FIITJEE JEE (Main), 2013 oom Program 4 in Top 10, 10 in ...fiitjee.com/AITSJEE2013/Concept Recapitulation Test III/Main/Paper... · A comet moves around the sun in an elliptical orbit.

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CONCEPT RECAPITULATION TEST (Set – III)

Time Allotted: 3 Hours Maximum Marks: 360 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 30) contains 30 multiple choice questions which have only one correct answer. Each question carries +4 marks for correct answer and – 1 mark for wrong answer.

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AITS-CRT(Set-III)-PCM-JEE(Main)/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. If a new planet is discovered rotating around the Sun with the orbital radius double that of earth,

then what will be its time period (in earth’s days)? (A) 1032 (B) 129 (C) 2920 (D) 45. 2. The air column in a pipe closed at one end is made to vibrate in its second overtone by a tuning

fork of frequency 440 Hz. The speed of sound in air is 330 m/s. End corrections may be neglected. Let P0 denote the mean pressure at any point in the pipe and ∆P0 the maximum amplitude of pressure variation. The maximum pressure at the closed end of the pipe is

(A) P P0+ ∆0 (B) P

P 0

2∆

+0

(C) P P02+ ∆0 (D) P02 3. A comet moves around the sun in an elliptical orbit. It is closest to the sun at a distance 1d and its

corresponding velocity is 1v , and if it is farthest from the sun at a distance 2d , then the corresponding velocity is

(A) 12

1

⋅v dd

(B) 11

2

⋅dvd

(C) 21

1

⋅dvd

(D) 11

2

⋅dvd

.

4. In a series LCR circuit the resistance 24R = Ω while the reactance corresponding to L and C

are 2Ω and 28Ω respectively at a certain frequency. The total impedance, if the frequency is doubled, is

(A) 30Ω (B) 28.1Ω (C) 34Ω (D) 26Ω .

Rough work



4

5. A smooth circular table is surrounded by a rim whose interior is vertical. A ball is projected along the table from a point on the rim in a direction making an angle θ to the radius through the point and returns to the point of projection after two impacts. If e be the coefficient of restitution, then

(A) 3

2etan

1 e e

θ = + +

(B) 2etan

1 eθ =

+

(C) 1tane

θ = (D) tan eθ =

6. The resistor in which the maximum heat is produced is given by (A) 2Ω (B) 3Ω (C) 4Ω (D) 12Ω

4 Ω 3 Ω

6 Ω 2 Ω

12 Ω

7. A parallel combination of two resistors of 1Ω each, is connected in series with a 1.5Ω resistor,

and two uncharged capacitances of 1.5µF and 3µF , also in series. The combination is connected to a 10 V battery. The initial current flowing in the circuit is (assume that the capacitors are initially uncharged)

(A) 5 A (B) 0 A (C) 0.3 A (D) 0.4 A. 8. The current in a coil changes from +5A to +2A in 0.03 s inducing a voltage of 8 V across it. The

initial energy stored in the coil was (A) 2 J (B) 1 J (C) 0.4 J (D) none of these 9. A sound wave of frequency 250 Hz covers a distance of 2000 meters in 10 seconds between

points A and B. Then the number of waves between A and B are (A) 250 (B) 100 (C) 2500 (D) 500

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5

10. Emf induced in coil by a change in the current from 5A to 10A in 0.1s is 10V. Then the change in the energy of the magnetic field in the coil is

(A) 2.5 J (B) 5.0 J (C) 7.5 J (D) 10 J 11. A simple pendulum consisting of a mass M attached to a string

of length L is released from rest at an angle α. A pin is located at a distance below the pivot point. When the pendulum swings down, the string hits the pin as shown in the figure. The maximum angle θ which string makes with the vertical after hitting the pin is

αL

θ

(A) 1 LcoscosL

− α + +

(B) 1 LcoscosL

− α + −

(C) 1 LcoscosL

− α − −

(D) 1 LcoscosL

− α − +

12. Water rises in a vertical capillary tube to a height 2cm. In another capillary tube whose radius is

one-third of it and which is inclined at 60° with the vertical, the water will occupy a length equal to (A) 2 cm (B) 6 cm (C) 8 cm (D) 12 cm 13. A block of wood of relative density 0.5 is placed 10 m inside a vessel containing water. The

vessel is accelerated upward with an acceleration of 10 m/s2. If the block is released at some instant then the time taken by the block to reach the surface of water from that instant is (consider g = 10 m/s2)

(A) 0.5 s (B) 1 s (C) 2 s (D) 4 s 14. (M – 1) divisions on the main scale of a vernier calipers coincide with M divisions on the vernier

scale. If each division on the main scale is of b unit, then least count of instrument is

(A) bM 1+

(B) bM

(C) bM 1−

(D) 1M

.

Rough work



6

15. A body cools from 85°C to 80°C in 10 minutes, when the temperature of the surrounding is 32.5°C. How much time will it take to cool down by the same amount (i.e. from 85°C to 80°C) if the temperature of the surroundings is 57.5°C ? Assume that Newton’s law of cooling is valid.

(A) 20 min (B) 5 min

(C) 203

min (D) 40 min

16. An ideal gas whose , and internal energy U at absolute zero temp. is equal to zero

= γ

p

v

CC

undergoes a reversible adiabatic compression. If , , .U p V T represent the internal energy, pressure, volume and temperature respectively of the ideal gas, then

(A) const.UV γ = (B) const.Upγ =

(C) 1

1 const.VU γ− = (D) 1 const.TU γ− = 17. An automobile travelling at 60 km/hr on a highway stops over a distance of 20 m after brakes are

applied. Traffic is moving on this highway at 72 km/hr. What is the minimum distance at which one driver should follow the other, if a driver requires a minimum of 1s to decide and apply the brakes?

(A) 20 m (B) 28.8 m (C) 48.8 m (D) 8.8 m. 18. A particle is projected from ground with speed u at angle θ with the horizontal. Radius of

curvature of the trajectory of the particle (A) is not minimum at highest point (B) is minimum at the point of projection (C) is same at all points

(D) varies from 2u

g cosθ to

2 2u cosg

θ

where aN is the component of acceleration in the direction perpendicular to velocity. 19. A body is moving down a long inclined plane of angle of inclination θ. The coefficient of friction

between the body and the plane varies as µ = 0.1 x, where x is the distance moved down the plane. The body will have the maximum velocity when it has travelled a distance x given by

(A) x = 10 tan θ (B) x = 5 tan θ

(C) 2 cot θ (D) 10xcot

=θ

Rough work



7

20. A particle is moving on a circular path of radius R with uniform angular speed ω. The magnitude

of average velocity of particle during time 2t3

π=

ω

(A) 3 R2

ωπ

(B) 3 R2

ωπ

(C) 3 3 R2

ωπ

(D) 2 R3

ωπ

21. A charged particle starts moving along the x–axis in a magnetic field which is given by 2

0ˆB x i= β

(A) the velocity of particle will decrease (B) the velocity of particle will increase (C) the velocity of particle will remain constant (D) the velocity of particle will change. 22. A long horizontal current carrying hollow conducting cylinder having current along its length is

kept east to west, a compass needle is kept inside the such that it is free to rotate in horizontal circle then the compass needle will be in a direction (do not ignore earth’s magnetic field)

(A) 45º north of east (B) 45º south of east (C) north–south direction (D) east–west direction. 23. An electron in a hydrogen atom makes a transition from first excited state to ground state. The

equivalent current due to circulating electron (A) increases 2 times (B) increases 4 times (C) increases 8 times (D) remains the same 24. In a sample of hydrogen like atoms all of which are in ground state, a photon beam containing

photons of various energies is passed. In absorption spectrum, five dark lines are observed. The number of bright lines in the emission spectrum will be (assume that all transitions take place)

(A) 5 (B) 10 (C) 15 (D) 20

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8

25. The electron in a hydrogen atom makes a transition n1 → n2 where n1 and n2 are the principal quantum numbers of the two states. Assume the Bohr model to be valid. The time period of the electron in the initial state is eight times that in the final state. The possible value of n1 and n2 are

(A) n1 = 6, n2 = 2 (B) n1 = 8, n2 = 2 (C) n1 = 8, n2 = 1 (D) n1 = 6, n2 = 3 26. If an equilateral prism minimum deviation is 30°, what will be the angle of incidence ? (A) 30° (B) 60° (C) 45° (D) 90° 27. In Young’s double slit experiment when wavelength used is 6000 Å and the screen is 40 cm from

the slits, the fringes are 0.012 cm apart. What is the distance between the slits? (A) 0.024 cm (B) 2.4 cm (C) 0.24 cm (D) 0.2 cm 28. A cubical block of wood of specific gravity 0.5 and a chunk of concrete of specific gravity 2.5 are

fastened together. The ratio of the mass of wood to the mass of concrete which makes the combination to float with its entire volume submerged under water is:

(A) 1/5 (B) 1/3 (C) 3/5 (D) 2/5 29. A steel rod of length 1 m rests on a smooth horizontal base. If it is heated from 0ºC to 100ºC. The

longitudinal strain developed in the rod is (Coefficient of linear expansion of steel = 1.2 × 10–5/ºC) (A) 1.2 × 10–3 (B) 1.2 × 10–7 (C) zero (D) none of these. 30. A string is wrapped several times round a uniform solid cylinder and then the end of the string is

held stationary while the cylinder is released from rest with no initial motion. The acceleration of the cylinder and tension in the string will be (assume that the cylinder remains horizontal while falling)

(A) 2 mggand3 3

(B) mggand2

(C) g mgand3 3

(D) g mgand2 3

Rough work



9

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. 1 amu = 112

(mass of 1 carbon atom). If 1 amu is redefined as: 1 amu = 13

(mass of 1 carbon

atom) then, the mass of one mole of carbon atoms in grams will (A) decrease four times (B) decreases two times (C) remain unchanged (D) increase four times 2. In Balmer series of lines of hydrogen spectrum, the second line from the red end corresponds to

which one of the following inner-orbit jumps of the electron for Bohr orbits in an atom of hydrogen?

(A) 2 → 1 (B) 3 → 2 (C) 4 → 2 (D) 5 → 2 3. The successive ionization energy values for an element ‘X’ are given below: (I.E. ionization energy) 1st I.E. = 410 KJ/mol 2nd I.E. = 820 KJ/mol 3rd I.E. = 1100 KJ/mol 4th I.E. = 2500 KJ/mol 5th I.E. = 2900 KJ/mol 6th I.E. = 3400 KJ/mol Number of valence electrons in atoms of ‘X’ are (A) 1 (B) 3 (C) 4 (D) 5 4. 3 3 2 3H BO Na CO Borax others+ → +

3 3H BO NaCl Borax others+ → + Which of these sodium salts is used for large scale production of Na2B4O7? (A) Na2CO3 because of its larger abundance compared to NaCl (B) Na2CO3 because CO2 is released as one of the products other than Na2B4O7 (C) Na2CO3 because enthalpy of solution of HCl dominates over that of carbonic acid (D) Na2CO3 because it is washing soda which in turn, produces colourless borax crystals

Rough work



10

5. Which of the following gives paracetamol on acetylation? (A) OH

NH2

(B) OH

NH2

(C) NH2

(D) OH

NH2 6. Natural rubber is not used in making footwear for polar regions because (A) natural rubber becomes soft at temperature lower than 10°C (B) Natural rubber becomes brittle at temperature lower than 10°C (C) Natural rubber melts at temperature lower than 10°C (D) Natural rubber becomes stronger at temperature lower than 10°C 7. Which base is present in RNA but not in DNA? (A) Uracil (B) Cytosine (C) Guanine (D) Thymine 8. 3 2

excessNH Cl A HCl+ → +

2A H O B Y+ → + A, B and Y are respectively: (A) NH4Cl, NH4OH, HClO4 (B) NCl3, NH4OH, HOCl (C) NH4Cl, NH4Cl, HCl (D) NH4Cl, NH4OH, HClO3 9. Cerium (Z = 58) is an important member of lanthanides. Which of the following statements about

Cerium is incorrect? (A) Common oxidation states of Cerium are +3 and +4 (B) +3 oxidation state of Cerium is more stable than +4 oxidation state (C) +4 oxidation state of cerium is not known in solutions (D) Cerium (IV) acts as an oxidising agent 10. Which of the following can be used instead of limestone in Fe extraction? (A) Glauber’s Salt (B) Gypsum (C) Dolomite (D) Carnallite

Rough work



11

11. Following reaction is balanced: a KMnO4 + bH2SO4 + cH2O2 → dK2SO4 + eMnSO4 + fH2O + gO2

What is the value of a b c2

+ +

?

(A) 1 (B) 3 (C) 5 (D) 7 12. An aqueous solution of 6.3 g oxalic acid dihydrate is made upto 250 ml. 40 ml of NaOH (unknown

normalilty) completely neutralises 10 ml of oxalic acid dehydrate solution. What is the normality of NaOH?

(A) 0.1 N (B) 0.2 N (C) 0.3 N (D) 0.4 N 13. What is the hybridisation of carbon and oxygen in electronic structure of ether? (Ether is acyclic

and aliphatic) (A) sp3 and sp3 (B) sp3 and sp3 (C) sp and sp (D) sp2 and sp2 14. A gas described by Van der Waals equation: (A) Does NOT behave similar to an ideal gas in the limit of large molar volumes (B) Behaves similar to an ideal gas in the limit of large pressures (C) Is characterised by Van der Waals coefficients that are independent of the identity of the gas

and the temperature (D) Has pressure that is lower than the pressure exerted by the same gas behaving ideally 15. NH2

2 4NaNO HBFHCl, 278 K A B→ →

The compounds ‘A’ and ‘B’ respectively are: (A) Nitrobenzene and fluorobenzene (B) benzene diazonium chloride and fluorobenzene (C) Phenol and benzene (D) nitrobenzene and chlorobenzene

Rough work



12

16. The major product of the following reaction is obtained from A;

PhS Nadim ethyl formamideA

− +

→

MePhS

F

NO2 A is

(A) BrMe

F

NO2

(B) BrMe

SPh

NO2 (C) MeCl

F

NO2

(D) SPhMe

SPh

NO2 17. OH

CH2OH

2 3

3

(i) K CO(ii) CH I→

(A) OH

CH2OCH3

(B) O

CH2OH

CH3

(C) CH3

CH2OH

(D) OCH3

CH2OH 18. If the boiling point of C2H5OH (molecular weight = 46) is 78°C, What is the boiling point of diethyl

ether (mol. Wt. = 74)? (A) 373 K (B) 351 K (C) 340 K (D) 248 K

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13

19. In the depression of the freezing point experiment, it is found that 1. The vapour pressure of the solution is more than that of pure solvent 2. The vapour pressure of the solution is less than that of pure solvent 3. Solute molecule do not solidify at freezing point to form solid mixture 4. Only solvent molecules solidify at freezing point When serial number of CORRECT statements from above are added (for example, if (1), (2), (3) and (4) are CORRECT, SUM = 1 + 2 + 3 + 4 = 10, The SUM is (A) 7 (B) 8 (C) 9 (D) 10 20.

Cl

CH2CH3

O

( )Na Hg / HClA→ →

Cl

Cl

CH2CH2CH3

(A) Cl2 (B) Br2 (C) HCl (D) Cl2/FeCl3 21. In order to find out the electrodes oxidation potencial for an electrode M(s)/Mn+

(aq) with concentration at different conditions, following expressions can be used

(1) 0 n10

2.303RTE E log MnF

+ = −

(2) 0 n10

2.303RTE E log MnF

+ = +

(3) ( )( )( )

( )0 0 n

102.303 298 8.314

At 25 C, E E log Mn 96500

+ = −

(4) ( )( )( )

( )0 0 n

102.303 298 8.314

At 25 C, E E log 1/ Mn 96500

+ = −

(E0 = standard oxidation potential of electrode when serial number of CORRECT statements from above are added, for example

if (1), (2), (3) and (4) are correct, SUM = 1 + 2 + 3 + 4 = 10, the sum is (A) 4 (B) 5 (C) 6 (D) 10

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14

22.

C

CH3

CH 3

2

1. O2. Zn / H O A B→ +

2NaOH/ IA Yellow precipitate→ What is A?

(A)

CH2 C CH3

O (B)

CH2 CH2 C CH3

O

(C)

CH CH3

OH

(D)

C CH3

O

23.

C

OH

H

C CH 2H / H O Pr oduct+

→

This product gives (A) Tollens test (B) Iodoform test (C) Lucas Test (D) Fruity smell 24. The Ksp of PbCO3 and MgCO3 are 1.5 × 10-15 and 1 × 10-15 respectively at 298 K. The

concentration of Pb2+ ions in a saturated solution containing MgCO3 and PbCO3 is (A) 1.5 × 10-4 M (B) 3 × 10-8 M (C) 2 × 10-8 M (D) 2.5 × 10-8 M 25. Have a look at the following statements: (1) The coordination number of Cs+ in CsCl is 8 whereas that of Cl– is (3 × 2) = 6 (2) A unit cell of an ionic crystal shares some of its ions with other unit cells (3) The length of a unit cell in PQ (rock salt structure) is 552 pm (rP+ = 95 pm, rϕ = 181 pm) When serial number of CORRECT statements from above are added (for examples, if (1), (2) &

(3) are correct, SUM = 1 + 2 + 3 are correct, SUM = 1 + 2 + 3 = 6, then SUM is (A) 3 (B) 4 (C) 5 (D) 6 26. Cl

Br

CH CH C C C H

O3

2

1. O2. Zn / H O products→

How many chiral products are formed? (A) 8 (B) 2 (C) 16 (D) 4

Rough work



15

27. Which of the following statements is CORRECT? (A) The correct order of hybridisation of the central atom in the following species NH3, PCl5 and

BCl3 is sp3, sp3d2 and sp2 respectively (B) The hybridisation of atomic orbitals of nitrogen in 2NO+ , 3NO− and 4NH+ are sp, sp2 and sp3

respectively (C) SO2 molecule has a linear structure like CO2 molecules (D) The geometry of H2S is linear 28.

CH3 C CH2 CH3

O4

2 4

LiAlHKCNH SO A→ → CH3 C CH2 CH3

OH

CH2NH2 Which of these statements is CORRECT? (A) One of these chemical substances (reactant, A and product) gives Tollens test

(B)

C CH3

OH

CH3

CN is formed (C) The final mixture is optically active (D) The final mixture is racemic and therefore optically inactive 29.

CH3 Ph NH C Ph

O

(para) OH−

→ NH2 CH3 ( )183

1. H2. CH O H

(C) D+

+ →

(C) and (D) are (A)

COO C O CH3

O18

18

(C) (D)

(B) COO C O CH3

O

(C) (D)18

(C)

COOC O CH3

O

(D)(C)18

(D)

COOC O CH3

O

(D)(C)

30. A solution contains 10 ml, 0.1 N NaOH and 10 ml, 0.05 N H2SO4 ((2M) H2SO4 = (1N) H2SO4),

pOH of this solution is (A) less than 7 (B) 7 (C) 0 (D) greater than 7

Rough work



16

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. There are two numbers x making the value of the determinant 1 2 52 x 10 4 2x

−− equal to 86. The sum

of these two numbers, is (A) −4 (B) 5 (C) −3 (D) 9 2. A function f(x) takes a domain D onto a range R if for each y ∈ R, there is some x ∈ D for which

f(x) = y. Number of function that can be defined from the domain D = 1, 2, 3 onto the range R = 4, 5 is

(A) 5 (B) 6 (C) 7 (D) 8 3. The planes 2x − 3y + z = 4 and x + 2y − 5z = 11 intersect in a line L. Then a vector parallel to L, is (A) ˆ ˆ ˆ13i 11j 7k+ + (B) ˆ ˆ ˆ13i 11j 7k+ −

(C) ˆ ˆ ˆ13i 11j 7k− + (D) ˆ ˆ ˆi 2j 5k+ − 4. Let f(x, y) = xy2 if x and y satisfy x2 + y2 = 9, then the minimum value of f(x, y) is (A) 0 (B) 3 3− (C) 6 3− (D) 3 6−

Rough work



17

5. ( )

3

101x 0

1 3x 1 xlim1 x 1 101x→

+ − −

+ − − has the value equal to

(A) 35050

− (B) 15050

−

(C) 15050

(D) 14950

6. Let ABCDEFGHIJKL be a regular dodecagon, then the value of AB AFAF AB

+ is

(A) 4 (B) 2 3 (C) 2 2 (D) 2 7. Number of rectangles with sides parallel to the coordinate axes whose vertices are all of the form

(a, b) with a and b integers such that 0 ≤ a, b ≤ n, is (n ∈ N)

(A) ( )22n n 14+

(B) ( )2 2n 1 n4

−

(C) ( )2n 1

4+

(D) n2

8. Number of roots of the function ( )( )3

1f x 3x sinxx 1

= − ++

is

(A) 0 (B) 1 (C) 2 (D) more than 2 9. If p(x) = ax2 + 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

Rough work



18

10. Let f(x) be a function that has a continuous derivative on [a, b], f(a) and f(b) have opposite signs, and f′(x) ≠ 0 for all numbers x between a and b, (a < x < b). Number of solutions does the equation f(x) = 0 have (a < x < b)

(A) 1 (B) 0 (C) 2 (D) cannot be determined 11. If x and y are real numbers and x2 + y2 = 1, then the maximum value of (x + y)2 is (A) 3 (B) 2

(C) 32

(D) 5

12. The value of the defined integral ( )( ) ( )a 2

0

dx a 01 x 1 x

∞

>+ +∫ is

(A) 4π (B)

2π

(C) π (D) some function of a

13. Let a, b, c are non zero constant number than r

a b ccos cos cosr r rlim

b csin sinr r

→∞

− equals

(A) 2 2 2a b c

2bc+ − (B)

2 2 2c a b2bc

+ −

(C) 2 2 2b c a

2bc+ − (D) independent of a, b and c

14. A curve y = f(x) such that f″(x) = 4x at each point (x, y) on it and crosses the x-axis at (−2, 0) at an

angle of 45°. The value of f(1) is (A) −5 (B) −15

(C) 553

− (D) 353

−

Rough work



19

15. The minimum value of the function ( )2 2 2 2

sin x cos x tanx cot xf x1 cos x 1 sin x sec x 1 cosec x 1

= + + +− − − −

as x varies over all numbers in the largest possible domain of f(x) is (A) 4 (B) −2 (C) 0 (D) 2 16. A non zero polynomial with real coefficients has the property that f(x) = f′*(x).f″(x). The leading

coefficient of f(x) is

(A) 16

(B) 19

(C) 112

(D) 118

17. Let z1, z2, z3 be complex numbers such that z1 + z2 + z3 = 0 and |z1| = |z2| = |z3| = 1, then

2 2 21 2 3z z z+ + is

(A) greater than zero (B) equal to 3 (C) equal to zero (D) equal to 1 18. The latus rectum of a conic section is the width of the curve through the focus. The positive

difference between the lengths of the latus rectum of 3y = x2 + 4x − 9 and x2 + 4y2 − 6x + 16y = 24 is

(A) 12

(B) 2

(C) 32

(D) 52

19. For some non zero vector V , if the sum of V and the vector obtained from V by rotating it by

an angle 2α equals to the vector obtained from V by rotating it by α then the value of α, is

(A) 2n3π

π ± (B) n3π

π ±

(C) 22n3π

π ± (D) 2n3π

π ±

Rough work



20

20. If

2

2

a1 cosa tankcosab

a w pcosasin2

− − =

+

where k, w and p have no common factor other than 1, then

the value k2 + w2 + p2 is equal to (A) 3 (B) 4 (C) 5 (D) 6 21. If x and y are real numbers such that x2 + y2 = 8, the maximum possible value of x − y, is (A) 2 (B) 2

(C) 22

(D) 4

22. Let u(x) and v(x) are differentiable functions such that ( )( )

u x7

v x= . If

( )( )

u xp

v x′

=′

and ( )( )

u xv x

′

= q,

then p qp q

+−

has the value equal to

(A) 1 (B) 0 (C) 7 (D) −7 23. Which of the following statement is/are true concerning the general cubic f(x) = ax3 + bx2 + cx + d (a ≠ 0 and a, b, c, d ∈ R) I. The cubic always has at lest one real root II. The cubic always has exactly one point of inflection (A) only I (B) only II (C) both I and II are true (D) neither I nor II is true 24. Through the focus of the parabola y2 = 2px(p > 0) a line is drawn which intersects the curve at

A(x1, y1) and B(x2, y2). The ratio 1 2

1 2

y yx x

equals

(A) 2 (B) −1 (C) −4 (D) some function of p

Rough work



21

25. If ( ) +→∞

=− + −

n

n n 1 nn

n.3 1lim3n x 2 n.3 3

, then the range of x is (n ∈ N)

(A) [2, 5) (B) (1, 5) (C) (−1, 5) (D) (−∞, ∞)

26. f : R → R is defined as ( )2x 2mx 1 for x 0f x

mx 1 for x 0 + − ≤

= − >

If f(x) is one-one then m must lie in the interval (A) (−∞, 0) (B) (−∞, 0] (C) (0, ∞) (D) [0, ∞)

27. ( )( )35/63 3 3 i+ is an integer where i 1= − . The value of the integer is equal to

(A) 24 (B) −24 (C) −22 (D) −21 28. Let n be the smallest positive integer larger than 150 so that number n

151C is divisible by n150C

but is not equal to it. The sum of the digits of n, is (A) 5 (B) 8 (C) 9 (D) 11 29. Consider the sequence 8A + 2B, 6A + B, 4A, 2A − B, …… which term of this sequence will have

a coefficient of A which is twice the coefficients of B ? (A) 10th (B) 14th (C) 17th (D) none of these

30. Let F(x) be the primitive of 3x 2x 9

+

− w.r.t. x. If F(10) = 60 then the value of F(13), is

(A) 66 (B) 132 (C) 248 (D) 264

Rough work

FIITJEE JEE (Main), 2013 oom Program 4 in Top 10, 10 in ...fiitjee.com/AITSJEE2013/Concept Recapitulation Test III/Main/Paper... · A comet moves around the sun in an elliptical orbit.

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