fiitjee

CONCEPT RECAPITULATION TEST

(Set – III)

Time Allotted: 3 Hours Maximum Marks: 360

ƒ Please read the instructions care f u l l y. Y o u a r e a l l o t t ed 5 m i n u t es

specific ally for this purpose.

ƒ You are not allo wed to leave the Examination Hall 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.

ALL INDIA TEST SERIES

FIITJEE

JEE (Main), 2013

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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.

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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) P₀ + ∆P₀ (B) P P0 2 ∆ + 0 (C) P₀+ 2∆P₀ (D) 2P₀

3. A comet moves around the sun in an elliptical orbit. It is closest to the sun at a distance d₁ and its corresponding velocity is v₁, and if it is farthest from the sun at a distance d₂, then the corresponding velocity is (A) 1 2 1 ⋅ v d d (B) 1 1 2 ⋅d v d (C) 2 1 1 ⋅ d v d (D) 1 1 2 ⋅ d v d .

4. In a series LCR circuit the resistance R= Ω24 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.

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) tan e3 ₂ 1 e e   θ = _ _ + +   (B) 2 e tan 1 e θ = + (C) tan 1 e θ = (D) tanθ = e

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

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 A 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

α L A

θ

(A) _cos 1 L cos

L −  α +   ₊    A A (B) 1 L cos cos L −  α +   ₋    A A (C) cos 1 L cos L −  α −   ₋    A A (D) 1 L cos cos L −  α −   ₊    A A

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) b M 1+ (B) b M (C) b M 1− (D) 1 M. Rough work

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) 20

3 min (D) 40 min

16. An ideal gas _whose = γ, and internal energy U at absolute zero temp. is equal to zero_

 

p v

C C

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) UVγ =const. (B) Upγ ₌const.

(C)

1

1 _const.

VUγ− _{= (D) TU}γ−1_=const.

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 2

u

g cos

θ

to 2 2

u cos

g

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) x 10

cot =

θ Rough work

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 t 2

3 π = ω (A) 3 R 2 ω π (B) 3 R 2 ω π (C) 3 3 R 2 ω π (D) 2 R 3 ω π

21. A charged particle starts moving along the x–axis in a magnetic field which is given by BG = β₀x i2ˆ

(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

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) 2gandmg 3 3 (B) mg gand 2 (C) gandmg 3 3 (D) g mg and 2 3 Rough work

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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 = 1

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

1

3 (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. H BO_{3 3}+Na CO_{2 3}→Borax others+

3 3

H 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

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} excess NH + Cl → +A HCl 2 A 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

11. Following reaction is balanced:

a KMnO4 + bH2SO4 + cH2O2 → dK2SO4 + eMnSO4 + fH2O + gO2

What is the value of a b c 2 + +      ? (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 sp}3 _{(B) sp}3 _{and sp}3

(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 4

NaNO HBF

HCl, 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

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

PhS Na dim ethyl formamide

A− + → Me PhS F NO2 A is (A) Me Br F NO₂ (B) Me Br SPh NO₂ (C) Cl Me F NO₂ (D) Me SPh 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

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 / HCl A → → 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) E E0 2.303RTlog₁₀ Mn nF +   = − _{ } (2) 0 n 10 2.303RT E E log M nF +   = + _{ } (3)

(

)(

)(

)

(

)

0 0 n 10 2.303 298 8.314 At 25 C, E E log M n 96500 +   = − _{ } (4)

(

)(

)(

)

(

)

0 0 n 10 2.303 298 8.314 At 25 C, E E log 1/ M n 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

22. C CH₃ CH 3 2 1. O 2. Zn / H O A B → + 2 NaOH / I A→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 →H / H O+ 2 _{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 O 3 2 1. O 2. Zn / H O products →

How many chiral products are formed?

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 NO₂+, NO₃− and NH₄+ are sp, sp2 and sp3 respectively

(C) SO2 molecule has a linear structure like CO2 molecules

(D) The geometry of H2S is linear

28. C H₃ C CH₂ CH₃ O 4 2 4 LiAlH KCN H SO A → → H3C 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 C H3 CN _{is formed}

(C) The final mixture is optically active

(D) The final mixture is racemic and therefore optically inactive 29. C H₃ Ph NH C Ph O (para) →OH− H₂N CH₃ 18

( )

3 1. H 2. CH O H (C) + D + → (C) and (D) are (A) COO C O CH3 O 18 18 (C) (D) (B) COO C O CH3 O (C) (D) 18 (C) COO C O CH3 O (D) (C) 18 (D) COO C O CH3 O (D) (C)

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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 5

2 x 1

0 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) 13i 11j 7kˆ+ ˆ+ ˆ (B) 13i 11j 7kˆ+ ˆ− ˆ

(C) 13i 11j 7kˆ− ˆ+ ˆ (D) ˆi 2j 5k+ ˆ− ˆ 4. Let f(x, y) = xy2 _{if x and y satisfy x}2 _{+ y}2 _{= 9, then the minimum value of f(x, y) is}

(A) 0 (B) −3 3

(C) −6 3 (D) −3 6

5.

(

)

3 101 x 0 1 3x 1 x lim 1 x 1 101x → + − −

+ − − has the value equal to

(A) 3 5050 − (B) 1 5050 − (C) 1 5050 (D) 1 4950

6. Let ABCDEFGHIJKL be a regular dodecagon, then the value of AB AF AF+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)

(

)

2 2 n n 1 4 + (B)

(

)

2 2 n 1 n 4 − (C)

(

)

2 n 1 4 + (D) n2

8. Number of roots of the function

( )

(

)

3 1 f x 3x sin x x 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

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) 3

2 (D) 5

12. The value of the defined integral

(

a

)(

2

)

(

)

0 dx a 0 1 x 1 x ∞ > + +

∫

is (A) 4 π (B) 2 π (C) π (D) some function of a

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

r

a b c

cos cos cos

r r r lim b c sin sin r r →∞ − equals (A) 2 2 2 a b c 2bc + − _{(B) c}2 _a2 _b2 2bc + − (C) 2 2 2 b 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) 55 3 − (D) 35 3 − Rough work

15. The minimum value of the function

( )

2 2 2 2

sin x cos x tan x cot x

f x

1 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) 1 6 (B) 1 9 (C) 1 12 (D) 1 18

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

2 2 2

1 2 3

z +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 x}2 _{+ 4y}2 _{− 6x + 16y =}

24 is (A) 1 2 (B) 2 (C) 3 2 (D) 5 2

19. For some non zero vector VG , if the sum of VG and the vector obtained from VG by rotating it by an angle 2α equals to the vector obtained from VG by rotating it by α then the value of α, is

(A) 2n 3 π π ± (B) n 3 π π ± (C) 2n 2 3 π π ± (D) n 2 3 π π ± Rough work

20. If 2 2 a 1 cosa tan k cosa b a w pcosa sin 2   − − _{ }   = +      

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) 2

2 (D) 4

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

( )

( )

u x 7 v x = . If

( )

( )

u x p v x ′ = ′ and

( )

( )

u x v x ′         = q, then p q p 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 y x x equals (A) 2 (B) −1 (C) −4 (D) some function of p Rough work

25. If

(

)

+ →∞ _{− + −} = n n n 1 n n n.3 1 lim 3

n 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 f x

( )

x2 2mx 1 for x 0 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.

(

3₃₊

( )

₃5/6 _i

)

3 _{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 151

C is divisible by n 150

C 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 2 x 9 +

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

(A) 66 (B) 132

(C) 248 (D) 264