JEE-MAIN MODEL GRAND TEST

Time: 9.00 AM to 12.00 Noon Max Marks:360

PHYSICS 1. Consider a light beam incident from air to a glass slab at Brewster’s angle as shown in

figure. A Polaroid is placed in the path of the emergent ray at point P and rotated

about an axis passing through the centre and perpendicular to the plane of the

Polaroid.

1) For a particular orientation there shall be darkness as observed through the Polaroid 2) The intensity of light as seen through the Polaroid shall be independent of the

rotation 3) The intensity of light as seen through the Polaroid shall go through a minimum but

not zero for two orientations of the Polaroid. 4) The intensity of light as seen through the Polaroid shall go through a minimum for

four orientations of the Polaroid. 2. A bob B of mass 1 kg is suspended from the ceiling of a toy train as shown in the

figure. The train oscillates simple harmonically in horizontal direction with angular

frequency 5 /rad s and amplitude 0.1a m . What is the ratio of maximum and

minimum tensions in the string AB during the motion 2 0& tan 3710 3 / 4mg s

1) 2 2) 3 3) 4 4) 1

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3. Two springs of force constant 100 N/m and 150 N/m are in series as shown. The block

is pulled by a distance of 2.5 cm to the right from equilibrium position. What is the

ratio of work done by the spring at left to the work done by the spring at right?

1) 32

2) 23

3) 0.2 4) 0.5

4. Three masses are connected with a spring & a string as shown. They are initially at rest, with spring at its natural length & string too at its original length. Find the maximum extension in the spring after the forces start acting as shown.

1) /F K 2) 2 /F K 3) / 2F K 4) 4 /F K 5. Uniform rod AB is hinged at end A in horizontal position as shown in the figure. The

other end is connected a block through a mass-less string m as shown. The pulley is

smooth and mass-less. Masses of block and rod is same and is equal to m. The

acceleration of block just after release from this position is

1) 6 /13g 2) / 4g 3) 3 / 8g 4) / 5g 6. A particle of mass m is moving in yz-plane with a uniform velocity v with its

trajectory running parallel to ve y-axis and intersecting z-axis at z a in figure. The change in its angular momentum about the origin as it bounces elastically from a wall at y = constant is.( xe is the unit vector along positive x-axis)

1) xˆmva e 2) xˆ2mva e 3) xˆymv e 4) xˆ2ymve

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7. A particle performs harmonic oscillations along a straight line with a period T and

amplitude a. The mean velocity of the particle averaged over the time interval during

which it travels a distance 2a starting from the extreme positions is

1) aT

2) 2aT

3) 3aT

4) 2aT

8. The drawing shows a top view of a frictionless horizontal surface, where there are two

identical springs with particles of mass 1 2m and m attached to them. Each spring has a

spring constant of 1200 N/m. The particles are pulled to the right and then released

from the positions shown in the drawing. How much time passes before the particles

are again side by side for the first time if 1 23.0 27m kg and m kg

1) sec40 2) sec

20 3) 3 sec

40 4) 3 sec

10

9. The earth is an approximate sphere. If the interior contained matter which is not of the

same density everywhere, then on the surface of the earth, the acceleration due to

gravity

1) Will be directed towards the center but not the same everywhere

2) Will have the same value everywhere but not directed towards the Centre

3) Will be same everywhere in magnitude directed towards the center

4) Cannot be zero at any point.

10. A sphere of brass released in a long liquid column attains a terminal speed 0v . If the

terminal speed attained by the sphere of marble of the same radius and released in the

same liquid is 0nv , then the value of n will be. Given: The specific gravities of brass,

marble and the liquid are 8.5,2.5 and 0.8 respectively.

1) 517

2) 1777

3) 1131

4) 175

11. Which of the following statements are true for wave motion?

1) Mechanical transverse waves can propagate through all mediums

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2) Longitudinal mechanical waves can propagate through solids only

3) Mechanical transverse waves can propagate through solids

4) Longitudinal mechanical waves can propagate through vacuum.

12. A rope hangs from a rigid support. A pulse is set by jiggling the bottom end. We want

to design a rope in which velocity v of pulse is independent of z, the distance of the

pulse from fixed end of the rope. If the rope is very long the desired function for mass

per unit length z in terms of 0 (mass per unit length of the rope at the top

0 , ,z g v and z is ( ‘g’ is acceleration due to gravity)

1) 2/

0g v zz e 2)

2/0

g v zz e

3) 0 2logegz zv

4) 2

0vz e zg

13. The maximum load a wire of length L and cross sectional area A can withstand without breaking is W. The maximum load that another wire of same material, length

2L and area of cross section A can withstand without breaking is

1) 2W 2) 2

W 3) 4W 4) W 14. An ideal gas undergoes four different processes from the same initial state in figure.

Four processes are adiabatic, isothermal, isobaric and isochoric. Out of 1,2,3 and 4

which one is adiabatic

1) 4 2) 3 3) 2 4)1

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15. A cubic vessel (with faces horizontal + vertical) contains an ideal gas at NTP. The

vessel is being carried by a rocket which is moving at a speed of 500 1ms in vertical

direction. The pressure of the gas inside the vessel as observed by us on the ground

1) remains the same because 1500ms is very much smaller than rmsv of the gas

2) remains the same because motion of the vessel as a whole does not affect the

relative motion of the gas molecules and the walls

3) will increase by a factor equal to 22 2rms rmsv 500 /v where rmsv was the original

mean square velocity of the gas

4) will be different on the top wall and bottom wall of the vessel

16. The magnitude of the electric field intensity at point B (2,0,0) due to a dipole moment,

ˆ ˆP i 3j

kept at origin is (assume that the point B is at large distance from the dipole

and k = 0

14

)(All quantities are in S.I units)

1) 13k8

2) 13k4

3) 7k8

4) 7k4

17. In the circuit shown, the switch is shifted from position 1 2 at t=0, The switch was

initially in position 1 since a long time. The graph between charge on capacitor C and

time ‘t’ is

1)

2)

3)

4)

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18. Two conducting large plate 1 2P & P are placed parallel to each other at very small separation ‘d’. The plate area of either face of plate is A. A charge +2Q is given to plate 1P & Q to the plate 2P (neglect ends effects). If plate 1 2P & P are now connected by conducting wire, then total amount of heat produced is

1)

2

0

4Q d3 A

2) 2

0

9 Q d8 A

3) 2

0

3Q d4 A

4)

2

0

8Q d9 A

19. Two volt meters of range 20.0V and 30.0V have to be constructed with a

galvanometer. The resistance connected in series with the galvanometer is 1680 for

the 20.0V range and 2930 for the 30.0V range. The resistance of the galvanometer

and the full scale current are respectively.

1) 320 and 8mA 2) 70 and 10mA

3) 820 and 10mA 4) 820 and 8mA

20. A square loop of side 2 cm carrying current 0I is placed in x-y plane in a magnitude

field B= ˆ ˆ4i 3j T. Find the unit vector along the axis about which it will start

rotating.

1) ˆ ˆ4 j 3i5 2)

ˆ ˆ4 j 3i5

3) ˆ ˆ4 j 3i5

4) ˆ ˆ4 j 3i5

21. The self inductance L of a solenoid of length l and area of cross-section A, with a fixed

number of turns N increases as

1) l and A increase. 2) l decreases and A increases.

3) l increases and A decreases. 4) both l and A decrease.

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22. The mutual inductance between the rectangular loop and the long straight wire as

shown in figure is M. Then

1) M=zero 2) 0a cM ln 12 b

3) 0b a cM ln2 b

4) 0a bM ln 12 c

23. A coil, a capacitor and an AC source of voltage 24 V (rms) are connected in series. By

varying the frequency of the source, a maximum rms current of 6 A is observed. If this

coil is connected to a DC battery of emf 12 V and internal resistance 2 , current

through it will be (in amp)

1) 1A 2) 2 A 3)3 A 4) 4 A

24. Spherical wave fronts shown in figure, strike a plane mirror. Reflected wave fronts

will be as shown in

1)

2)

3)

4)

25. An object approaches a convergent lens from the left of the lens with a uniform speed

15ms and stops at the focus. The image.

1) Moves away from the lens with an uniform speed 5m/s

2) Moves away from the lens with an uniform acceleration

3) Moves away from the lens with a non- uniform acceleration

4) moves towards the lens with a non-uniform acceleration.

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26. The half lives of a radioactive sample are 30years and 60 years from -emission and

-emission respectively. If the sample decays both by -emission and -emission

simultaneously, the time after which, only one-fourth of the sample remain is

1) 10 years 2) 20 years 3) 40 years 4) 45 years

27. A conical pendulum consists of a mass ‘M’ suspended from a string of length ‘l’. The

mass executes a circle of radius ‘R’ in a horizontal plane with speed ‘v’. At time ‘t’, the

mass is at position ˆRi and has velocity ˆv j . At time ‘t’, the angular momentum vector

of the mass ‘M’ about the point from which the string suspended is

1) ˆMvR k 2) ˆMvl k

3) 2 2

ˆˆl R RMvl i kl l

? ?? ?? ?? ?? ?

4) 2 2

ˆˆl R RMvl i kl l

? ??? ?? ?? ?? ?

28. In the figure assuming the diodes to be ideal

1) 1D is forward biased and 2D is reverse biased and hence current flows from A to B

2) 2D is forward biased and 1D is reverse biased and hence no current flows from B to

A

3) 1 2D and D are both forward biased and hence current flows from A to B

4) 1 2D and D are both reverse biased and hence no current flows from A to B

29. A 100 m long antenna is mounted on a 500 m tall building. The complex can become

a transmission tower for waves with .

1) 400m 2) 25m 3) 150m 4) 2400m

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30. In a vernier calipers having 10Vsd , the vernier constant is 0.1 mm. when the jaws are

closed, zero of vernier lies to the left of zero of main and 7th Vsd coincides with a

main scale division. When a cylinder is placed between the jaws the main scale

reading was 7.7 cm and vernier scale read 8 divisions. What is the diameter of the

cylinder?

1) 78.1 mm 2) 77.5 mm 3) 77.8 mm 4) 78.5 mm

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CHEMISTRY

31.

OH O - Li+

+What is/are the product(s) of the above reaction?

1) O

2)

O - Li+

OH

+

3)

4) OH OH

32. Identify the correct statement of the following

a) Hypo forms super saturated solutions

b) on thermal decomposition hypo gives H2S, SO2 and S.

c) Dilute sodium thio sulphate, on reaction with AgNO3 finally gives black ppt. of

Ag2S

d) AgBr can be used in making photo graphic films.

1) A, B, C 2) B, C, D 3) A, C, D 4) A, B, C, D

33 Specify the coordination geometry around and hybridisation of N and B atoms in a

1 : 1 complex of BF3 and NH3

1) N : tetrahedral, sp3 ; B–tetrahedral, sp 3

2) N–Pyramidal, sp3; B–pyramidal, sp 3

3) N–pyramidal, sp3; B–planar, sp2

4) N–pyramidal, sp3; B–tetrahedral – sp3

34 Two components A and B form an ideal solution. The mole fractions of A and B in

ideal solution are XA and XB, while that of in vapour phase, these components have

their mole fractions as YA and YB. Then, the slope and intercept of plot of A A

1 1vsY x

will

be :

1) 0 0 0A B A0 0B B

P P P,

P P 2)

0 0 0B A B0 0A A

P P P.

P P 3)

0 0B B0 0 0A 3 A

P P,

P P P 4)

00 0 AA B 0

B

PP P ,

P

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35. Which of the following compounds posses a chiral centre?

1)

CH2

OHH

2)

CH2

OHH

3)

BrH

4) Cl Br

36 Which of the following has the minimum heat of dissociation:

1) [(CH3)3N BF3] 2) [(CH3)3N B(CH3)F2]

3) [(CH3)3N B(CH3)2F] 4) [(CH3)3N B(CH3)3]

37 Which among the following is most soluble in water?

1) CsClO4 2) NaClO4 3) LiClO4 4) KClO4

38 For Adiabatic free expansion (Pext = 0) of an ideal gas

1) Ssurrounding = 0 2)Ssurrounding < 0 3)Ssurrounding > 0 4) Ssystem = 0

39 C6H5–CO–CH2–CH2 –CH2–COOH C6H5–CO–CH2–CH2–CH2–CD2OH.

This conversion is done by

1) NaBH4 / H3O+ followed by LiAlD4 / H2O

2) LiAlD4 / H2O followed by NaBH4 / H3O+

3) (CH2OH)2 followed by LiAlD4 / H3O+

4) DMgBr / H3O+

40 On heating potassium ferrocyanide with conc. H2SO4 produces a neutral gas ‘A’. The

gas ‘A’ on treatment with caustic soda under high pressure produced ‘B’, what are ‘A’

and ‘B’ respectively.

1) CO2, Na2CO3 2) SO2, Na2SO4 3) CO, HCOONa 4) NO2, NaNO3

41 Among the following statements, the incorrect one is :

1) Calamine and siderite are carbonates

2) Argentite and cuprite are oxides

3) Zinc blende and pyrites are sulphides

4) Malachite and azurite are ores of copper

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42 Which of the following transition state is more stable in the nucleophilic substitution ?

1)

OHCl

2NO

2)

OHCl

2NO

3)

OHCl

3CH

4)

OHCl

2NO

43 An organic compound upon hydrolysis produces two compounds one product gave silver mirror test, other product reacts with Hinsberg reagent to produce an alkali insoluble product. The organic compound is

1) 3 2 3

||O

CH CH C NHCH 2) H C

O3CH

N3CH

3) 3NHC2CH3CH

O

4)

C 2CH 3CH

O

H NH 44 100 ml of a sample of hard water after passing through cation exchange resin, required

20ml of 0.05M NaOH for neutralisation. One litre of same sample of water on

treatment with sufficient lime gave 200mg of CaCO3. Assuming that the hardness is

only due to Ca+2 ions. Find the degree of permanent hardness of water.

1)300ppm 2)150ppm 3) 100ppm 4) 200ppm

45 Calculate the pH at which Mg(OH)2 begins to precipitate from solution containing

0.1 M Mg+2 ions. Ksp for Mg(OH)2 is 1.0 x 10–11

1) 4 2) 9 3) 5 4) 8

46 Arrange the following bromides in the increasing order of reactivity towards AgNO3

Br

A)

B)

Br

C)

Br

O

1) C A B 2) C A B 3) B A C 4) A B C 47 Shape of 4MnO and hybridization of Mn in 4MnO are respectively

1) Tetrahedral, sp3 2) Tetrahedral, d3s

3) Sq. planar, dsp2 4) sq. planar sp2d

48. Bond dissociation energy of XY, X2 and Y2 (all diatomic molecules) are in the ratio

1:1: 0.5 and Hf of XY is –200 kJ mol–1. The bond dissociation energy of X2 will be:

1) 800 kJ mol–1 2) 200 kJ mol–1

3) 300 kJ mol–1 4) 400 kJ mol–1

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49. Which of the following pairs of structures do not represent tautomers?

1) OH

CHO

and

O

CHOH

2) C

CHO

O

OH and O

C

O

HOH

3) N

H

O

andN

OH

4)

CHO

H OH

OH H

CH2OH

and

CH2OH

C O

H OH

CH2OH 50. Consider a titration of potassium dichromate solution with acidified Mohr’s salt

solution using diphenylamine as indicator. The number of moles of Mohr’s salt

required per mole of dichromate is

1) 3 2) 4 3)5 4)6

51 The compound insoluble in acetic acid is

1) Calcium oxide 2) Calcium carbonate

3) Calcium oxalate 4) Calcium hydroxide

52

O

OH-

O

O + CHX3

X2 Which of the following is correct comparison of rate of haloform reaction with various

halogens ?

1) rCl2 > rBr2 > rI2 2) rI2 > rBr2 > rCl2

3) rBr2 > rCl2 > rI2 4) rCl2 rBr2 rI2

53. A complex of certain metal has the magnetic moment of 4.91 BM whereas another

complex of the same metal with same oxidation state has zero magnetic moment. The

metal ion could be…… and if that metal ion forms a complex with EDTA, then its

EAN would be……

1) Co2+,36 2) Mn2+,38 3) Fe2+,36 4) Ag+ ,36

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54 Of the following metals, which one cannot be obtained by electrolysis of the aqueous

solution of its salt is

1) Ag 2)Mg 3) Cu 4)Au

55. The major product of the following reaction is

O

RCH2OH

H (anhydrous)+

(1) A hemiacetal (2) An acetal (3) An ether (4) An ester

56 The compressibility of a gas is less than unity at STP. Therefore,

1) 22.4mV L 2) 22.4mV L 3) 22.4mV L 4) 44.8mV L

57 Nitrogen dioxide cannot be obtained by heating

1) KNO3 2) Pb(NO3)2 3) Cu(NO3)2 4) AgNO3

58. The IUPAC name of O

CN

is

1) 2-methyl-3-(1-methylethyl)-4-oxopentanenitrile

2) 4-cyano–3-(1-methylethyl)–2–pentanone

3) 3-acetyl–2–cyano–4–methylpentane

4) 3–ethanoyl–2–methyl–3–(1–methylethyl) pentanenitrile

59. For the electrochemical cell, M |M+|| X–| X, E(M+|M) = 0.44 V and E

(X/X–) = 0.33 V. From this data one can deduce that

1) M + X M+ + X– is the spontaneous reaction

2) M+ + X– M + X is the spontaneous reaction

3) Ecell = 0.77 V

4) Ecell = –0.77 V

60 The exhausted permutit is generally regenerated by percolating through it a solution

of:

1) Sodium chloride 2) Calcium chloride

3) Magnesium chloride 4) Potassium chloride

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MATHEMATICS 61. The number of real solutions of 1 + ex + |1 ex| = ex(ex 1) is

(1) 0 (2) 1 (3) 2 (4) > 2

62. A solution set for 4 sin2 x – 8 sin x + 3 0 is

(1)

34,

3 (2)

65,

6 (3) ]2,0[ (4)

,6

63. If 1sin 1x then 21 1sin sinx x can be equal to

(1) -1 (2) 14

(3) 34

(4) 45

64. ~(p q) (~ p q) is logically equivalent to

(1) p (2) q (3) ~p (4) ~q

65. Assertion : If A is a skew symmetric matrix of order 3, then its determinant must be

zero.

Reason: If A is a square matrix of order n and k is any scalar then |kA| = kn|A|.

(1) Both A and R are individually true and R is the correct explanation of A.

(2) Both A and R are individually true but R is not the correct explanation of A.

(3) A is true but R is false. (4) A is false but R is true.

66. If the tangents drawn from a point on the hyperbola x2 y2 = a2 b2 to the

ellipse 2

2

2

2

by

ax

= 1 make angle and with the transverse axis of the

hyperbola, then

(1) tan tan = 1 (2) tan + tan = 1

(3) tan . tan = 1 (4) tan tan = 1

67. Assertion : If 1

0xsin dxe then 200dxe

200

0xsin

Reason:If dxe1

0]x[x then 200dxe

200

0]x[x

(1) Both A and R are individually true and R is the correct explanation of A.

(2) Both A and R are individually true but R is not the correct explanation of A.

(3) A is true but R is false.

(4) A is false but R is true.

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68. Assertion : The elimination of arbitrary constants in y = )eCCC( 4C321 x results into a

differential equation of first order xdxdy = y

Reason: Elimination of n independent arbitrary constants 1 2, ,......, nc c c in the equation

1 2, , , ,.... 0nf x y c c c gives differential equation of order n.

(1) Both A and R are individually true and R is the correct explanation of A.

(2) Both A and R are individually true but R is not the correct explanation of A.

(3) A is true but R is false. (4) A is false but R is true.

69. If f(x) = x [x], x( 0) R where [x] is the greatest integer less than or equal to

x, then the number of solutions of f(x) +f

x1 = 1 are

(1) 0 (2) 1 (3) infinite (4) 2

70. Assertion : Length of latus rectum of parabola (6x + 8y + 7)2 = 4(8x + 6y + 3)

is 4.

Reason:Length of latus rectum of parabola y2 = 4ax is 4a.

(1) Both A and R are individually true and R is the correct explanation of A.

(2) Both A and R are individually true but R is not the correct explanation of A.

(3) A is true but R is false. (4) A is false but R is true.

71. f : NN, f(n) = sum of digits of n, then f is

(1) bijective (2) surjective (3) injective (4) f is not a function

72. If the roots of z3 + iz2 + 2i = 0 represent the vertices of a triangle in the argand plane,

then its area is

(1) 2 (2) 4

73 (3) 273 (4) 4

73. The number of roots of equation x2+x+3+2 sin x = 0, where x [–,] is

(1) 0 (2) 2 (3) 3 (4) 4

74. If x > 0, y > 0, z > 0 and minimum value of x(y2 + z2) + y(z2 + x2) + z (x2 + y2) is xyz

then is

(1) 1 (2) 2 (3) 4 (4) 6

75. The value of 30C0 30C10 – 30C1 30C11 + 30C2 30C12 – ………………..+ 30C20 30C30 is

(1) 60C30 (2) 40C30 (3) 60C20 (4) 30C10

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76. Let R be a relation on a set A such that 1a, b ( , )R b a R and 1R R then R is

1) Reflexive 2) symmetric 3) transitive 4) an equivalence relation

77. The number of permutations of the letters of the word H I N D U S T A N such

that neither the pattern ‘H I N’ nor ‘D U S’ nor ‘T A N’ appears are

(1) 166680 (2) 181434 (3) 166674 (4) 169194

78. If two events A and B are such that P(A) = 0.3, P(B) = 0.4 and P(A B) = 0.5,

then

'BABP =

(1) 41 (2)

51 (3)

53 (4)

52

79. xlog1

1xxlog

4tanlim

is

(1) e–1 (2) e2 (3) 2 (4) e

80. let f : R R is a differentiable function and f(1) = 4, then the value of

)x(f

41x

dt1x

t2lim

is

(1) 8 f (1) (2) 4 f (1) (3) 2 f (1) (4) f (1)

81. f (x) =

x

0

1x

x

,3,1x,dt)t(f)x(gand3x2,ex

2x1,e21x0,e

then

1) g(x) has local maxima at x = 1+ ln 2 and local minima at x = e

2) g(x) has local minima at x = e and local maxima at x =1

3) g(x) has no local minima 4) g(x) has no local maxima

82. If

dx

1x

cx)1xlog(be2

2x = )1xlog(e2c 2x then values of b and c can be

(1) b = 1, c = 2 (2) b = 31 , c =

21 (3) b =

31 , c = 1 (4) b = 2, c = 3

83. Value of dx}x{

dx]x[

n

0

n

0

where [x] and {x} are integral and fractional parts of x and

n N, is equal to

(1) 1n

1

(2) n1 (3) n (4) n – 1

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84. Area of the region bounded by the curve y = ex and lines x = 0 and y = e is

(1) e – 1 (2) e

1dy)y1eln( (3)

e

1dy)y1ln( (4)

e

1dy)yeln(

85. The bisectors BD and CF of a triangle ABC have equations y = x and x = 10.

If A is (3, 5) then equation of BC is

(1) 5y – 2x = 11 (2) 6y –5x = 17 (3) 6y – x = 13 (4) none of these

86. Tangents drawn from P(1,8) to the circle x2 + y2 – 6x – 4y – 11 = 0 touch the

circle at A and B. The equation of circumcircle of triangle PAB is

(1) x2 + y2 + 4x – 6y + 19 = 0 (2) x2 + y2 – 4x – 10y + 19 = 0

(3) x2 + y2 – 2x + 6y – 29 = 0 (4) x2 + y2 – 6x – 4y + 19 = 0

87. In a battle 72% of the combatants lost one eye 78% an ear, 75% an arm, 80% a leg and

x% lost all the four . The minimum value of x is

1) 10 2) 12 3) 15 4) 5

88. Let P(3, 2, 6) be a point in space and Q be a point on the line

)k5ji3()k2ji(r

. Then the value of for which PQ is parallel to the plane

x – 4y + 3z = 1 is

(1) 41 (2)

41

(3) 81 (4)

81

89. Angle between the vectors a and b , where c,b,a are unit vectors satisfying

0c3ba is

(1) 2 (2) 6

(3) 4

(4)

3

90. If the standard deviation of 10 observations 1 2 10, ,.......,x x x is 4 and that of another set of

10 observations 1 2 10, ,......,y y y is 3 and also i i iX x x y y . x is mean of all ix ' s

and y is mean all iy ' s . 10

1

80ii

X

Then standard deviation of observations

1 1 2 2 10 10, ,........,x y x y x y is

1) 1 2) 3 3) 5 4) 5

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JEE-MAIN MODEL GRAND TEST

KEY

PHYSICS

1) 3 2) 1 3) 1 4) 2 5) 3 6) 2

7) 3 8) 3 9) 4 10) 2 11) 3 12) 1

13) 4 14) 3 15) 2 16) 3 17) 2 18) 2

19) 4 20) 3 21) 2 22) 4 23) 2 24) 3

25) 3 26) 3 27) 3 28) 2 29) 1 30) 1

CHEMISTRY

31) 2 32) 3 33) 1 34) 2 35) 2 36) 4

37) 2 38) 1 39) 3 40) 3 41) 2 42) 1

43) 2 44)1 3 45) 2 46) 2 47) 2 48) 1

49) 4 50) 4 51) 3 52) 2 53) 3 54) 2

55) 2 56) 2 57) 1 58) 2 59) 2 60) 1

MATHEMATICS

61) 2 62) 2 63) 2 64) 3 65) 1 66) 3

67) 4 68) 1 69) 3 70) 4 71) 2 72) 1

73) 1 74) 4 75) 4 76) 2 77) 4 78) 1

79) 2 80) 1 81) 1 82) 1 83) 4 84) 2

85) 4 86) 2 87) 4 88) 1 89) 4 90) 2

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JEE-MAIN MODEL GRAND TEST

SOLUTIONS

PHYSICS

3. 100 150 60 /

250eqk N m

2.5 360100 2eqF k x N

For left spring 13

2 100x

For right spring 23

2 150x

2 2

2 2

1 3 1100150 32 2 100100 21 3 1150

2 2 150

So

4. 21 2 1 2 1 2

1 22

FF x x K m m m mK

5. For block, mg-T=ma …(i) for rod;

2

2 3l mlT x l mg x …(ii)

a l …(iii) From (i), (ii) & (iii); we get

38ga

6. The initial velocity is i yˆv ve and, after reflection from the wall, the final velocity is

f yˆv ve . The trajectory is described as y zˆ ˆr ye ae . Hence the change in angular

momentum is f i xˆr m v v 2mvae . Hence the answer is (b) 7. cosx a t

sinV a t

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/6

0/6

0

sin3

T

T

a tav

Tdt

]

8. 11

k 1200w 20 rad /sm 3

21200 20 rad /s

27 3

20 t3

20t

202 20 t3

6 3t sec80 40

10. 20

8.5 0.8 g2v rg

2

0

2.5 0.8 g2nv rg

n=17/77

12. zF 0

(T dT) gdz T 0

dT gdz .........(i)

2also T v

2dT d v 2vdvd

As v is independent of z

dv=0

2dT v d ……(ii)

From equation (1) and (2) we get

z

20

du g dzv

or 20e g /v z

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14. c. adiabatic A is isobaric process, D is isochoric. Of B and C, B has the smaller slope (magnitude) hence is isothermal. Remaing process is adiabatic

15. Comment for discussion: This brings in concepts of relative motion and that when collision takes place, it is the relative velocity which changes.

16.

2 21 2E E E

= k 3 48

= k78

17.

0 0qV 2 2iR vC

qiR 2RC

also dqidt

q t

C 0

dq dtq

R 2RC

q

C

qlnR 2RC

t1

2RC

ln 2 C q t

C 2RC

t/2RCq 2 C Ce

t/2RCq C 2 e

18. 0ACd

2 21.5 2.25

2 2i

Q Q dUC C

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2

0

98iQ dU

A

After connecting

0fU

Heat = 2

0

9i f

QU Ud A

19. (1680+r) I = 20

(2930+r)I = 30

2 x 2930 + 2r = 3 x 1680 + 3r

r = 820 ]

20. Magnetic moment is in divection of xza

and ˆ ˆB 4i 3 j

4 3ˆ ˆB i j5 5

iA

40

ˆ4x10 x I k

ˆˆ xB

4 3ˆ ˆˆ j i5 5

Thus the unit vector of the axis is

22. 0x

i x adx M i2 x

0 b caM ln2 c

26. 1130

2160

1 2

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120

1/20

t/t

NN2

1/20 0

t/t

N N4 2

1/2t 2t = 0.6932

2 0.693 20t yrs = 27.72 yrs 30. zero error =-(10-7)x0.1 =-0.3 mm

Diameter = 77.0+8 x 0.1-(0.3)

=78.1 mm]

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CHEMISTRY

31. R – OH < Ph – OH (Acidic Strength) 32. Conceptual 33. Conceptual 34. Conceptual 35. 1, 3, 4 contains plane of symmetry

H OH

CH2

Chiral center*

36. Conceptual 37. Conceptual

38. Free explanation 0surrS 0sysS 39.

OHOH Ph-C-CH2-CH2-CH2-COOHPh-C-CH2-CH2-CH2-COOH

OOO

LiAlD4/H3O+

Ph-C-CH2-CH2-CH2-CD2OH

O

40. Conceptual 41 42. Conceptual 43.

H-C-N

OCH3

CH3

H3O+H-COOH + CH3-N-H Ph-SO2Cl

CH3 - N - S - Ph

Silver mirror test

NaOH

O

O

CH3 CH3

44

45. 22 1110Mg OH

112 1010 10 5

0.1OHOH P

PH=9

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46.

O

Stability orderof C

47 48. 2 2 2X Y xy

2 21 12 2

x y xy 200H

21 12 2

BEX 2 . 200BEY B EXY

49. Conceptual.

50. 2 2 3

2 7Cr O Fe Cr Fe le

6e

3

51. Conceptual 52. formation of enol is rate determining step. 53 54. Conceptual 55.

O-CH2-R Acetal 56. VRz

Vi

57. Conceptual 58 59. 0.11 .oM X M X E Cell V 60. Conceptual

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MATHEMATICS 62. (2 sin x – 1)(2 sin x – 3) 0 sin x <

23 2 sin x – 3 < 0 for all x

2 sin x – 1 0 sin x

21

x

65,

6

64. ~(p q) (~ p q) (~ p ~ q) (~ p q) ~ p (~ q q) ~ p t ~p 65. A = A |A| = |–A| = (–1)n |A| If n is odd |A| = –|A| 2|A| = 0 |A| = 0 67. If period of f(x) is T then

T

0

nT

0dx)x(fndx)x(f

Period of ex – [x] is 1 Statement II is true. But period of esin x 1 Statement I is false. 68.(a) y = x)eCCC( 4C

321 y = Ax A

dxdy

y = dxdyx

70. 6x + 8y + 7 = 0 and 8x + 6y + 3 = 0 are not perpendicular Latusrectum 4. 71.(b) f(21) = f(12) = 3

f is not 1 – 1 for every y N there exists x N s.t. f(x) =y

f is onto (surjective) 72. z3 + iz2 + 2i = 0 (z – i ) (z2 + 2iz – 2 ) =0 z =i,

28i4i2 2

z = i, 1 – i, –1–i area of triangle with vertices (0,1), (1, –1), (–1,–1) 2

111111110

21

73. 0xsin2341

21x

2

411xsin2

21x

2

LHS 0 0411xsin2

811xsin

0411xsin2

not possible no solution

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74. If numbers are xy2, x z2 , yz2, yx2, zx2, zy2 then AM GM

6

xyz6zyx6yxzxzyzyx

zy.zx.yx.yz.xz.xy6

yxzxzyzyx

61

666222222

61

222222222222

75. difference of suffices is same =10 and terms are alternately positive and negative we take expansion of (1 – x )30 and (x + 1)30 , multiply and equate cofficient of x20 on both sides. (1 – x )30 (x + 1)30 = (1 x2)30

coff of x20 = 30C10 77. total permutations =

!2!9

No. of permutations containing ‘H I N’ = 7! No. of permutations containing ‘D U S’ =

!2!7

No. of permutations containing ‘T A N’ =7! No. of permutations containing ‘H I N’ and ‘D U S’ = 5 ! No. of permutations containing ‘D U S’ and ‘T A N’ = 5 ! No. of permutations containing ‘T A N’ and ‘H I N’ = 5 ! No. of permutations containing ‘H I N’, ‘D U S’ and ‘T A N’ = 3 ! Required permutations =

!3!5!5!5!7

!2!7!7

!2!9

=169194 79. 1 form

limit = xlog

11xlog4

tan1x

lim

e

= x1

x1xlog

4sec

e

2

1xlim

= 242sec

ee

(using L Hospital Rule ) 80.

1x

tdt1x

t2 2

)x(f

4

22

1x4)x(fdt

1xt2

form

00

1x4)x(flim

1xtdt2lim

)x(f

4

22

1x1x

)x('f)x(f2lim1x

=2 f (1) f ‘ (1) = 8 f ‘ (1) 81. g(x) = f(x) (By Leibnitz rule) g(x) = 0 when 2 – ex – 1 = 0 or when x – e = 0 x = 1 + ln 2, x = e At x = 1 + ln 2, g(x) = –ex – 1 = –eln 2 = – 2 Local maxima at x = 1 + ln 2 At x = e, g(x) = 1 > 0 Local minima at x = e 82. Using dx))x('f)x(f(ex = exf(x), we have

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dx1x

cx)1xlog(be2

2x

= ex.b log (x2 + 1) where 2b = c

= )1xlog(2c.e 2x

2b = c which only (a) satisfies 83.

n

0dx]x[ =

n

1n

2

1

1

0dx]x[.........dx]x[dx]x[

=

n

1n

2

1

1

0dx).1n(.........dx.1dx.0

= 0 + 1 + 2 + ….. + (n – 1) = 2

n)1n(

n

0dx}x{ =

1

0dx}x{n =

2nxdxn

1

0

Required value = 1n

2n2

n)1n(

84. y = ex x = log y Area =

e

1ydylog

= dy)ye1log(e

1

85. Reflection of A in BD and CF lies on BC. Reflection of A in BD i.e. y = x is (5, 3) Reflection of A in CF ie. x = 10 is (17, 5) Equation of BC is y – 3 = )5x(

51735

i.e. x – 6y + 13 = 0 or 6y – x = 13 86. Circumcircle of PAB has CP as diameter its equation is (x – 1)(x – 3) + (y –8)(y –2) = 0 x2 + y2 – 4x – 10y + 19 = 0 87. 100 28 22 25 20 88. Q is any point on the line coordinates of Q are (–3 + 1, – 1, 5 +2) Direction ratios of PQ are –3 – 2, – 3, 5 – 4 PQ || plane PQ is perpendicular to normal

1(–3 – 2) – 4( – 3) + 3(5 – 4) = 0 =

41

89.

c3ba Squaring

222 |c|3b.a2|b||a|

2 + 3cos|b||a|2

cos = 21 =

3

90. Let 10 10 102 22

1 1

2

1

1 1 1 210 10 10 10x y i i i i i

i i iE x y x y E x x x x y yy y

y = e e

1 0

1

A (3, 5)

F D

CB

A

B

C (3,2)

P (1,8)

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