FIITJEE JEE (Main), 2014

From Classroom/Integrated School Programs 7 in Top 20, 23 in Top 100, 54 in Top 300, 106 in Top 500 All India Ranks & 2314 Students from Classroom

CONCEPT RECAPITULATION TEST - II

/Integrated School Programs & 3723 Students from All Programs have been Awarded a Rank in JEE (Advanced), 2013

Time Allotted: 3 Hours Maximum Marks: 432

Pl ea s e r ea d t h e i n s t r u c t i o n s c a r ef u l l y. Yo u a r e a l l o t t ed 5 m i n ut es
s p ec i f i c a ll y f o r t h i s p u r p o s e.
Yo u a r e n o t a ll o wed t o l ea v e t h e E xa m i n at i o n Ha ll b ef o r e t h e en d o f
ALL INDIA TEST SERIES

t h e t es t .

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 03 and 10 to 12) 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.

Section-A (04 to 09 and 13 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.

Name of the Candidate

Enrolment No.

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Useful Data

PHYSICS
2
Acceleration due to gravity g = 10 m/s

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

3 3
Density of water water = 10 kg/m

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.

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Physics 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. Two similar blocks A and B are riveted rigidly with a rigid rod k = 0.25
of length L such that block A can slide along an inclined A
plane as shown in the figure. If masses of the both blocks
and the rod are each equal to m and kinetic friction between L
A and inclined plane is k = 0.25 then the magnitude of
acceleration of centre of mass of the block rod system is B
(A) 6 m/s2 (B) 4 m/s2
2
(C) 5 m/s (D) 2 m/s2 = sin1(3/5)

2. A wedge of mass M and dimensions as shown in the figure

is lying completely inside a region (I) of zero pressure. The Region - I
pressure inside region II is 1 Pa. The surfaces are smooth. Region - II
The wedge is imparted an initial velocity v0 so that it just h
travels a distance b inside region II before finally coming to a
rest. The value of v0 is b

abh 3 abh
(A) (B)
M 4 M
4 abh 4 abh
(C) (D)
3 M 3 M

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3. A block A of mass m and a box B of same mass are B

lying on a horizontal surface as shown in the above
figure. The coefficient of kinetic friction between the L
block and horizontal surface is A, while that A
between the box and surface is B. Given that B >
A. At t = 0, the position is as shown in the figure
and both are simultaneously given a velocity v0 to
the right. The time after which block will collide with
the right end of the box is
2 L 2 L
(A) (B)
B A g B A g
4 L 4 L
(C) (D)
B A g B A g

4. It is observed that only 0.39% of the original radioactive sample remains undecayed after eight
hours. Hence choose the incorrect statement.
(A) The half life of that substance is 1 hr.
(B) The mean life of that substance is 1/ln 2 hr
(C) The decay constant of the substance is ln 2 hr1
(D) The half life of the substance is 1/2 hr

5. The figure shows a hollow circular tube fixed to a frictionless

horizontal table. It is the top view of the arrangement. A ball A
is shot into end A of the tube to leave the other end B at high
speed. The balls diameter is slightly less than the internal
diameter of the tube. Which of the paths below will the ball B
follow on the table after it leaves the tube?

(A) (B)

(C) (D) None of these

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6. A mass m on the end of an ideal spring stretches the spring

to a length when at rest. The mass is now set into motion
so that it executes up and down vibrations while swinging
back and forth as a simple pendulum. The mass moves in a
figure eight pattern in a vertical plane a shown in the figure. m
The force constant of the spring assuming small oscillations
is m
mg 2mg
(A) (B)
k
4mg mg
(C) (D)
2

7. When an ultrasonic waves travels from air into water,

(A) waves bends towards the normal
(B) frequency of waves changes from air to water.
(C) speed of ultrasonic wave is greater in water than that in air.
(D) speed of ultrasonic wave is less in water than that in air.

8. A narrow tunnel is made through the earth of mass M and radius R from the centre to the surface.
The speed (in m/s) with which a particle should be projected from the centre of the earth through
the tunnel so that it escapes to the space is
(A) 2gR (B) gR
(C) 3gR (D) 2 gR

9. A cubical block of volume v and density 3 is placed inside a

liquid of density and attached to a spring of spring k C
constant k as shown in the figure. Assuming ideal spring and A
pulley and spring is attached at A which is at R/2 from
centre. The compression in the spring at equilibrium is

4vg 2vg
(A) (B)
k k
3vg vg
(C) (D)
k k

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10. A conductor having uniform linear charge density is placed in y

circular form in 3 quadrants with different polarities as shown in
+
the figure. Let E0 be the electric field at O. Then the angle ++
+
+
made by E0 with the line along y-axis is + R
+
1 + x
(A) tan1 (B) tan1 1

O
2

1
(C) tan1 (D) None
3

11. The pulley and the string are light and all the surfaces are frictionless. 1 kg
The system is initially held at rest and The string is just taut. The 1kg
block on the plane is suddenly imparted a velocity of 5m/s to the left, 1 kg
and the system is simultaneously released. Find the displacement of
the 1kg block (on the plane) after 3s. [Take g = 10 m/s2]
(A) 10 m (B) 20 m
(C) 30 m (D) 40 m

12. A light thread is wound on a disk of mass m and other end of m

thread is connected to a block of mass m, which is placed on a
rough ground as shown in diagram. Find the minimum value
of coefficient of friction for which block remain at rest: m
1 1 r
(A) (B)
3 4
1 1
(C) (D)
2 5

13. A milli ammeter of range 5 mA and resistance 0.5 is joined 0.5 5 mA

in a circuit as shown. Find the value of current for which
meter gives full scale deflection when A & B are used as 0.1 0.2 0.3
Terminal:
(A) 50 mA (B) 5 mA
(C) 100 mA (D) 1 A
S B C D

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14. In the circuit shown, the batteries E1 = E2 = 1 V, E3 = 2.5 V E1 R1

and the resistances R1 = 10 , R2 = 20 . The potential
difference across the capacitor is E2 C
(A) 0.2 V (B) 0.4 V
R2
(C) 0.5 V (D) 1.2 V E3

15. A horizontal spring mass system of mass M performs simple harmonic motion of amplitude a0
and time period T0. When the mass M passes through mean position another stationary mass M
sticks to it and both move together. If a and T be new amplitude and time period. Then
a
(A) a 0 (B) a a0 2
2
T
(C) T 0 (D) None
2

16. System is shown in figure. Assume the cylinder 2u m/s

u m/s
remains in contact with the two wedges. The velocity
of cylinder is 30o 30o

u 13
(A) 19 4 3 m/s (B) v m/s
2 2
(C) 3v m/s (D) 7u m/s

17. Consider the following statements for a particle moving in an elliptic orbit under the influence of a
central force:
(1) The radius vector covers equal area in equal time.
(2) The motion takes place in a plane.
(3) The angular momentum is constant of motion.
Which of options given above are correct?
(A) 1 & 2 only (B) 2 & 3 only
(C) 1 & 3 only (D) all

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18. A person is standing on a weighing

machine placed on the floor of an elevator.
The motion of the elevator is shown in the Velocity
adjacent diagram. The maximum and the (upward)
minimum weights recorded are 66 kg and
57 kg. The magnitude of the upward 10 40 60
accelerations is [Take g = 10 m/s2.] Time in seconds
(A) 1 m/s2 (B) 0.5 m/s2
(C) 2 m/s2 (D) 0.25 m/s2

19. When a charge particle of charge 2 c and mass 1 gm is released at origin in gravity free space

having E E0 j and B B0 j (E0 = 103 N/c and B0 = 5T). Then its speed
(A) depends only of X co-ordinate (B) depends only on Y co-ordinate
(C) depends only on z co-ordinate (D) depends only on X, Y, Z co-ordinate

20. A neutral metallic finite block is placed at large but finite distance from a large charged sheet in
the middle space in front of sheet. Then the block will be
(A) Attracted towards the sheet (B) Repelled away from the sheet
(C) Depend on nature of charge on the sheet (D) Zero force on the block

21. Two equal masses hang on either side of a pulley at the same
height from the ground. The mass on the right is given a
horizontal speed, after some time.
(A) The mass on the left will be nearer to ground.
(B) The mass on the right will be nearer to ground. u
(C) Both the masses will be at equal distance from the ground.
(D) Nothing can be said regarding their positions. Ground

22. O is the centre of mass of a body of mass M as shown in the figure.

A, B, C are three different point on the body. OB = 8 cm, OC = 10 A
O
cm, BC = 6 cm and OA = 10 cm. Which of the following can be
written by using parallel axis theorem? I0 is the moment of inertia B C
about the axis passing through point O and perpendicular to plane of
object.
(A) IB = IC + M(BC)2 (B) IC = IB + M(BC)2
(C) IA = I0 + M(OB)2 (D) None of these

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23. A solid sphere of radius R, and dielectric constant k has spherical cavity of radius R/4. A point
charge q1 is placed in the cavity. Another charge q2 is placed outside the sphere at a distance of
r from q. Then Coulombic force of interaction between them is found to be F1. When the same
charges are separated by same distance in vacuum then the force of interaction between them is
found to be F2 then
(A) F1 = F2/k (B) F2 = F1/k
1
(C) F1. F2 = (D) F1 = F2
k

24. The relation between R and r (internal resistance of the battery) for R R
which the power consumed in the external part of the circuit is
R R R
maximum.
r R
(A) R = r (B) R 4R
2
(C) R = 2r (D) R = 1.5 r r

25. An equilateral triangular loop having a resistance R and length of each side is x x x x
x x x x x x
dB x x x x x x
placed in a magnetic field which is varying at = 1 T/s. The induced current x x x x x x
dt x x x x x x
in the loop will be x x x
3 2 4 2
(A) (B)
4 R 3 R
3 R 4 R
(C) (D)
4 2 3
2

26. The temperature of a mono-atomic gas in an uniform container of

A B
length L varies linearly from T0 to TL as shown in the figure. If the
molecular weight of the gas is M0, then the time taken by a wave T0 TL
pulse in travelling from end A to end B is L

2L 3M 3(TL T0 )
(A) (B)
( TL T0 ) 5R 5RM0L
2L 3M M0
(C) (D) L
( TL T0 ) 5R 2R(TL T0 )

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27. 12 identical rods made of same material are arranged in the T U

form of a cube. The temperature of P and R are maintained P Q
at 900 C and 300C respectively. Then the temperature of 0
90 C a
point V, when steady state is reached,
(A) 65C (B) 60C W
V
(C) 20C (D) 50C
0
S R 30 C

28. A spring block system with mass of block m and spring constant K (all m
the surfaces of block are perfectly reflecting and smooth) is placed on a
smooth horizontal plane as shown in the diagram. A light beam of
intensity I is switched on from rightwards. Find the amplitude of
oscillations of the block.
I 2I
(A) (B)
KC KC
4I
(C) (D) Zero
KC

29. A rod of mass M and length L is placed on a smooth horizontal table and is hit by a ball moving
horizontally and perpendicular to length of rod and sticks to it. Then conservation of angular
momentum can be applied
(A) About any point on the rod (B) About a point at the centre of the rod
(C) About end point of the rod (D) None

30. Magnetic force on a spiral carrying current I0 and placed in magnetic r2

I0
field B0 parallel to the plane of spiral as shown in diagram, will be
nearly (initial point, final point and centre lie on a line) B0
r1
(A) Zero (B) I0B0 r2 r1
I0B0 r2
(C) I0B0 r2 r1 (D)
r1

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Chemistry 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.

7
1. The pressure (P) and density () of a diatomic gas changes from P1 1 to P22. If
5
1 P
32, the value of 1 is:
2 P2
1
(A) 32 (B)
32
1
(C) 128 (D)
164

2. 0.2 mol of NH4Cl are introduced into an empty container of 10 litre and heated to 327oC to attain


equilibrium as NH4 Cl s
2
NH3 g HCl g , Kp = 0.36 atm . The quantity of solid NH4Cl left
is:
(A) 0.02 mole (B) 0.078 mole
(C) 0.095 mole (D) 0.035 mole

3. Which of the following alcohols will give same alkene on reaction with conc. H2SO4?
OH
OH
OH OH

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

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4. The correct set of reagents for the following conversion is:

O

CH3 2 CHCOOH O

O
(A) P4/I2, Na, conc. H2SO4 (B) P2O5, LiAlH4
(C) P4/I2, Na, P2O5/ (D) P2O5/, H2O, P4/I2, Na

5. Order of basic strength of the following compounds is:

O O

O CH2 NH2
(I) (II) (III) (IV)
(A) IV > II > I > III (B) III > II > IV > I
(C) II > III > IV > I (D) II > III > I > IV

6. When K2CrO4 is added to CuSO4 solution there is formation of CuCrO4 and CuCr2O7. Formation
of CuCr2O7 is due to:
(A) basic nature of CuSO4 solution which converts CrO24 to Cr2 O72
(B) acidic nature of CuSO4 solution which converts CrO24 to Cr2 O72
(C) CuSO4 has a typical property of converting CrO24 to Cr2 O72
(D) none of the above is correct

7. Amorphous boron is extracted from borax by following steps:

X Y Z
Borax H3BO3 B2 O3 Boron
What are (X) and (Z)?
(A) H2SO4, Al (B) HCl, C
(C) HCl, Fe (D) H2SO4, Na

8. Heat of neutralisation of strong acid and strong base under 1 atm and 25oC is 13.7 kcal. If
standard Gibbs energy change for dissociation of water to H+ and OH is 19.14 kcal, the
change in standard entropy for dissociation of water in cal k-1 mol-1 is:
(A) 18.25 (B) 110.2
(C) - 18.25 (D) none of these

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9. A metal crystallises in bcc lattice. The % of fraction of edge length not covered by atom is:
(A) 10.4% (B) 13.4%
(C) 12.4% (D) 11.4%
-33
10. E0Al3 / Al 1.66 V and Ksp of Al(OH)3 = 1.0 10 . The reaction potential of above couple at
pH = 14 is:
(A) - 2. 31 V (B) + 2.31 V
(C) + 1.01 V (D) - 1.01 V

11. Which of the following is correct option of reagent for the given conversion?

CH4 COOH

(A) Br2 / h, MgCl, Br2 / h, HCOOH

(B) Cl2 / h, MgCl, Br2 / h, KCN / H3 O
(C) Br2 / h, MgCl, Br2 / h, NaNH2 , HCN, H3 O
(D) Cl2 / h,CH3MgBr,Br2 / h,CH3 COOH

12. Which is correct statements about P4O6 and P4O10?

(A) Both form oxyacids H3PO3 and H3PO4 respectively.
(B) In P4O4 each P is joined to three O and in P4O10 each P is joined to four O atoms.
(C) Both of these
(D) None of these

13. 50 mL of a solution of Na2CO3 neutralizes 49.35 mL of 4 N HCl. The reaction is represented as

CO32 2H CO2 H2 O
The density of this Na2CO3 solution is 1.25 g mL-1. The percentage of Na2CO3 in it is:
(A) 47.7 (B) 37.7
(C) 26.7 (D) 16.7

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14. In the extraction of copper, the reaction which does not take place in Bessemer converter is:
(A) 2CuFeS2 O2 Cu2 S 2FeS SO 2 (B) 2Cu2O Cu2S 6Cu SO2
(C) 2Cu2S 3O 2
2Cu2 O 2SO2 (D) 2FeS 3O2 2FeO 2SO 2

15. In the following compounds of manganese what is the distribution of electrons on d-orbitals of
manganese?
2+ 4-
(i) [Mn(H2O)6] (ii) [Mn(CN)6]
3 2 5 0
(A) t 2g eg in both (B) t 2g eg in both
(C) t 32g e2 in (i) and t 52g eg0 in (ii) (D) t 52g eg0 in i and t 2g
3
eg2 ii

16. O
Cl C O
LiAlH4 Conc. H2 SO 4
C Cl
X

Cl C
O
The final product (X) is:

(A) (B)

(C) (D)

17.

H3 O
A B
O
Compounds A and B can be differentiated by
(A) 2, 4-DNP (B) Fehling solution
(C) Lucas reagent (D) NaHSO3

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18. OH

O

H
Major product

(A) O (B)
O

(C) (D)
HO

HO

19. Arrange the following reactions ion decreasing order of electrophilic addition reaction:
H3C H3C H3C
HCl
C HCl
CH2
, C CH2 , C
HCl
CH2
H3C H3C O H3C NH
(I) (II) (III)
(A) I > II > III (B) III > II > I
(C) II > III > I (D) I = II = III

20. Reductive ozonolysis of alkene C10H14 gave CH3 CO CH3, CH3 COCHO and
OCH CH2CO CHO in 1 : 1 : 1 ratio. Which of the following is the most likely structure of
alkene.

(A) (B)

(C) (D)

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21. Temperature of 4 moles of an ideal gas is raised from 300 K to 350 K. What is value of
-1 -1
H E for this process? (R = 8.314 J J mol K )
(A) 0 (B) 415 J
(C) 41.5 (D) 1660 J

22. Ksp of CdS is 8.0 10-27 and that of H2S is 1 10-22. 1 10-14 M CdCl2 solution is precipitated on
passing H2S when pH is about
(A) 4 (B) 6
(C) 5 (D) 7

23.
For the following equilibrium reaction N2 O 4 g 2NO2 g , NO2 is 50% of total volume at
given temperature. Hence, vapour density of equilibrium mixture is:
(A) 34.5 (B) 25.0
(C) 23.0 (D) 20.0

HO
CH2 CH3
24. To synthesize
CH3
(A)

Which of the following reactants cannot be suitable:

(A) O (B) O

CH2 CH3 , CH3MgBr CH3 , CH 3CH 2MgBr

(C) CHO (D) O MgBr

CH 3CHMgBr
, H3C CH2 CH3 ,
H2 C CH3

25. The number of stereoisomers possible for the compound

H3C CH CH CH(OH)COOH is

(A) 2 (B) 4
(C) 6 (D) 8

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26. Aspartame, an artificial sweetener is peptide and has the following structure:

O COOH
NH C
O C NH2
OCH3
Free amino acids obtained by the hydrolysis of aspartame are
(A) (B) O COOH
H2N
NH2 NH2
HOOC
(C) O (D) Both (A) and (C)
O C OH
HO C
NH2

27. Haworth projection of -D-glucose is

(A) CH2OH (B)

CH 2OH
O
H H O
H H OH
4 1 H
OH H
OH OH OH H
OH H
3 2
H OH
H OH
(C) Both (A) and (B) (D) None of these

28. Which of the following is having highest bondlength?

(A) NO (B) NO

(C) CN (D) CN

29. A 0.001 molal solution of Pt(NH3)4Cl4 in aqueous solution decreases the freezing point by
o -1
0.0054 C. If Kf = 1.86 K kg mol . What is the formula of the complex:
(A) [Pt(NH3)4Cl4] (B) [Pt(NH3)5Cl]Cl3
(C) [Pt(NH3)4Cl2]Cl2 (D) [Pt(NH3)3Cl3]Cl.NH3

30. Holms signal can be given by using

(A) CaC2 + CaCN2 (B) CaC2 + Ca3P2
(C) CaC2 + CaCO3 (D) Ca3P2 + CaCN2

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Mathematics 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 points A(3, 4) and B(4, 3) lies on same or opposite side of the line ax + by + c = 0, (a, b, c
R+) in which origin lies and P1 and P2 are length of perpendicular from A and B to the line such
that 2P1 + 3P2 = 10 then the line ax + by + c = 0 touches the circle
2 2
2 2 18 17
(A) (x 18) + (y 17) = 4 (B) x y 4
5 5
2 2
18 17
(C) x y 4 (D) (x 3)2 + (y 6)2 = 4
5 5

2. If a hyperbola whose foci are (2, 4) and (4, 6) touches yaxis then equation of hyperbola is
2 x 3y 14 2 x 3y 7 2 x 3y 8 2
(A) 3x y 8 20 (B) 1
4 2 8
3x y 8 2 x 3y 14 2 x 3y 14 2 3x y 8 2
(C) 1 (D) 20
2 8 2 8

x 4 14x 2 25
3. f x , zeroes are of the form a b , a, b z then a + b is equal to
5
x 2
x
(A) 5 (B) 6
(C) 7 (D) 8

4. The number 916238457 is an example of nine digit number which contain each of the digits 1 to 9
exactly one, it also has the property that the digit 1 to 5 occur in their natural order, while the digit
1 to 6 do not. Number of such numbers are
(A) 2268 (B) 2520
(C) 2975 (D) 1560

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5. Let z = 18 + 26i, where z = a + ib (a, b R) is the cube root of z having least positive argument
then the value of ab(a + b) is
(A) 18 (B) 12
(C) 10 (D) 10

6. A parabola touches two given straight lines originating from a given point. The locus of midpoint
of portion of any tangent. Which is intercepted between the given straight line is
(A) parabola (B) ellipse
(C) straight line (D) hyperbola

7. z1, z2 and z3 are three points on a circle centred at origin. A point z is chosen on the circle such
that the line joining z and z1 is perpendicular to the line joining z2 and z3. Which of the following is
true?
(A) zz1 + z2z3 = 0 (B) z2 z12 + z2z3 = 0
2
(C) z + z12 + z2z3 = 0 (D) zz1 z2z3 = 0

1
n
tan1 nx
8. Let Cn dx then lim n2 Cn equals
1
sin1 nx n

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

9. We have 21 identical balls available with us which we need to be distributed amongst 3 boys A, B
and C such that A always gets an even number of balls. The number of possible ways of doing
this is
(A) 112 (B) 120
(C) 126 (D) 132

1 x
10. The equation x a1 / x a = 2 a, a 1; a, x R has
x
(A) only one solution as x = 1 (B) only positive solution as x = 1
(C) infinite solutions (D) none of these

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11. A card from a pack of 52 cards is lost. From the remaining cards two cards are drawn at random
and are found to be spades. What is the probability that the missing card was a spade
1 11
(A) (B)
5 50
6 1
(C) (D)
25 4

12. The latus rectum of a conic section is the width of perpendicular line segment through the focus.
The positive difference between the lengths of the latus rectum of 3y = x 2 + 4x 9 and x2 + 4y2
6x + 16y = 24 is
1
(A) (B) 2
2
3 5
(C) (D)
2 2

13. Let A be a (n n) matrix with |A| = 4, B is the adjoint of the matrix 2A such that |B| = 1024. What
is the value of n?
(A) 3 (B) 4
(C) 5 (D) more than 5

14. Suppose that the domain of the function f(x) is set D and the range is the set R, where D and R
x f x
are the subsets of real numbers. Consider the functions: f(2x), f(x + 2), 2f(x), f , 2 . If m
2 2
is the number of functions listed above that must have the same domain as f(x) and n is the
number of functions that must have the same range as f(x) then the ordered pair (m, n) is
(A) (1, 5) (B) (2, 3)
(C) (3, 2) (D) (3, 3)

15. Let a1 , a2 .. an be the sides of a regular polygon inscribed in a circle of unit radius. If

a1 a2 a2 a3 ..... an a1 = a1 a2 a2 a3 ..... an a1 smallest possible value of n is
(A) 4 (B) 6
(C) 8 (D) 12

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2n+1
16. Let us consider f(x) = 1 x + [x] [1 x] and g(x) = 1 x lim (cos x)
n
(A) the graph of f(x) is a straight line passing through origin.
The graph of g(x) is straight line passing through origin
(B) the graph of both f(x) and g(x) is a straight line not passing through origin
(C) the graph of f(x) is a straight line passing through origin whereas for that of g(x) is not passing
through origin
(D) will not be a straight line for either of the curve

17. Let f(x) = [x] + {x}2 : R R, then area of figure bounded by y = f 1(x), y = 0 between the ordinates
1
x= and x = 5 is (where [.] and {.} represent the greatest integers and fractional part functions
2
respectively)
1 40
(A) 40 2 1 (B)
3 2 3
40 1
(C) (D) 40 2 1
3 2 3 2

sec 4 sec 4 k
18. What is the minimum value of 2
2
for , B , k z?
tan tan 2
(A) 2 (B) 4
(C) 8 (D) 16

19. Consider straight line ax + by = c where abc R+ and a, b, c are distinct. This line meets the
coordinate axes at P and Q respectively. If area of OPQ, O being origin does not depend upon
a, b and c then
(A) a, b, c are in GP (B) a, c b are in GP
(C) a, b, c are in AP (D) a, c, b are in AP

20. In a ABC, a, c, A are fixed. The third side may have two possible values say b1 and b2. It is
c
given that b2 = 2b1. Then find the value of 1 8 sin2 A
a
(A) 1 (B) 2
(C) 3 (D) 4

21. If A is an invertible idempotent matrix and B = 7A7 + 6A6 + 5A5 + .. + A then |B| is equal to
(A) 7 (B) 14
(C) 28 (D) 35

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22. Find the number of triangles formed by the lines represented by x 3 x2 x 2 = 0 and x y2 + 2 x y
+ 4 x 2 y2 4 y 8 = 0
(A) one (B) two
(C) three (D) none of these

3 2
23. Let f(x) = sin x + sin x for <x< and > 0. The interval in which should lie in order
2 2
that f(x) has exactly one minimum and one maximum is
3
(A) 0 < < (B) 0 < < 1
2
(C) 1 < < 2 (D) 1 < < 3/2

24. If P(x) = ax 2 + bx + c leaves a remainder of 4 when divided by x, a remainder of 3 when divided

by (x + 1) and a remainder of 1 when divided by (x 1) then P(2) is
(A) 3 (B) 6
(C) 3 (D) 6

25. Let L1 : 2x + 3y + p 3 = 0 and L2 : 2x + 3y + p + 3 = 0 be two lines and p z.

Let C : x2 + y2 + 6x + 10y + 30 = 0. If it is given that at least one of the lines L1 L2 is chord of C the
probability that both are chords of C is
2 3
(A) (B)
7 7
4 5
(C) (D)
11 11

n 3n 1
26. If lim 2
then the range of x is (n N)
n n x 2 n 3 n1 n
3 3
(A) [2, 5] (B) (1, 5)
(C) (1, 5) (D) (, )

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27. The coordinate plane is constrained by the hyperbola x2 y2 = 4 so that only the region exterior of
the hyperbola is accessible what are the possible values of the ordinates of the center of a
circular disc of radius 4 where centres lies on the yaxis?

(A) , 2 2,
(B) , 2 3 2 3,
(C) , 6 6,
(D) , 2 6 2 6,

28. If x2 sin x 2 x 2 2 sin x then
4 4
(A) x 0, 2 (B) x 2, 2
(C) x R (D) x 2, 0

2
29. A plane P makes intercepts with the axes, the sum of whose square is a constant equal to K .
The foot of the perpendicular from the origin P is (, , ) the value of in terms of K is
K K
(A) (B)
3 3 2 3
K 2K
(C) (D)
3 3

x2 e
30. Let f x In 2 then the range of g x sin f x cos f x is
x 1

(A) 1, 23/4 (B) 1, 21/2
(C) 1, 22 (D) (1, 2)

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