all india test series

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

CONCEPT RECAPITULATION TEST (Set – I)

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.

ALL

IND

IA T

ES

T S

ER

IES

FIITJEE JEE (Main), 2013Fr

om L

ong

Term

Cla

ssro

om P

rogr

ams

and

Med

ium

/ Sh

ort C

lass

room

Pro

gram

4 in

Top

10,

10

in T

op 2

0, 4

3 in

Top

100

, 75

in T

op 2

00, 1

59 in

Top

500

Ran

ks &

354

2 t

ota

l s

ele

cti

on

s i

n I

IT-J

EE

20

12

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 ×10−34 J-s

Charge of electron e = 1.6 × 10−19 C

Mass of electron me = 9.1 × 10−31 kg

Permittivity of free space ε0 = 8.85 × 10−12 C2/N-m2

Density of water ρwater = 103 kg/m3

Atmospheric pressure Pa = 105 N/m2

Gas constant R = 8.314 J K−1 mol−1

CHEMISTRY

Gas Constant R = 8.314 J K−1 mol−1 = 0.0821 Lit atm K−1 mol−1 = 1.987 ≈ 2 Cal K−1 mol−1 Avogadro's Number Na = 6.023 × 1023 Planck’s constant h = 6.625 × 10−34 J⋅s = 6.625 × 10–27 erg⋅s 1 Faraday = 96500 coulomb 1 calorie = 4.2 joule 1 amu = 1.66 × 10–27 kg 1 eV = 1.6 × 10–19 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. A small block of mass m slides along a frictionless loop inside loop

track as shown in figure. The minimum value of the ratio R/r so that the block may not lose contact at the highest point the inner loop is

(A) 7/2 (B) 2 (C) 5/2 (D) 3

Rr

B

A

2. Two blocks A and B each of mass m are connected by a massless spring of natural length and spring constant k. The blocks are initially resting on a smooth horizontal floor with the spring 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 along the line joining A and B collides with A. Then (assuming the collision 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) mv2 when 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) m6 kπ (B) m

k

(C) 2 m3 kπ (D) m

4 kπ

Rough work



4

4. A force ( )ˆ ˆF yi xj N= − + 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 when the particle undergoes one complete revolution is (assume x, y are in m)

(A) zero (B) 32 Jπ (C) 8 Jπ (D) 16 J 5. A uniform rod AB of mass m and length is at rest on a smooth horizontal surface. An impulse J

is applied to the end B perpendicular to the rod in horizontal direction. Speed of the point A of the rod after giving impulse is

(A) J2m

(B) J2m

(C) Jm

(D) J2m

6. In figure there are two sliders and they can slide on two

frictionless parallel wires in uniform magnetic field B, which is present everywhere. The mass of each slider is m, resistance R and 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) 0v4

(B) 0v2

(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



5

8. A conductor of variable cross-section is connected across a battery. Let us consider two cross-sections A and a. Let V and v be drift speeds of electrons at those cross-sections respectively then

(A) AV > av (B) AV = av (C) Av = aV (D) AV < av

av

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/s2 (B) 36 m/s2 (C) 15 m/s2 (D) 45 m/s2

10. A charged plate of charge Q is kept in between two plates. Both the

plates have an initial charge of Q and 3Q respectively. The final charges on q3 & q5 surfaces of the plate are

(A) ,2 2Q 3Q

− − (B) ,2 2Q 5Q

− −

(C) ,2 23Q 5Q

− − (D) ,2 23Q 3Q

− −

q1

q2q3

q4q5

q6

3Q

Q

Q

11. A particle of mass 0.002 kg and a charge 1 µC 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. The speed 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 ms–1 (D) 100 ms–1

12. The switch is closed at time t = 0. The potential difference across

inductor and resistor (at t = 0), respectively, is (A) 5V, 5V (B) 8V, 6V (C) 10V, 0V (D) 0V, 10V

10V

5H 5Ω

Rough work



6

13. Acceleration vs time graph is shown in the figure for a particle moving along a straight line. The particle is initially at rest. Find the time instant(s) when the particle comes to rest?

(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 4 t(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 (ρwater = 1 × 103 kg/m3, g = 10 m/s2)

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

m=10 kg

15. The minimum horizontal acceleration of the container so that

the pressure at the point A of the container becomes atmospheric is

(A) g23

(B) g

34

(C) g (D) g43

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 after a 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 constant

pressure P0. The heat absorbed by the gas in the process is (A) 2P0V0 (B) 4P0V0

(C) 3P0V0 (D) P0V0

Rough work



7

18. Two identical conducting rods AB and CD are connected to a circular conducting ring at two diametrically opposite points B and C. The radius of the ring is equal to the length of rods AB and CD. The area of cross-section, and thermal conductivity of the rod and ring are equal. Points A and D are maintained at temperatures of 100ºC and 0ºC. Temperature of point C will be

A B C D100ºC 0ºC

(A) 62ºC (B) 37ºC (C) 28ºC (D) 45ºC

19. Two balls of same material and finish have their diameters in the ratio 2 : 1. Both are heated to

the same temperature and allowed to cool by radiation. Rate of cooling of big ball as compared to smaller 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 are

connected in series as shown in the figure. Thermal conductivities of their materials are K, 2K and 1.5K. If the temperatures of free end of first and the last bar are 200ºC and 18ºC. The value of θ1 and θ2 are (in steady state)

K 2K 1.5K200ºC 18ºC

θ1 θ2

(A) 120ºC, 80ºC (B) 116ºC, 80ºC (C) 116ºC, 74ºC (D) 120ºC, 74ºC 21. 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? (Assume motion is 1 dimension)

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

t84−1O

4

a

Rough work



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 inclined plane perpendicularly.

(A) 2gH5

(B) 2gH7

(C) gH5

(D) gH7

90°

H

30°90°

V0

23. A uniform bar of square cross-section is lying along a

frictionless horizontal surface. A horizontal force is applied 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 friction force on block Q when the spring regains its natural length is

Q

P µs

K

(A) 13

KA2 (B) 23

KA2

(C) 12

KA2 (D) 14

KA2

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

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

(A) 0 I (1 sin )4 dµ

− θπ

(B) 0µ 0 I (1 sin )4 dµ

+ θπ

(C) 0 I (1 cos )4 dµ

− θπ

(D) 0 I (1 cos )4 dµ

+ θπ

I

p

θ

dI

Rough work



9

26. An electron moving with a velocity 1ˆV 1i= m/s at a point in a magnetic field experiences a force

1ˆF 2 jN= − . If the electron is moving with a velocity 2

ˆV 1j= m/s at the same point, it experiences a

force 2ˆF 2i= N. The force that an electron would experience if it were moving with a velocity

3ˆV 2k= m/s at the same point is

(A) zero (B) ˆ2kN

(C) ˆ2k− N (D) information is insufficient 27. X–ray 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 Kα lines is λz for target A and λ1 and λ2 for two impurities

z z

1 2

14 &4

λ λ= =

λ λ

Screening constant of Kα 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 is z 12

+

.

28. All electrons ejected from the surface by incident light of wavelength 200 nm can be stopped

before travelling 1 m in direction of uniform electric field of 4 N/C. The work function of the surface is (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 curved

surfaces. 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



10

CChheemmiissttrryy PART – II

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. In Bohr’s model of hydrogen atom the ratio between period of revolution of an electron in the orbit

n = 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

kH

c+

=

= ∑ (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

NaOH X∆

→

Product X is (A) NO2 O

O

(B) NO2

O CH3

O

(C)

O O

CH3NO2

OH

(D)

C

O

CH2 C CH3

O

O2N

Rough Work



11

5. CH3

H OH

D

MeS

O

O

Cl

( )3CH COONa

2→ ( )( )

2OH / H O Ester hydrolysis3

−

→ ( ) ( )Acidification34 CH COOH P→+ +

( )0X+

The product P has optical rotation (A) (+ X0) (B) (– X0) (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 its

10 ml portion required 12.4 mL 0.1 M HCl solution to reach the equivalence point. In an another experiment, 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 % in the 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 Rt− (D) t (R2 – R1)

Rough Work



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 pH 8.12 mL 4.57 16.24 ml (equivalence point) 7.02

OH

COOH

2H O+

OH

COO

3 a2H O : K++

OH

COO

2H O+

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

E2 PE

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



13

12. The equation for the reaction in the figure below is ( ) ( ) ( )5 3 2AB g heat AB g B g+ +

0.8

0.6

0

0.2

0.4

1 2 3

AB2

B2

AB5

Time (min)

Con

c. (M

mol

/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 <120° in molecule OCF2 (D) The axial FSF bond angle in SF4 = 180 14.

NH2

COOH

H

OHH

COOH

2NaNO X∆

→

What is X?

(A) OH

COOH

H

OHH

COOH

(B) H

COOH

OH

OHH

COOH(C) COOH

CH2

C

COOH

O

(D) COOH

C

CH3

O

Rough Work



14

15. Cl

Cl

( )2Mg Et O1 eq Pr oduct 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



15

18. The linkage between the two monosaccaride unit in lactose is (A) C1 of β—D glucose and C4 is β—D galactose (B) C1 of β—D galactose and C4 of β—D glucose (C) C1 of α—D galactose and C4 of β—D glucose (D) C1 of α—D glucose and C4 of α—D galactose 19. The product P is

OH

CH3I

H

D

I

*( ) ( )

18aq K OH/ 2 eq / H acidification P

+∆→→

(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 Pν ∆ ∆→→→→→

O

Me

The major product P is

(A)

CH2

Me

HO

(B)

CH2

MeOH

(C) CH2 Me

(D)

CH2

Me

Rough Work



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) EA–B < [ ]A A B B1 E E2 − −+ (E is representing bond energy

(C) The two π bonds in CO2 molecules are coplanar (D) The largest ion among Se2– O2–, F– and Rb+ is Se2– 23. The IUPAC nomenclature oxidation state and hybridisation of the given complex

K2 [Cr(NO)(NH3)(CN)4] with µ = 1.73 BM respectively are (A) potassiumanninetetracyano (C) nitrosyl chromium (I), +1, d2sp3 (B) Potassium ammine tetracyano (c) nitrosylchromium (I), +2, sp3d2 (C) Potassiumamminetracyano (c) nitrosonium chromium (I), +2, sp3d2 (D) Potassium amminetetra cyano (c) nitrosonium chromium (I), +1, d2sp3 24.

( )4 4

YX NH OH Gelatinous ppt. NH Cl+ → +

Y NaOH Soluble.+ → Identify ‘x’ (A) FeCl3 (B) CrCl3 (C) AlCl3 (D) SbCl3 25. CHO

NO2

( ) ( ) ( )2 4

3

Ac O,AcONa FeSO HCl/NH

reductionB 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



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 2Fe e Fe+ + ++ → The standard potential (in volt) corresponding to the reaction (i) and (ii) are E1 and E2

respectively. The value (in volt) of standard potential corresponding to the reaction 3Fe 3e Fe is+ −+ →

(A) E1 + E2 (B) 1 22E E3+

(C) 1 2E 2E2+

(D) 1 2E E3+

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 hydroxide

which decomposes to give an insoluble brown layer of its oxide. X is a powerful cantery and breaks 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 NBSii Mg/ether ii H

A B+→ →

O CH3

(A)

OH

(B)

OH

(C) OH

(D)

OH

Rough Work



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

2esin x∫ [2 ln cosec x + sin 2x] dx is equal to

(A) −2ecotx ln (cosec x) + c (B) ecotx ln x + c (C) ecotx (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 total

surface area may be minimum is

(A) 7 (B) 227

(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



19

5. Let N be any four digit number say x1 x2 x3 x4. Then maximum value of 1 2 3 4

Nx x x x+ + +

is equal

to

(A) 1000 (B) 11114

(C) 800 (D) None of these 6. ABCD is a quadrilateral with side lengths AB = 4, BC = 10, CD = 6 and AD = 6, and diagonal BD = 8 units. If the incircles of triangles ABD and BCD touch BD at P and Q respectively, then

area of quadrilateral C1PC2Q (where C1 and C2 are incentres of triangle ABD and BCD respectively), is

(A) 1532

+ sq. units (B) 3 sq. units

(C) 156

sq. units (D) none of these

7. If ( )f x dx 1∞

−∞

=∫ then 1f x dxx

∞

−∞

− ∫ is equal to

(A) 0 (B) 1 (C) −1 (D) 2

8. In a ∆ABC, if base BC is unity. If sin A2

= x1, sin B2

= x2, cos A2

= x3, cos B2

= x4 such that

2006200731

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 (integral

point is a point both of whose coordinates are integers), then ( )2r

f rlimr→∞

is equal to

(A) 1 (B) π (C) 2π (D) π/2

Rough work



20

10. f(x) + 1f 1x

−

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

(A) 1 (B) 14

(C) 34

(D) 12

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 three

distinct 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 the

perpendicular 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 eax2 – e2ax + ea – 1 = 0 lies between

1 and 2 are given by

(A) ln (5 17) 5 17 a ln 4 4

− +< < (B) 0 < a < 100

(C) ln 54

< a < ln 103

(D) none of these

15. If a, b, c , are three nonzero vectors, no two of which are collinear, a 2b+ is collinear with c

and b 3c+ is collinear with a , then | a 2b 6c+ + | 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) 2n –(k – m) (B) 2k − m − 2 (C) 2k − m − 1 (D) 2k − m + 1

Rough work



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 period

of f(x) is (A) 2 (B) 4 (C) 8 (D) 12

19. The value of 3 5

2 4cos x cos xsin 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 tan–1 (sin x) + c (D) none of these

20. If A = sin

21

t dt1 t

θ

+∫ and B = cosecθ

21

dtt(1 t )+∫ , then the value of

2

A B 2

2 2

A A B

e e B 1

1 A B 1

−

+ −

is

(A) sin θ (B) cosec θ (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/a2 – y2/b2 = 1, the product of whose

slopes is c2, lies on the curve (A) y2 – b2 = c2 (x2 + a2) (B) y2 + a2 = c2 (x2 – b2) (C) y2 + b2 = c2 (x2 – a2) (D) y2 – a2 = c2 (9x2 + b2) 23. If α ≤ sin–1x + cos–1x + tan–1x ≤ β, then

(A) α = 4π , β = 3

4π (B) α = −π, β = 2π

(C) α = 0, β = π (D) none of these

Rough work



22

24. If I be the centre of the circle inscribed in the ∆ABC whose sides BC, CA and AB are of lengths α, β and γ respectively, then α IA + β IB + γ IC is

(A) unit vector (B) null vector (C) vector of magnitude α + β + γ (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 the

extreme values of abc is equal to, (A) 3600 (B) 144 (C) 3456 (D) None of these

27. If I(n) =/ 2

n

0

.sin dπ

θ θ θ∫ , n ∈ N, n > 3, then the value of I(n) – ( )n 1 I n 2n−

− is

(A) 21

n (B) 1

n 1−

(C) 21

n 1− (D) 1

n 1+

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

of θ, 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 of xr + ys is (A) 2 (B) 0

(C) 1 (D) 12

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

slides along the circle x2 + y2 = a2, 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

all india test series

Documents