide

ntra

yP

Q R

15. A steel wire of cross-sectional area 1 10-4 m2 is clamped firmly at each end when itstemperature is 25C so that it cannot contract on cooling. The tension in the wire when it iscooled to 0C is ( = 10-5 /C, Y = 2 1011 N/m2 )

(A) 5000 N (B) 7000 N(C) 7400 N (D) 4700 N

16. At time t = 0, a horizontal disc starts rotating with angular acceleration 1 rad/sec2

about an axisperpendicular to its plane and passing though its center. A small block is lying on this disc at adistance 0.5 m from center, coefficient of friction between surface of block and disc is 0.255. Theblock will start slipping on the disc at time t, is approximately equal to

(A) 2 3 s (B) 2 2 s

(C) 6 s (D) 5 s

17. Identify a spring mass system which has a different time periodcompared to the system shown in figure

(A)

K

m

V=const.

(B)

K

m

K

m

(C)

K

m

(D) none of these

Rough work


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7

18. It is found that an increase in pressure of 100 kPa causes a certain volume of water to decreaseby 5 10

3percent of its original volume. Then the speed of sound in the water is about (density

of water = 1 103

kg/m3)

(A) 330 m/s (B) 140 m/s(C) 1400 m/s (D) 1800 m/s

19. A rodA

of weightW

is hanging from ceiling. Extension in the rod due to its own weight is 1l

.Another identical rod B but negligible mass is also hanging from ceiling. A force of magnitudeequal to W is acted on the free end of the rod. The extension in the rod B is

2l . Then 1 2/l l isequal to

(A)1

2(B) 2

(C) 3 (D)1

3

20. Assume that the escape velocity for a body projected vertically upward from the earths surface is11 km/s. If the body is projected at an angle of 45

owith the vertical the escape velocity will be

(assume that the earth is not rotating and there is no air resistance)

(A)11

km/s2

(B) 11 2 km/s

(C)11

km/s2

(D) 11 km/s

21. In an imaginary atmosphere, the air exerts a small force F on any particle in the direction of theparticles motion. A particle of mass m projected upward takes a time t1 in reaching themaximum height and t2 in the return journey to the original point then(A) t1 < t2(B) t1 > t2(C) t1 = t2(D) The relation between t1 and t2 depends on the mass of the particle

Rough work


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8

22. Water in a bucket is whirled in a vertical circle with a string attached to it. The water does not falldown even when the bucket is inverted at the top of its path. We conclude that in this position

(A)r

mvmg

2

= (B) mg is greater thanr

mv2

(C) mg is not greater than

r

mv2(D) mg is not less than

r

mv2

23. A shot of mass m penetrates a thickness t of a fixed plate of mass M. If M were free to move andthe resistance supposed to be uniform, find the thickness penetrated

(A)Mt

M m+(B)

mt

M m+

(C)t

M m+(D)

t

M 2m+

24. The magnetic field (B) on the axis of a circular current carrying wire varies with distance (z) fromthe centre of the wire as (radius of wire = r; z >> r)

(A)1

Bz (B) 2

1B

z

(C)3

1B

z (D) B z .

25. Two identical circular coils each of radius r and having number of turns n are lying inperpendicular planes, such that they have common centre. The magnetic field at the centre of the

coils, if they carry currents equal to i and 3 i respectively is

(A) 02 ni

r

(B) 0

ni

r

(C) 0ni

2r

(D) 0

3 ni

r

Rough work


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9

26. As compared to12

C atoms,14

C atom has(A) two extra protons and two extra electrons (B) two extra protons but no extra electron(C) two extra neutrons but no extra electron (D) two extra neutrons and two extra electrons

27. Which of the following is a wrong description of biding energy of a nucleus ?(A) it is the energy required to break a nucleus into its constituent nucleons.(B) it is the energy made available when free nucleons combined to from a nucleus.(C) it is the sum of the rest mass energies of its nucleons minus the rest mass energy of the

nucleus.(D) it is the sum of the kinetic energy of all the nucleons in the nucleus.

28. An -particle is bombarded on 14N. As a result, a 17O nucleus is formed and a particle is emitted.This particle is a(A) neutron (B) proton(C) electron (D) positron.

29. The wavelength of sound emerging from a moving source is observed to decrease from its staticvalue in a certain direction from the source. Then, the source(A) must be moving in that direction(B) must be moving in the opposite direction

(C) must be moving at an acute angle with this direction(D) must be moving at an obtuse angle with this direction.

30. A car sounding its horn at 480 Hz moves towards a cliff at 20 m/s. The driver of the car hears(velocity of sound = 320 m/s)(A) No beats(B) beats at a frequency of 64 Hz(C) beats at a frequency of 30 Hz(D) beats at a frequency of 32 Hz.

Rough work


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10

CChheemmiissttrryy PART II

SECTION A

Single Correct Choice Type


1. An element X (At. mass 25) exists as X4 in benzene. When 51 g of saturated solution of X inbenzene was added 50 g of benzene, the resulting solution showed a depression of freezingpoint of benzene by 0.55 K. the solubility of X per kilogram of benzene will be (Kf benzene = 5.5K Kg mol

1, degree of association of X in benzene = 1)

(A) 10 g (B) 20 g(C) 25 g (D) none

2. Consider the following galvanic cell at 25CPt, H2(1atm)|0.1 M HA || 0.1M BOH| H2 (1 atm), Pt

If Ka of HA = 1.0 105 and Kb of BOH = 4 107 the emf of cell is(A) 0.432 V (B) 0.041 V(C) 0.432 V (D) 0.041 V

3. 20 ml of 0.2 M HCN is titrated with 0.2 M NaOH. The pH at 20% neutralization and at the

equivalence point will be respectively (ka for HCN = 4 1010

)(A) 11.2 and 8.80 (B) 8.80 and 11.2(C) 9.4 and 11.2 (D) 8.80 and 12.05

Rough work


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11

4. In the following reaction (A) and (B) are isomerO

2

HO

H OB

ZZZZXYZZZZ

Identify B(A) O

(B) O

(C) O

(D) O

5. Pick out the incorrect statement of the following

(A) [Fe(CN)6]4 is inner orbital complex with zero magnetic moment(B) [Fe(CN)6]

3is low spin complex, magnetic moment being 1.73 BM

(C) Ni(CO)4 has tetrahedral geometry and magnetic moment to be 2.83 BM(D) [Ni(CN)4]

2has square planar geometry and magnetic moment to be 1.73 BM

6. The d-orbitals, which are stabilised in an octahedral magnetic field are:

(A) 2xy zd and d (B) 2 2 2x y zd and d

(C) xy xz yzd ,d and d (D) 2zd only

7. When a mole mercuric ions in V dm3

solution is shaken with excess of Hg(l), an equilibrium isestablished. The equilibrium constant can be represented by, if x mole of mercuric ions areconverted into mercurous ions

(A)( )

24x V

a x(B)

( )x

a x

(C)( )

2x

a x / v(D) None of these

Rough work


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12

8. Consider the following statements(I) Thermal stabilities of carbonates of alkaline earth metals increase down the group(II) Except LiOH, hydroxides of alkali metals are thermally stable(III) Li2CO3 is thermally stable whereas other carbonates of the other group are thermally

unstable.(IV) Solubilities of hydroxides of alkaline earth metals increase down the group.Pick out the correct set of the following:(A) I, II, III (B) II, III, IV(C) I, II, IV (D) All of the above

9. Which of the following molecules has unequal bond length(A) BF3 (B) PF3(C) PF5 (D) SF6

10. x/m represents the amount of a gas adsorbed per unit mass of solid adsorbent at the equilibriumpressure p, which of the following validates the applicability of the Langmuirs adsorptionisotherm?

(A) x/m v/s p linear graph (B) x/m v/s p1/n linear graph (n > 1)

(C) log x/m v/s log p linear graph (D)p

x /m

v/s p linear graph

11. In the following reaction

CH3 C CH2 CH2 CH3

CH3

Br

( ) ( )( )3 2 2

6 12 6 13 2

BH / T HF H O , OHEtO /ETOH6 14

C H Major product C H BHA B C C H O

The structure of C is(A)

CH3 C CH CH3

OHCH3

CH3

(B)

CH3 C CH2 CH2

OH

CH3

CH3

(C)

CH3 C CH2 CH

CH3

CH3

OH

(D)

CH3 CH CH CH2

CH3

CH3

OH

Rough work


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13

12. Identify the product (A) formed in the following reactionCH3 H

Br

( )3CH OH A HBr + +

(A)CH

3H

OH

(B)CH

3H

OCH3

(C) CH3 H

CH3

(D) CH3 OCH3

13. Select the most stable structure of the following

(A) (B)

(C) (D)

14. ( ) ( ) ( )2Ni/HNaCN acetic3 2 anhydrideCH CH Cl X Y Z

product (Z) is(A) CH3CH2CH2NH2 (B) CH3CH2CH2CONHCH3(C) CH3CH2CH2NHCOCH3 (D) CH3CH2CH2CONHCOCH3

Rough work


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14

15. In the following reactionCl

( ) ( )22 3

(i) CH OMg

Et O (ii) H OI II+

(A) (B) OH

(C)

OH

(D)

16. The radius ratio of Cs

+and Cl

ions is 0.90. It the radius of Cl

ion is 1.80 A, the false statement

regarding the CsCl structure is(A) The distance between the centres of Cl

ions is greater than 3.60 A

(B) The distance between the centres of two Cl

ions is 3.60 A(C) The distance between Cs

+and Cl

ions is 3.42 A

(D) The distance between two Cs+

ions is greaer than 3.24 A

17. In the following reaction

0

dil NaOH

5 CA B+

O

CH3 CH

OH

CH3

Identify A and B(A) O

CH3 CHO

and CH3 CHO

(B) O

and CH2 CHOCH3

(C) O

CH3

and CH3 CHO

(D) O

and CH2 CHOCH3

Rough work


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15

18. Which of the compounds exist mainly in the enol form?(A)

NH

NH

NH

(B) O

(C) O

OO

(D) All of these

19. 10 ml of a given sample of H2O2 was diluted to 100 ml. 20 ml of the diluted solution required 40ml of 0.02 M KMnO4 in acidic medium for decolourisation. The Volume strength of the givensample is(A) 5.6 Vol (B) 11.2 Vol(C) 16.8 Vol (D) 22.4 Vol

20. When excess of KI is added to aqueous CuSO4 solution. The solution acquires dark browncolouration. This is due to the formation on(A) CuI2(s) (B) Cu2I2(s)

(C) ( )3I aq (D) I2(s)

21. X + H2SO4 Y (a colourless gas which turns lime water milky)Y + K2Cr2O7 + H2SO4 Green solutionX and Y are

(A) 23 2CO , CO (B) Cl , HCl

(C) 2 2S , H S (D) 23 2SO , SO

Rough work


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16

22. A sulphide ore is generally roasted to oxide before reduction because(A) Enthalpy of formation of CO2 is more negative than that of CS2(B) A metal sulphide is generally more stable than metal oxide(C) No reducing agent is found suitable for reducing metal sulphide(D) A sulphide ore cannot be reduced at all

23. In qualitative analysis of a salt or salt mixture, Pb2+

is detected in both I and II groups. It isbecause(A) PbCl2 has too low ksp and its is sparingly soluble in water(B) PbCl2 has too high ksp and is soluble in water only(C) PbCl2 does not have too low ksp and gets only partially precipitated in presence of dilute HCl(D) PbS has very high ksp and gets precipitated on passing H2S in acidic medium

24. In the given reactionCH3

CH

CH3

NH2 2NaNOHCl

The product is/are

(A)OH

CH3

CH3

(B)

CH3

CH3

(C) CH3

CH CH2

(D) CH3

CH3

25. The EAN of a metal carbonyl, M(CO)x is 36. The atomic number of metal is 24. The value of x is:

(A) 6 (B) 5(C) 4 (D) 3

Rough work


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17

26. Mole fraction of methanol in water is 0.5. The concentration of the solution in terms of percent bymass of methanol is(A) 36 (B) 50(C) 64 (D) 72

27. Consider the reversible reactions

( ) ( ) ( ) ( )4 2 2 1(I) CH g H O g CO g 3H g K+ +ZZZX

YZZZ ( ) ( ) ( ) ( )2 2 2 2(II) CO g H O g CO g H g K+ +ZZZXYZZZ

( ) ( ) ( )2 2 4 2 3III CO 4H g CH g 2H O K+ +ZZZXYZZZ

Which of the following is true?(A) log K3 = log K1 + log K2 (B) log K1 + log K2 + log K3 = 0

(C) log K3 = log K1 log K2 (D) log K3 = ( )1 21

logK logK2

+

28. For 1 2K K

A B C , if C0 is the initial concentration of A and C1, C2 and C3 are concentration

of A, B and C respectively at any time t, then the value of K 1 and K2 can be determined usingequation

(A) 2k tk,t1 022 1

K CC e e

K K

= (B) 2k tk,t1 12

2 1

K CC e e

K K

=

(C) 2k tk,t1 021 2

K ,CC e e

K K

= (D) None of these

29. Which is the least stable form of glucose

(A) -D-Glucose (B) -D-glucose(C) open chain structure (D) All are equally stable

30. Corrundum is

(A) -Al2O3 (B) -Al2O3(C) Both the above (D) None of these

Rough work


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18

MMaatthheemmaattiiccss PART III

SECTION ASingle Correct Choice Type


1. If 2 2 2 2S 1 3 5 ..... 99= + + + + then the value of the sum ( )22 2 22 4 6 ..... 100+ + + + is(A) S + 2550 (B) S + 5050(C) 2S (D) 4S

2. A point (x, y) in the first quardant lies on a line with intercepts (a, 0) and (0, b), where a, b > 0.

Rectangle 1 has vertices (0, 0), (x, 0), (x, y) and (0, y) while rectangle 2 has vertices (x, y),(x, b), (a, b) and (a, y). The ratio of the area of1 to that of2 is(A) 1 (B) 2

(C)1

2(D) dependent on the value of a, b, x and y

3. The value of cot 70 + 4 cos 70 is equal to

(A) 1 (B) 3 1+

(C) 3 (D) 3 1

4. The units digit of ( )( )2013

32k

k 0

3 2k

=

+ is(A) 3 (B) 9(C) 2 (D) 1

5. If the graph of y = ax3

+ bx2

+ cx + d is symmetric about the line x = k then

(A) k = c (B) k =c

b

(C)c

a k 02b

+ + = (D) none of these

Rough work


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19

6. The roots of the equation x2

+ 2(a + b + c)x + 3k(ab + bc + ca) = 0 are real for all a, b, c R+,then K belongs to

(A)1 5

,3 3

(B)4 5

,3 3

(C)4

,3

(D)

5,

3

7. If f(n, x) =n

2

rr 1

x1 tan

2=

, then

(A) ( )n

tanxlim f n, x

x

2

=

(B) ( )n

xlim f n, x

tanx=

(C)

/ 42

0

tanxdx

16 x 4

<


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20

11. The number of even numbers from 1000 to 9999 both inclusive that do not have all four differentdigits(A) 2296 (B) 4500(C) 2204 (D) none of these

12. Number of zeros of the cubic f(x) = 2x3

+ 6x + k, k R, is

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

13. Number of ways in which 7 green bottles and 8 blue bottles can be arranged in a row if exactly,one pair of green bottle is side by side, is (Assume all bottles to alike except for the colours)(A) 84 (B) 360(C) 504 (D) none of these

14. The solution set of inequality

( ) ( ) ( )1 1 1 1tan x cot x tan x 1 2cot x 2 12 2

+ + +

> 1xlim sec x

2

is

(where [.] denotes the greatest integer function)(A) (tan 1, tan 2) (B) (cot1, cot2)

(C) (tan1, tan2) (D) (tan1, )

15. All the seven digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7, exactly once, and notdivisible by 5 are arranged in the increasing order. The 2004

thnumber is (in this list)

(A) 4315627 (B) 4316527(C) 4316572 (D) 4316257

16. In a plane there are two families of lines y = x + r, y = x + r where r {0, 1, 2, 3, 4,} then thenumber of squares of diagonal length 2 formed by the lines is(A) 6 (B) 9(C) 27 (D) 18

17. If (5, 12) and (24, 7) are the foci of a conic passing through the origin then eccentricity of conic is

(A) 38612

(B) 38614

(C)386

13(D)

386

15

Rough work


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21

18. You are given an unlimited supply of each of the digits 1, 2, 3, or 4. Using only these four digits,you construct n digit numbers. Such n digit numbers will be called BEAUTIFUL if it contains thedigit composite number either an even number times or not at all. Number of n digit BEAUTIFULnumbers are(A) 2

n+ 1 (B) 2

n + 1+ 2

(C) 2n 1

(2n

+ 1) (D) 2n + 2

+ 4

19. Four three noncoplanar vectors a, b, cGG G

. If a b b c c a G GG G G G

is equal to [ ]a b b c c a+ + +G GG G G G

,

then ( )a b c GG G

is equal to

(A) 1 (B) 2(C) 10 (D) none of these

20. A fair dice is thrown 3 times. The probability that the product of the three out comes is a primenumber, is

(A)1

24(B)

1

36

(C)1

32(D)

1

8

21. If the equation p sin 2x + cos x = p has a solution, then the range of p is

(A) (, ) (B) [1, 1](C) ], 0] (D) [0, [

22. For a unique value of p and q, the system of equations given by x + y + z = 6, x + 2y + 3z = 14,2x + 5y + pz = q has infinitely many solutions, then the value of (p + q) is equal to(A) 14 (B) 24(C) 34 (D) 44

23. Given a regular tetrahedron OABC (where O is origin) with side length 1. Let D, E be the midpoint

of the sides AB, OC respectively. The scalar product of two vectors DEJJJG

and ACJJJG

is equal to

(A) 0 (B) 1

(C)1

2(D) 2

Rough work


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24. Let W denote the words in the English dictionary. Define the relation R be

R = {(x, y) (W W) | the words x and y have atleast one letter in common}. Then R is(A) not reflexive, symmetric and transitive (B) reflexive, symmetric and transitive(C) reflexive, symmetric and not transitive (D) reflexive, not symmetric and transitive

25. From a fixed point on the circle x2

+ y2

+ 6x = 0 chords are drawn to the circle. The difference of

area of the locus of trisect point of the chords is(A) (B) 2(C) 3 (D) none of these

26. If ( )k k

lnsin x lnsin x2 n n 1

nf(x) lim n e e +

= where 0 < x < , n N and

/ 2

0

f(x)dx ln4k

= thenax

k+ b + cx

k= 0 is

(A) a quadratic equation (B) cubic equation(C) biquadratic equation (D) none of these

27. If f(x) = sin {[x + 5] + {x {x {x}}}} is a function for x 0,2

, where {.} and [.] denotes

fractional part and greatest integer functions respectively, then range of f(x) is(A) [0, sin 1[ (B) ]0, sin 1[ ]0, cos 1[(C) [0, sin 1[ ]0, cos 1[ (D) [0, cos 1[

28. Equation 3x2

8xy 3y2

= 0 and x 2y = 3 represents the sides of a triangle which is(A) right angled but not isosceles (B) isosceles(C) equilateral (D) right angled and isosceles

29. If ax2

+ by2

+ cz2

+ 2ayz + 2bzx + 2cxy can be solved into two linear factors then a3

+ b3

+ c3

is(A) abc (B) 2abc

(C) 2abc (D) none of these

30. The greatest value of ( )( )2 2

y 2 x x x= + + is, , R

(A) 4 2 2 2x + + (B) 2 22 3 +

(C) 2 2 + (D) +

Rough work

Concept Recap Test Mains 2

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