Fiitjee Aits

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 – V)

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 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

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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 block of metal weighing 2 kg is resting on a frictionless plane. It is

struck by a jet releasing water at a rate of 1 kg/s and at a speed of 5 m/s. The initial acceleration of the block is

Block2kg

(A) 53

m/s2 (B) 254

m/s2

(C) 258

m/s2 (D) 52

m/s2

2. A body is fired from point P and strikes at Q inside a smooth

circular wall as shown in the figure. It rebounds to point S (diametrically opposite to P). The coefficient of restitution will be

(A) cot α (B) 1 (C) tan α (D) tan2α P

Q

Sα

3. A wedge of mass M resting on a horizontal frictionless surface is

given a force F in the horizontal direction. The net horizontal force acting on the shaded portion of the wedge is

(A) F (B) 3F

(C) 6F

(D) zero

h3

h3 hF

h

Rough work



4

4. Conductor ABC which consist of two rigid quarter circular wires of radius

R, lies in X-Y plane and carries current I as shown. A uniform magnetic field B is switched on in the region that exert force F = 2 IRB0 k on conductor ABC. B can be

(A) )i(B

−20 (B) j

B20

(C) )ji(2

B0 + (D) both (B) and (C)

O

Y

C

R

R

B R

X I

I A

R

5. In the fusion reaction 2 2 3 1

1 1 2 0H H He n,+ → + the masses of deuteron, helium and neutron expressed in amu are 2.015, 3.017 and 1.009 respectively. If 1 kg of deuterium undergoes complete fusion, find the amount of total energy release (D) 1 amu = 931.5 MeV/c2

(A) 136.02 10 J≈ × (B) 135.6 10 J≈ × (C) 139.0 10 J≈ × (D) 130.9 10 J≈ × 6. The compound unstable nucleus 236

92 U often decays in accordance with the following reaction

236 140 9492 54 38U Xe Sr→ + + other particles

In the nuclear reaction presented above, the other particles might be (A) an alpha particle, which consists of two protons and two neutrons (B) two protons (C) one proton and one neutron (D) two neutrons 7. A point object is placed at a distance of 1000 mm from a concave mirror of focal length

400 mm. If the object is moved towards the mirror by 200 mm, the image moves by a distance of (A) 133.3 mm towards mirror (B) 133.3 mm away from the mirror (C) 30 mm towards mirror (D) 30 mm away from the mirror

Rough work



5

8. If an object is placed 20 cm in front of a half thin convex lens of

focal length 10 cm, as shown in figure, then co-ordinate of image taking object position as origin 20cm

2mm 0

y

x (A) [20 cm, 0.2 cm] (B) [40 cm, 0.4 cm] (C) [40 cm, − 0.2 cm] (D) [20 cm, 0.4 cm] 9. When an elastic material with Young’s modulus Y subjected to a stretching stress X, the elastic

energy stored per unit volume of the material is

(A) 2XY

(B) 2Y

2X

(C) 2X Y2

(D) 2X

2Y

10. From a disc of mass 2 kg and radius 4m a small disc of radius 1 m

with center O' is extracted. The new moment of inertia. about an axis passing through O perpendicular to plane of disc is

(A) 16 kg m2 (B) 12 kg m2

(C) 2255 kg m16

(D) 2247 kg m16

OO’

4m

2m

11. A wire frame AOPQB, lying in the horizontal plane, is free to rotate about a

vertical axis passing through center C of the same circle and ⊥ to plane of AOPQB. The mass M of the frame is uniformly distributed over its whole length. The moment of inertia of the frame about this axis, is (OA = QB = r and CP = r the radius of semicircular part)

(A) 2 14 3Mr3 6

+ π π +

(B) 2 rMr2r

π + π +

(C) 2 3Mr4

π

(D) 21 Mr2

O

A

P

B

Q

rC

r

r

Rough work



6

12. A block of mass m is suspended from a spring. Its frequency of oscillation is f .

The spring is cut into two identical halves and the same block is suspended from one of the two pieces of the spring, such that it just touches the other spring below in its equilibrium position. The frequency of small oscillations of the mass will be

(A) f (B) 2f

(C) 2f (D) 2(2f – 2f )

m

13. The angle between the directions of the particle velocity and the wave velocity, in a transverse

wave, is

(A) 4π

(B) zero

(C) 2π

(D) 2

π.

14. A block of 1 kg is kept on a rough surface of an elevator moving up with

constant velocity of 5 m/s. In 10 second work done by normal reaction (no sliding on incline surface)

(i) from ground frame is 320 J (ii) is equal to work done by friction force in elevator frame (iii) is equal to work done by friction in ground frame correct answer is

1kg

37º

5m/s

(A) (i) (B) (ii), (iii) (C) (i), (ii) (D) only (iii). 15. Average power delivered by an AC source when a resistor of resistance R, an inductor of

inductance L and a capacitor of capacitance C are connected in series is (choose the most appropriate option)

(A) minimum when the frequency of the source is 1

2π1LC

.

(B) maximum when the frequency of the source is 1

2π1LC

.

(C) zero when there is no resistor in the circuit. (D) both (B) & (C)

Rough work



7

16. A uniform rod AB of length 1 m is placed at edge of a smooth table as

shown in figure. It is hit horizontally at point B from left. If the magnitude of the displacement of the centre of mass in 1 s is 5 2 m, the angular speed of rod is

(A) 30 rad/s (B) 20 rad/s (C) 10 rad/s (D) 5 rad/s

B

TableA

17. Four resistors are connected in a square formation. Two batteries of emf, E and 2E are connected along the diagonals as shown in the figure. The current flowing in the resistor (R1) is (R1 = R)

(A) 43ER

(B) ER

(C) 32ER

(D) 3ER

R R

RR1

2E

E

18. The rate at which heat is dissipated in the 1.5 Ω resistance in the previous question is given by (A) 37.5 W (B) (37.5 W) . e-t/2, where t is the time in µS (C) (37.5 W) e-t, where t is time in µS (D) (50 W) e-t/2, where t is the time in µS 19. An electric dipole with dipole moment ˆ ˆp (3i 4 j)= + C-m, is kept in electric field ˆE 0.4kN / Ci= .

What is the torque acting on it & the potential energy of the dipole ? (A) ˆ1600(N m)k, 1200J× − (B) ˆ1600(N m)k, 1200J− ×

(C) ˆ1600(N m)k, 1200J− × − (D) ˆ1600(N m)k, 1200J× 20. In the arrangement of a pair of parallel plates having separation 1 cm as

shown. What is electric field in the region between the plates ? (A) 15 kN/C towards right (B) 15 kN/C towards left (C) 25 kN/C towards right (D) 25 kN/C towards left

1cm

-50V +200V

Rough work



8

21. In the figure ab and cd are two long conducting wires kept

parallel to each other at a separation in a uniform time varying magnetic field. The ends a and c are connected together by a resistor of resistance R. The magnetic induction B perpendicular to the plane of the wires varies with time according to the relation B = B0t, where B0 is a positive constant with proper unit and t is the time in second. A conducting wire PQ placed on the wires ab and cd and dragged on the wire with constant speed. v along

R F

a b

c d

P

Q

the length of the wires by applying a constant force F. Find the value of F in terms of the other given parameters. At time t = 0, PQ is very close to ac. Ignore any resistance (electrical as well as mechanical) other than R.

(A) 2 2 20B vt

R (B)

2 2 202B vtR

(C) 2 2 204B vtR

(D) 2 2 208B vtR

.

22. A small air bubble of radius r is found to form at a depth of H from the open surface of the liquid

contained in a beaker. If S is the surface tension and ρ , the density of the liquid and po, the atmospheric pressure the pressure inside the bubble is

(A) 4Sr

+ ρ gH + po (B) 2Sr

– ρ gH + po

(C) 4Sr

– ρ gH + po (D) 2Sr

+ ρ gH + po

23. Three identical spheres having mass M and radius R each are kept in contact at

rest as shown in figure. On a frictionless horizontal plane. The net force acting on any one sphere is

(A) 2

2GMR

(B) 2

2GM34R

(C) 2

2GM2R

(D) zero.

(where G is universal gravitational constant)

Rough work



9

24. Two adiabatic vessels, each containing the same mass m of water but at different temperatures, are connected by a rod of length L, cross–section A, and thermal conductivity K. the ends of the rod are inserted into the vessels, while the rest of the rod is insulated so that there is negligible loss of heat into the atmosphere. The specific heat capacity of water is s, while that of the rod is negligible.

The temperature difference between the two vessels reduces to 1e

of its original value after a

time, ∆t. The thermal conductivity (K) of the rod may be expressed by

(A) msLA t∆

(B) emsLA t∆

(C) msL2eA t∆

(D) msL2A t∆

.

25. The heat (Q) supplied to a solid, which is otherwise thermally isolated from its surroundings, is

plotted as a function of its absolute temperature, θ. It is found that they are related by the equation,

Q = aθ2 + bθ4. (a, b are constants). The heat capacity of the solid is given by

(A) 3 5

a b3 5θ θ

+ (B) 3a bθ + θ

(C) 3

a b3 5θ θ

+ (D) 2aθ + 4bθ3.

26. The K.E. (K) of a particle moving along a circle of radius R depends on the distance covered

s as 2=K as . The force acting on particle is

(A) 22as

R (B) 1/ 22

2

1

+

as

sR

(C) 1/ 22

22 1

+

sasR

(D) none of these.

Rough work



10

27. Two particles are simultaneously projected with the same speed in the same vertical plane, but perpendicular to each other, in a uniform gravitational field. Their times of flight are 1T , 2T while the horizontal ranges are 1R , 2R . Then (choose one option)

(A) 1 2R R= (B) 1 2 1 2R R gTT+ =

(C) 1 2

1 2

R RT T

= (D) both (A) and (B) are true

28. A block of mass 3

10kg is placed on a rough horizontal surface as

shown in the figure. A force of 1 N is applied at on end of the block and the block remains stationary. The normal force exerted by the surface on the block acts (g = 10 m/s2)

(A) through the centre of mass of the block. (B) through point A. (C) through point B. (D) through the point at a distance 5 cm. from A.

30ºF = 1N

20 c

m

20 cmB A

29. A uniform conducting rectangular loop of sides , b and mass

m carrying current i is hanging horizontally with the help of two vertical strings. There exists a uniform horizontal magnetic field B which is parallel to the longer side of loop. The value of tension which is least is

(A) mg Bb2− (B) mg Bb

2+

b

B

(C) mg 2iBb2

− (D) mg 2Bb2+ .

30. The mean lives of a radio-active substances are 1620 years and 405 years for α-emission and β-

emission respectively. Find out the time during which three fourth of a sample will decay if it is decaying both by α-emission and β-emission simultaneously

(A) 324 years (B) 449 years (C) 480 years (D) 425 years

Rough work



11

CChheemmiissttrryy PART – II

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 an atomic BCC, what fraction of edge is not covered by atoms: (A) 0.144 (B) 0.130 (C) 0.279 (D) 0.134 2. A complex [CoA6]3+ red coloured while [CoB6]3+ is green coloured, which of the following is correct

statement? (A) Ligand A is producing larger crystal field splitting (B) [CoB6]3+ is easily oxidized (C) Both are equally oxidized (D) Ligand and B is producing larger crystal field splitting 3. A solution of H2O2 labelled as ’20 V’ was left open. Due to this, some H2O2 decomposed and

volume strength of the solution decreased. To determined the new volume strength of the H2O2 solution, 10 ml of the solution was taken and it was diluted to 100 ml. 10 ml of this diluted solution required 25 ml of 0.0245 M KMnO4 solution for titration under acidic condition. Calculate the new volume strength of H2O2 solution?

(A) 17.15 V (B) 18.4 V (C) 19.4 V (D) 16.5 V 4. A white crystalline substance dissolves in water. On passing H2S in this solution, a black

precipitate is obtained. The black ppt. dissolves completely in hot HNO3. On adding a few drops of concentrated H2SO4, a white precipitate is obtained which is soluble in ammonium acetate. The white precipitate is that of:

(A) BaSO4 (B) SrSO4 (C) PbSO4 (D) Ag2SO4 5. When 1 mole of A (g) is introduced in a closed 1 L vessel maintained at constant temperature.

The following equilibria are established: ( ) ( ) ( )

1cA g B g 2C g ; K ?+ =

( ) ( ) ( )2cC g 2D g 3B g ; K ?+ =

The pressure at equilibrium is 136

times the initial pressure.

Calculate 1 2c cK & K if

[ ][ ]

eq.

eq.

C 4A 9

= .

(A) 0.14 and 0.14 (B) 0.28 and 0.18 (C) 0.19 and 0.16 (D) 0.11 and 0.14

Rough Work



12

6. For the reaction, ( ) ( )2 4 2N O g 2NO g , if percentage dissociation of N2O4 are 20%, 45%, 65% and 80% then the sequence of observed densities at given extent of dissociation are:

(A) (d20 = d45) > (d65 = d80) (B) d80 > d65 > d45 > d20

(C) d20 > d45 > d65 > d80 (D) d20 = d45 = d65 = d80 7. One mole of naphthalene was burnt in oxygen gas at constant volume to give carbondioxide gas

and liquid water at 25oC. The heat evolved was found to be 5138.8 kJ. Calculate the enthalpy of reaction at constant pressure:

(A) - 5143.8 kJ (B) +6538.3 kJ (C) - 4148 kJ (D) - 3398 kJ 8. For the set of reactions: (i) 1

1

K

KA B C

−+ (ii) 2KC B D+ →

[ ][ ] [ ] [ ][ ]1 1 2K A B K C K C B−− − is equal to:

(A) [ ]d Adt

− (B) [ ]d Bdt

−

(C) [ ]d Cdt

(D) [ ]d Ddt

9.

( )+ →2 41. H SO2 2. conc.HCl2

CH OEt Product.

OH

NO2 (A)

Cl

OH

NO2

(B)

CH2Cl

OH

NO2

(C)

CH2Cl

NO2

(D)

OH

CH2Cl

Rough Work



13

10.

CH

O

OMe

2

KCN,EtOHH O Product is:→

(A) C HC

O OH

OMe

MeO

(B) C HC

O OH

MeO (C)

C HC

O OH

OMe

(D) C HC

O OH

OMe

OMe

11. ( )3 2CH CO O

HX Y+→ →

O

O Li

X and Y are: (A)

[ ]X

O O;

OOCOCH3

[ ]Y

(B)

[ ]X

O

OH;

OOCOCH3

[ ]Y

(C)

[ ]X

OO ;

O

OCH3

O[ ]Y

(D)

[ ]X

OOH

;

OOCOCH3

[ ]Y

Rough Work



14

12. For the cell: Pt | H2(g) | sol X || KCl (saturated)| Hg2Cl2(s) | Hg|Pt The observed emf at 25oC was 612 mV. When solution X was replaced by a standard phosphate

buffer whose assigned pH is 6.86, the emf was 741 mV. Find pH of the solution X. (A) 4.68 (B) 5.13 (C) 6.12 (D) 4.24 13. The pKa of acetylsalicylic acid (aspirin) is 3.5. The pH of gastric juice in human stomach is about

2 – 3 and the pH in the small instestine is about 8. Aspirin will be: (A) Unionised in the small intestine and the stomach (B) Completely ionized in the small intestine and in the stomach (C) Ionised in the stomach and almost unionized in the small intestine (D) Ionised in the small intestine and almost unionized in the stomach 14. Calculate wavelength of He atom whose speed is equal to the r.m.s. at 20oC? (A) 7.64 × 10-12 m (B) 5.28 × 10-11 m (C) 7.38 × 10-11 m (D) 2.28 × 10-10 m 15. Which one of the following ylides will be most stable and least reactive for nucleophilic addition

reaction with aldehyde or ketones?

(A) ( )( ) ( )

6 5 33C H P CH

+ −

− − (B) ( )( ) ( )

6 5 33C H P CH CH

+ −

− −

(C) ( ) ( )

3 2 3Ph P CH CH CH+ −

− − − (D) ( ) ( )

3 2 5Ph P CH COOC H+ −

− − 16. In the Perkin reaction which one of the following intermediates gives compound (I):

H5C6 CH CH C O C CH3

O O

(I)

(A)

H5C6 CH CH2 C O C CH3

O OO

(B)

H5C6 HC

CH2

O C

O

CH3

COO

(C)

H5C6 HC

CH2

O C

O

CH2

COOH

(D)H5C6 HC

CH2

O COCH3

C

O

O C

O

CH3

Rough Work



15

17. When the deuterium labeled compound (I) is subjected to dehydrohalogenation:

H BrCH3 D

H H2 5C H O Na X

− +

→

The only product X is: (A) Methylcyclohexene (B) 3-Methylcyclohexene (C) Cycloheptene (D) 2-Methylcyclohexene 18. Which of the following is/are non-reducing sugars? (A) Fructose (B) Glucose (C) Sucrose (D) Lactose 19. Which of the following statement is not true? (A) O-F bond length in OF2 is less than O-F bond length in O2F2. (B) In 3HCO− , all C – O bond length are not identical. (C) In diborane, two different B – H bond lengths are observed although the hybridization of both

boron atoms are same. (D) In hydrazine, the N – N bond length is larger than normal N – N bond length. 20. How many stereoisomers are possible for the following molecule

H

CH3

CH CH CHCOOH

(A) 2 (B) 3 (C) 4 (D) 6 21. The vapour pressure of water at 20oC is 17.54 mm Hg. What will be the vapour pressure of the

water, in the apparatus shown after the piston is lowered, decreasing the volume of the gas above the liquid to one half of its initial volume (assume T constant):

(A) 8.77 mm Hg (B) 17.54 mm Hg (C) 35.08 mm Hg (D) Between 8.77 and 17.54 mm Hg. Water

vapourLiquid water

Rough Work



16

22. Which one of the following is most stable?

(A) H

HH

HH Br

NH2

(B) H

HH

HH Br

NH2

(C) H

HH

HH Br

NH2

(D)H

HH

H

NH2

Br

H

23. The acid strength order is:

II

COOH

(a)

I

COOH

I(b)

COOH

I

I(c)

COOH

II(d)

(A) (a) > (b) > (c) > (d) (B) (a) > (c) > (b) > (d) (C) (a) > (b) > (d) > (c) (D) (b) > (a) > (d) > (c) 24. In a space shuttle, the CO2 output per astronaut has been estimated as 44 g per hour. An

experimental catalytic converter reduces CO2 at a rate of 600 mL (STP) per min into H2O. What fraction of the time would such a converter have to operate in order to keep up the CO2 output of one astronaut?

(A) 0.622 (B) 0.782 (C) 0.382 (D) 0.411 25. The correct statements are: The planar shape of N(SiH3)3 is explained by the 1. Type of hybrid orbitals of nitrogen. 2. Additional p dπ − π overlap along the N-Si bond. 3. Higher electronegativity of nitrogen. (A) 1, 2, and 3 (B) 1 and 2 (C) 2 and 3 (D) 1 and 3 26. The compound (X) obtained in the following reaction is most likely to be:

( ) ( )3Li/NH X→

(A)

(B)

(C) (D)CH

Rough Work



17

27. In the acid catalysed dehydration of

OH

the product(s) formed is/are:

CH2

(I)

CH3

(II) (III)

CH3

(IV)

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

28.

CCH3O

PhO

( )( )

4a NaIOb H ,

Product is:+ ∆→

(A)

O

(B) O

HOOC Ph

(C) O

COOH

(D)

Ph

COOH

OH

29. Consider the following statement: S1 : Fluorine does not form any polyhalide as it has low F – F bond energy. S2 : The chlorine has the most negative electron gain enthalpy. S3 : The first ionization potential of N is greater than O. Which of the above statements are correct? (A) S1, S2 and S3 (B) S1 and S2 (C) S1 and S3 (D) S2 and S3 30. ( )2

4MnO 1mole− in neutral aqueous medium, disproportionates to:

(A) 23

mole of 4MnO− and 13

mole of MnO2 (B) 13

mole of 4MnO− and 23

mole of MnO2

(C) 13

mole of Mn2O7 and 13

mole of MnO2 (D) 23

mole of Mn2O7 and 13

mole of MnO2

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. The curve y = 2ax bx+ has minimum at (2, –12) then the values of a, b are (A) 3, –12 (B) –3, –12 (C) –3, 12 (D) 3, 12 2. The point on the curve 2y x 4x 3= + + which is nearest from the line y = 3x + 2 is

(A) 1 5,2 4

(B) 1 5,2 4

−

(C) 52,3

−

(D) 52,3

3. The sides of the greatest rectangle that can be inscribed in the ellipse 2 2

2 2x y 1a b

+ = are

(A) a 2 , b 2 (B) a , b

(C) a, b (D) a b,2 2

4. The maximum area of the triangle with vertices at (a, 0), (a cos θ, b sin θ) and (a cos θ, –b sin θ)

is

(A) 3 3 ab4

(B) 3 ab

(C) 3ab4

(D) 3 3ab

Rough work



19

5. If the perimeter of a sector of a circle is 16 cms then it’s maximum area is (A) 16 sq. cms (B) 32 sq. cms (C) 48 sq. cms (D) 64 sq. cms

6. The shortest distance of the point (0, c) from the curve 2y x= , where c ≥ 12

is

(A) c 4− (B) 1c4

−

(C) 4c 1− (D) c 1−

7. The function ( ) sinxf xx

= is

(A) increasing in 0,2π

(B) decreasing in 0,2π

(C) stationary at x = π/2 (D) both increasing and decreasing at x = π/2

8. If xy x tan2

= , then dydx

is equal to

(A) 1 sin xcos x+ (B) x cos x

sinx+

(C) x sin x1 cos x

++

(D) 1 cos xx sin x++

9. If f is a differentiable function with ( )f 1 8= , ( ) 1f 18

′ = . If f is invertible and 1g f −= , then

(A) ( )g 1 8′ = (B) ( ) 1g 18

′ =

(C) ( )g 8 8′ = (D) ( ) 1g 88

′ =

Rough work



20

10. Let ( ) xf x1 x tan x

=+

, then ( )f x has

(A) one point of minima in 0,2π

(B) one point of maxima in 0,2π

(C) no critical point in 0,2π

(D) exactly one critical point in ,2 2

−π π

11. The critical points of ( ) ( ) ( )2 / 3f x x 2 2x 1= − + is(are) (A) 1 only (B) 1, 2 (C) 2 only (D) 0, 1, 2 12. The height of the right circular cylinder of maximum volume that can be inscribed in a sphere of

radius ‘a’ is

(A) a3

(B) 2a3

(C) 2a

(D) a 2

13. The stationary point of 2 250y xx

= + is

(A) (1, 51) (B) (5, 1) (C) (5, 25) (D) (5, 75)

14. The function ( ) 2f x 2x 4x 1= − + in the interval 93,2

has minimum value

(A) –1 (B) 7 (C) 1 (D) 17

15. The turning points of ( ) 1 1f x sin x sin2x sin3x2 3

= + + in (0, π) are at

(A) x = π/4 only (B) x = π/4, π/3 (C) x = π/4, 2π/3 only (D) x = π/4, 2π/3, 3π/4

Rough work



21

16. If a > b, then the maximum and minimum values of 2 2asin x bcos x+ is (A) a, b (B) b, a

(C) 2 2a b+ , – 2 2a b+ (D) ab , – ab

17. The greatest value of ( ) 22

2f x 2xx

= + , for [ ) ( ]x 2,0 0,2∈ − ∪ and ( )f 0 1= is

(A) 172

(B) 1

(C) 0 (D) 112

18. If x and y are strictly positive such that x + y = 1, then the minimum value of x log x + y log y is (A) log 2 (B) –log 2 (C) 2 log 2 (D) 0 19. The minimum value of E(x, y) = cos x + cos y + 2 cos (x + y) where x, y ∈ R is equal to

(A) 92

− (B) 94

−

(C) 32

− (D) 34

−

20. If A, B, C are interior angles of triangle ABC such that (cos A + cos B + cos C)2 + (sin A + sin B + sin C)2 = 9 then number of possible triangle is (A) 0 (B) 1 (C) 3 (D) infinite 21. Let a, b, c ∈ R and 1 be a root of the equation ax2 + bx + c = 0, then the equation 4ax2 + 3bx + 2c = 0 has (A) imaginary roots (B) real and equal roots (C) real and unequal roots (D) rational roots

Rough work



22

22. Sandra is writing her Christmas list. If she writes n items in her list, the probability of getting each

item is 12n

for example if she writes 3 items, she will get each item with a probability of 16

. If

Sandra write infinitely many gifts on her list then the probability of her getting no gifts, is

(A) e–1 (B) 12e

(C) 1/ 2e− (D) 12

23. If the roots of cubic equation x3 – 6x2 + β1x – β2 = 0 are in A.P. with positive integral common

difference, them the maximum value of (β1 + β2), is (A) 17 (B) 18 (C) 19 (D) 20 24. Let α1 and α2 he two real values of α for which the numbers 2α2, α4, 24 taken in that order from

an arithmetic progression if β1 and β2 are two real values of β for which the numbers 1, β2, 6 – β2 taken in that order form a geometric progression, then the value of ( )2 2 2 2

1 2 1 2α + α + β + β is equal to (A) 10 (B) 11 (C) 12 (D) 13 25. Let f(x) = x2 + ax + b cos x, a being an integer and b is a real number. Find the number of ordered

pairs (a, b) for which the equations f(x) = 0 and f(f(x)) ≡ 0 have the same (non–empty) set of real roots

(A) 1 (B) 2 (C) 3 (D) 4 26. Number of integral values of ‘a’ for which every solution of the inequality x2 + 1 > 0 is also the

solution of the inequality (a – 1)x2 – (a + |a – 1| + 2)x + 1 ≥ 0, is (A) 0 (B) 1 (C) 2 (D) 3

Rough work



23

27. The expression ( ) ( ) ( ) ( ) ( ) ( )2 2 2 22 210 10 10 10 10 100 1 2 8 9 10C C C .......... C C C− + − + − + is equal to

(A) 0 (B) ( )2105C

(C) 105C− (D) 9

52 C⋅ 28. If C0, C1, ….., C2012 are binomial coefficients in the expansion of (1 + x)2012 and a0, a1, ….., a2012

are real numbers in arithmetic progression then value of a0C0 – a1C1 + a2C2 – a3C3 + ….. + a2012C2012, is

(A) 1 (B) 2012 (C) –1 (D) 0

29. If 3tan94

θ = (where 0 < θ < 18π ) then the value of (3 cosec 3θ – 4 sec 3θ) is equal to

(A) 3 (B) 4 (C) 5 (D) 10 30. Let n be the number of ordered quadruples (x1, x2, x3, x4) of positive odd integers that satisfy

4

ii 1

x 98=

=∑ , then the value of n100

is equal to

(A) 144 (B) 169 (C) 196 (D) 225

Rough work

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