2019 apr jee-main q p k s - sri chaitanya · 2019-04-15 · 2019_jee-main question paper_key &...

2019_Apr_Jee-Main Question Paper_Key & Solutions

2019_Jee-Main Question Paper_Key & Solutions

Sri Chaitanya IIT Academy # 304, Kasetty Hegihts, Ayappa Society, Madhapur, Hyderabad – 500081

www. srichaitanya.net, [email protected]

PHYSICS 1. The magnetic field of a plane electromagnetic wave is given by:

0 1cos cosB B i kz t B j kz t

Where 50 3 10B T and 6

1 2 10B T .

The rms value of the force experienced by a stationary charge 410Q C at z = 0 is

1. 0.6 N 2. 0.1 N 3. 23 10 N 4. 0.9 N

Ans: 1

Sol: 2 2

0 0

2 2rmsCB CBF g

5 6

83 10 2 1010 3 102 2

0.63

2. A solid sphere of mass ‘M’ and radius ‘a’ is surrounded by a uniform concentric

spherical shell of thickness 2a and mass 2M. The gravitational field at distance ‘3a’

from the centre will be:

1. 2

23GMa

2. 29GM

a 3. 2

29GMa

4. 23GM

a

Ans: 4

Sol: Field at

2 2

23 3

G MGMPa a

P 3a Ma

Field = 23GM

a

3. A moving coil galvanometer has resistance 50 and it indicates full deflection at 4

mA current. A voltmeter is made using this galvanometer and a 5 k resistance.

The maximum voltage, that can be measured using this voltmeter, will be close to:

1. 10 V 2. 15 V 3. 40 V 4. 20V

Ans: 4




Sol: max 4I MA

50GR

In voltmeter

max s KV I R S

20V

4. For a given gas at 1 atm pressure, rms speed of the molecules is 200 m/s at 127ºC. At

2 atm pressure and at 227ºC, the rms speed of the molecules will be:

1. 100 /m s 2. 80 /m s 3. 80 5 /m s 4. 100 5 /m s

Ans: 4

Sol: rmsV T

2 500200 400V

2 100 5V

5. Taking the wavelength of first Balmer line in hydrogen spectrum (n = 3 to n = 2) as

660 nm, the wavelength of the 2nd Balmer line (n = 4 to n = 2) will be:

1. 642.7 nm 2. 889.2 nm 3. 488.9 nm 4. 388.9 nm

Ans: 3

Sol: 1st line Balmer

N=N NH 36 6605R

2

1 1 14 16

R

2

163R

1 5 66036R

216 5 66013 36

2 488.9

6. The pressure wave, 20.01sin 1000 3P t x Nm , corresponds to the sound produced

by a vibrating blade on a day when atmospheric temperature is 0ºC. On some other




day when temperature is T, the speed of sound produced by the same blade and at

the same frequency is found to be 1336ms . Approximate value of T is:

1. 12ºC 2. 11ºC 3. 15ºC 4. 4ºC

Ans: 4

Sol: soundV T

2336 31000 273

T

2 277T K

2 4ºT C

7. The following bodies are made to roll up (without slipping) the same inclined plane

from a horizontal plane : (i) a ring of radius R, (ii) a solid cylinder of radius 2R and

(iii) a solid sphere of radius 4R . If, in each case, the speed of the center of mass at the

bottom of the incline is same, theratio of the maximum heights they climb is:

1. 2 : 3 : 4 2. 4 : 3 : 2 3. 10 : 15 : 7 4. 14 : 15 : 20

Ans: 3

Sol: 21 12 cmMgh mV K

1 21 1:1 :12 5

Ratio

20 :15 :14

8. An NPN transistor is used in common emitter configuration as an amplifier with

1k load resistance. Signal voltage of 10 mV is applied across the base-emitter. This

produces a 3 mA change in the collector current and 15 A change in the base

current of the amplifier. The input resistance and voltage gain are:

1. 0.33 ,300k 2. 0.67 ,300k 3. 0.33 ,1.5k 4. 0.67 ,200k

Ans: 2

Sol: 200C

B

II




B iI R V

23iR

Voltage gain = C C

B I

I RI R

Voltage gain = 300

9. A ball is thrown vertically up (taken as +z-axis) from the ground. The correct

momentum – height (p-h) diagram is:

1. 2.

3. 4.

Ans: 4

Sol: 2 2 2v u gh

2 2 2v v gh

2p A Bh

10. The electric field of light wave is given as 3 147

210 cos 2 6 105 10

x NE t xC

.

The light falls on a metal plate of work function 2eV. The stopping potential of the

photo-electrons is:Given, E (in eV) 0

12375

Ain

1. 0.48 V 2. 2.48 V 3. 0.72V 4. 2.0

Ans: 1




Sol: 0 .hv hv s potential

12375 .5000

w s potential

2.48 2 .ev s potential

0.4stop potential eV

11. A string is clamped at both the ends and it is vibrating in its 4th harmonic. The

equation of the stationary wave is 0.3sin 0.157 cos 200Y x t . The length of the

string is: (All quantities are in SI units.)

1. 40 m 2. 20 m 3. 80 m 4. 60 m

Ans: 3

Sol: ln 4th harmonic

4. 22

2 2 3.14 20 400.157

mk

2 80m

12. A signal cosA t is transmitted using 0 0sin t as carrier wave. The correct

amplitude modulated (AM) signal is:

1. 0 0sin cost A t

2. 0 0 0 0sin sin sin2 2A At t t

3. 0 0cos sinA t t

4. 0 0sin 1 0.01 sinA t t

Ans: 2

Sol: Mododulater wave = 0 0sin cosC rmsA V t t w

= 0 0cos sinV A t t

0 0 0 0sin sin sin2 2A AV t t t




13. In the density measurement of a cube, the mass and edge length are measured as

10.00 0.10 kg and 0.10 0.01 m , respectively. The error in the measurement of

density is:

1. 30.31 /kg m 2. 30.07 /kg m 3. 30.01 /kg m 4. 30.10 /kg m

Ans: 1

Sol: 10 0.1mass kg

Length = 0.1 0.01 m

Denisty = 3

m DL

0.1 0.01310 0.1

=0.01+0.3

= 0.31 kg/m3

14. A simple pendulum oscillating in air has period T. The bob of the pendulum is

completely immersed in a non-viscous liquid. The density of the liquid is 116

th of the

material of the bob. If the bob is inside liquid all the time, its period of oscillation in

this liquid is:

1. 1210

T 2. 1214

T 3. 1414

T 4. 1415

T

Ans: 4

Sol: 1Tg

1

1 1

1516

T gT g

15' 4

TT

4'15TT




15. If ‘M’ is the mass of water that rises in a capillary tube of radius ‘r’, then mass of

water which will rise in a capillary tube of radius ‘2r’ is:

1. 4 M 2. M 3. 2M 4. 2 M

Ans: 4

Sol: 2Threg

2mass r h

mass r

So final mass rise = 2M

16. Following figure shows two process A and B for a gas. If AQ and BQ are the

amount of heat absorbed by the system in two cases, and AU and BU are changes

in internal energies, respectively, then:

1. ,A B A BQ Q U U 2. ,A B A BQ Q U U

3. ,A B A BQ Q U U 4. ,A B A BQ Q U U

Ans: 1

Sol: A BU U is U is state function

A A AQ U W

B B BQ U w

A B A BQ Q w w

0

A BQ Q




17. The total number of turns and cross-section area in a solenoid is fixed. However, its

length L is varied by adjusting the separation between windings. The inductance of

solenoid will be proportional to:

1. 21 / L 2. L 3. 2L 4. 1/L

Ans: 4

Sol: WKT

Self inductance 2L n volume

2

2

N A KL

2N A

L

Self inductance 1L

18. A uniform cable of mass ‘M’ and length ‘L’ is placed on a horizontal surface such

that its 1 th

n

part is hanging below the edge of the surface. To lift the hanging part

of the cable upto the surface, the work done should be:

1. 2

gLn 2. n gL 3. 22

gLn

4. 2

2 gLn

Ans: 3

Sol: The resultant is the centre of mass of the part AB is shifter to top i.e. 2lhn

U mg hcm

22 2M L L MgLgL n N n

22MgLU

n




19. A body of mass 2 kg makes an elastic collision with a second body at rest and

continues to move in the original direction but with one fourth of its original speed.

What is the mass of the second body?

1. 1.2 kg 2. 1.8 kg 3. 1.0 kg 4. 1.5 kg

Ans: 1

Sol: Momentum conservation

22 2 14vv m

132vmv

Elatic collision

Energy conservation

2

2 21

1 1 12 22 2 4 2

vv mv

22 2 212 15 915

16 8 4mvv v v

m m

1.2m kg

20. A capacitor with capacitance 5 F is charged to5 C . If the plates are pulled apart

to reduce the capacitance to 2 F , how much work is done?

1. 63.75 10 J 2. 62.55 10 J 3. 66.25 10 J 4. 62.16 10 J

Ans: 1

Sol: work done by force F S

2

0

32 2

q dA

2

0

34

q dA

63 5 10

4x

63.75 10 J




21. A rectangular coil (Dimension5 2.5cm cm ) with 100 turns, carrying a current of 3A

in the clock-wise direction, is kept centered at the origin and in the X-Z plane. A

magnetic field of 1 T is applied along X-axis. If the coil is tilted through 45º about

Z-axis, then the torque on the coil is

1. 0.38 Nm 2. 0.42 Nm 3. 0.55 Nm 4. 0.27 Nm

Ans: 4

Sol: Torque M B

= sin 4Nxi A B J

= 4 1100 3 12.5 10 1 12

0.265

22. The figure shows a Young’s double slit experimental setup. It is observed that when

a thin transparent sheet of thickness t and refractive index is put in front of one

of the slits, the central maximum gets shifted by a distance equal to n fringe widths.

If the wavelength of light used is ,t will be:

1. 2

1nD

a

2.

2

1D

a

3.

1nD

a

4.

1D

a

Ans: 3

Sol: 1 2 1 0S P S P t

2 11 t S P S P

1 t nl

1

nlt

23. A wire of resistance R is bent to form a square ABCD as shown in the figure. The

effective resistance between E and C is: (E is mid-point of arm CD)




A B

CD E

1. R 2. 34

R 3. 764

R 4. 116

R

Ans: 3

Sol: 4 4 4 8AB AD BC CR R R RR R R RO R

Reff 4 4 4 8

4 4 4 8

R R R R

R R R R

778 864

R RR

R

24. A rigid square loop of side ‘a’ and carrying current 2I is lying on a horizontal

surface near a long current 1I carrying wire in the same plane as shown in figure.

The net force on the loop due to the wire will be:

1. Attractive and equal to 0 1 2

3I I

2. Zero

3. Repulsive and equal to 0 1 2

4I I

4. Repulsive and equal to 0 1 2

2I I

Ans: 3

Sol: 0AB DCF F

2ADF BI a

022

I I

(right wards)




1 2

2I I

2'BCF B I a

0 1 2

4I I

(up + wards)

0 1 2 0 1 2

2 4netI I I IF

0 1 2

4I I

(right wards ev (repulsive)

25. The stream of a river is flowing with a speed of 2 km/h. A swimmer can swim at a

speed of 4 km/h. What should be the direction of the swimmer with respect to the

flow of the river to cross the river straight?

1. 150º 2. 60º 3. 90º 4. 120º

Ans: 4

Sol: to cross the river straight, Valong the river = 0

Vsteam = Vswimmer cos

2 = 14cos cos2

60º

Direction w.r.t flow of river = 180º-60º = 120º

26. A stationary horizontal disc is free to rotate about its axis. When a torque is applied

on it, its kinetic energy as a function of , where is the angle by which it has

rotated, is given as 2k . If its moment of inertia is I then the angular acceleration of

the disc is:

1. 2kI 2.

2kI 3.

4kI 4. k

I

Ans: 1

Sol: 2KE K

212

IW K

Differentiating on both sides w.r.t to




2 212

d dI kd d

22 kI KI

27. A system of three charges are placed as shown in the figure:

If D > > d, the potential energy of the system is best given by:

1. 2

20

14

q qQdd D

2. 2

20

14 2

q qQdd D

3. 2

20

1 24

q qQdd D

4. 2

20

14

q qQdd D

Ans: 4

Sol: The potential energy of +q, -q system is 2

1kqUd

The system of +q, -q act as an electric dipole as d<<D

Hence, Potential energy between Q and dipole is 2 2

kpU Qr

Hence, 1 20

14

U U U

2

2q d dd D

28. An HCl molecule has rotational, translational and vibrational motions. If the rms

velocity of HCl molecules in its gaseous phase is v

, m is its mass and Bk is Boltzman

constant, then its temperature will be :

1. 2

6 B

mvk

2. 2

7 B

mvk

3. 2

3 B

mvk

4. 2

5 B

mvk

Ans: 1

Sol: 23 3rms

KT KTV Vm m

2

3mvT

k




29. A concave mirror for face viewing has focal length of 0.4 m. The distance at which

you hold the mirror from your face in order to see your image upright with a

magnification of 5 is:

1. 0.24 m 2. 0.16 m 3. 1.60 m 4. 0.32 m

Ans: 4

Sol: 1 1 1f u v 1 1 1

0.4u v

5vmu

54v

1 1 15 0.4u u

1.6 0.325

u

30. Determine the charge on the capacitor in the following circuit:

1. 10 C 2. 200 C 3. 60 C 4. 2 C

Ans: 2

Sol: 4 12 616TR

9

72 89TI A

1248 2

16I A

10 2I A

10 20V V

20 20 10 200C CV V Q F F




CHEMISTRY 31. Among the following, the set of parameters that represents path functions, is:

A) q+w B) q C) w D) H-TS

1. A, B and C 2. B, C and D 3. B and C 4. A and D

Ans: 3

Sol: Except q, w all other thermodynamic variables are state functions q, w are path

functions.

32. The correct IUPAC name of following compound is

2NO

3CHCl

1. 2-methyl – 5 - nitro 1- chlorobenzene

2. 2-+chloro – 1- methyl -4 - nitrobenzene

3. 5 – chloro – 4 – methyl – 1 – nitrobenzene

4. 3- chloro – 4 – methyl – 1 – nitrobenzene

Ans: 2

Sol:

3CH

2NO

Cl12

34

2NO - Nitro Cl - Chloro 3CH - Methyl

33. The degenerate orbitals of 3

2 6Cr H O

are

1. yzd and 2zd 2. 2and dx xzd 3. and dxz yzd 4. 2 2and dxyx yd




Ans: 3

Sol: ,zx yzd d are degenerate in octahedral splitting.

34. The organic compound that gives following qualitative analysis is:

Test Inference

a) Dil. HCl Insoluble

b) NaOH solution Soluble

c) Br2/ water Decolourization

1.

2NH

2.

OH

3.

2NH

4.

OH

Ans: 3

Sol:

on

insoluble in Dil. HCl, Soluble in NaOH solution , & Decolourization of

2Br / water

35. Excessive release of 2CO into the atmosphere results in

1. depletion of ozone 2. polar vertex

3. formation of smog 4. global warming

Ans: 4

Sol: Excessive release of 2CO into the atmosphere results in global warming.

36. The element having greatest difference between its first and second ionization

energies, is

1. K 2. Ca 3. Ba 4. Sc

Ans: 1

Sol: In given options k has greatest difference between its first & second ionisation

energies.




37. The major product of the following reaction is:

21.

2 .PBr

KOH alcOH

O

1. O 2. O

3. O

OH

4. O

Ans: 4

Sol:

BrOH

O

3PBr .alc KOH

OO

38. The number of water molecules not coordinated to copper ion directly in

24.5CuSOl H O , is

1. 2 2. 3 3. 1 4. 4

Ans: 3

Sol: In 4 2.5CnSO H O , four water molecules are coordinates to 2Cu ion and one water

molecules is outside the coordinate sphere.

Number of water molecules not coordinated to copper ion is

39. 60 ,C an allotrope of carbon contains:

1. 18 hexagons and 14 pentagons 2. 20 hexagons and 12 pentagons

3. 12 hexagons and 20 pentagons 4. 16 hexagons and 16 pentagons

Ans: 2

Sol: In 60C , 102n hexagonal, 12 pentagonal

20 hexagonal, 12 petagonal




40. Aniline dissolved in dilute HCl is reacted with sodium nitrite at 0ºC. this solution

was added dropwise to a solution containing equimolar mixture of aniline and

phenol in dil. HCl. The structure of the major product is

1.

N=N NH2

2.

N=N O

3.

N=N OH

4.

N=N NH

Ans: 1

Sol:

2. ,0ºNaNO

dil HCl C

2NH2N Cl

In equimolar mixture of aniline, phenol in dil.HCl, amiline in more reactive, as

slightly acidic medium supports.

41. The major product of the following reaction is

1 .3

i DCl equivii DlCh C CH

1. 3 2CH C l Cl CHD 2. 3CH CD Cl CHD l

3. 3 2CH CD CH Cl l 4. 3CH CD l CHD Cl

Ans: 1

Sol: 3

i DCl eqilateralentii DICH C C H

3CH C C H

I

Cl

D

D

42. The standard Gibbs energy for the given cell reaction in kJ mol-1 at 298 K is:

2 2 , º 2 298Zn s Cu aq Zn aq Cu s E V at K

1. -192 2. 384 3. -384 4.192

Ans: 3

Sol: 22aqZn s Cu aq Zn Cu s




0º cellG nFE

2 96000 2

4 96kJ 384kJ

43. The given plots represent the variation of the concentration of a reactant R with

time for two different reactions (i) and (ii). The respective orders of the reactions

are

ln R

time

(i)

R

time

(ii)

1. 0,1 2. 0,2 3. 1,0 4. 1,1

Ans: 3

Sol:

ln R

time

R

time

T is proportional [R] it is 1st order t is proportional to [R] it is 0th order

44. The increasing order of reactivity of the following compounds towards aromatic

electrophilic substitution reaction is

A B C D

Cl OMe Me CN

1. D<A<C<B 2. B<C<A<D 3. D<B<A<C 4. A<B<C<D Ans: 1




Sol:

Cl OMe Me CN

B > C > A > D(dueto + M, Directing)

45. The correct order of the oxidation states of nitrogen in 2 2 2 3, ,NO N O NO and N O is

1. 2 2 3 2NO NO N O N O 2. 2 2 3 2N O N O NO NO

3. 2 2 3 2N O NO N O NO 4. 2 2 3 2NO N O NO N O

Ans: 3

Sol: Compound Oridation state

No +2

2N O +1

2NO +4

2 3N O +3

46. Match the catalysts (colum I) with products (column II).

Column I Column II

Catalyst Product

A) 2 5V O i) Polyethylene

B) 4 3/TiCl Al Me ii) ethanal

C) 2PdCl iii) 2 4H SO

D) Iron oxide iv) 3NH

1. a-iii, b-i, c-ii, d-iv 2. a-ii, b-iii, c-i, d-iv

3. a-iv, b-iii, c-ii, d-i 4. a-iii, b-iv, c-i, d-ii

Ans: 1

Sol: Catalyst Product 2 5V O 2 4H SO 4 3/TiCl Al Me Polyethelene

2PdCl Ethanal Iron oxide 3NH




47. Which of the following statement is not true about sucrose? 1. On hydrolysis, it produces glucose and fructose

2. It is also named as invert sugar

3. It is non-reducing sugar

4. The glycosidic linkage is present between C1 of glucose and C1 of fructose

Ans: 4

Sol: coshydrolysisSucrose glu e fructose

48. The aerosol is a kind of colloid in which:

1. Solid is dispersed in gas 2. liquid is dispersed in water

3. gas is dispersed in liquid 4. gas is dispersed in solid

Ans: 1

Sol: aerosol is a kind of colloid is disersion phase(s) dispersion medium(g)


2 3CH CH

4

3

alkaline KMnOiii H O

1.

2CH CHO

2.

COOH

3.

2CH COOH

4.

3COCH

Ans: 2

Sol:

CH2CH3

4

3

i alkaline KMnOii H O

COOH




50. Consider the van der walls constants, a and b, for the following gases

6 2

2 3 1

/ 1.3 0.2 5.1 4.1

/ 10 3.2 1.7 1.0 5.0

Gas Ar Ne Kr Xe

a atmdm mol

b dm mol

Which gas is expected to have the highest critical temperature

1. Kr 2. Ar 3. Ne 4. Xe

Ans: 1

Sol: To have highest critical temperature a should be high b should below so, it is Kr

51. Among the following, the molecule expected to be stabilized by anion formation is:

2 2 2, , ,C O NO F

1. NO 2. 2C 3. 2F 4. 2O

Ans: 2

Sol: C2 has vacant and if e—enters into this if will be stabilized

52. The ore that contains the metal in the form of fluoride is

1. cryolite 2. Malachite 3. Magnetite 4. Sphalerite

Ans: 1

Sol: Cryolife Na3AlF6

Malachite CuCO3.CU(OH)2

Magnetite Fe3O4

Sphalarite ZnS

53. The one that will show optical activity is:

(en = ethane – 1,2, -diamine)

1. 2. 3. 4.

Ans: 4

Sol: Given which will show optical activity

The Compound should not have plans of symmetry only compounds in options




54. For any given series of spectral lines of atomic hydrogen, let max minv v v be the

difference in maximum and minimum frequencies in 1cm . The ratio balmerv is:

1. 4:1 2. 9:4 3. 5:4 4. 27.5

Ans: 2

Sol: 2 2 2 2

2 2 2 2

1 1 1 191 1 2

1 1 1 1 42 2 3

vyman

Balmer

R RVV R R

55. The major product of the following reaction is:

1. 2.

3. 4. Ans: 3 Sol: 56. The osmotic pressure of a dilute solution of an ionic compound XY in water is four

times that of a solution of 0.01 M BaCl2 in water. Assuming complete dissociation of the given ionic compounds is water, the concentration of XY (in mol L-1) in solution is

1. 44 10 2. 416 10 3. 24 10 4. 26 10

Ans: 4




Sol: 2 – Chloro – 1 – methyl - 4 – nitro benzene

2 3 0.01 4C RT RT

for 2BaCli-factor for xy

C = 0.06 M 57. For a reaction, 2 2 33 2 :N g H g NH g identify dihydrogen 2H as a limiting

reagent in the following reaction mixtures.

1. 35g of N2 + 8g of H2 2. 28g of N2+6g of H2

3. 56g of N2 + 10g of H2 4. 14g of N2+ 4g of H2

Ans: 3

Sol: 2 2 33 2N H NH 56g 10 g 2 mol 5 mol Actually 6 mol of 2H is required but only 5 mol present so it is limiting reagent. 58. Magnesium powder burns in air to give:

1. 3 3 22and MgMg NO N 2. 3 2and MgMgO N

3. MgO only 4. 3 2and MgMgO NO

Ans: 2

Sol: 2 3 2Mh O MgO Mg N


43 2 3

LiAIHCH CH CHCO CH

1. 3 2 2 2 3CH CH CH CO CH 2. 3 2 2 2CH CH CH CH OH

3. 3 2 2CH CH CH CHO 4. 3 2CH CH CHCH OH

Ans: 4

Sol:

O

O4LiAlH

OH

60. Liquid ‘M’ and liquid ‘N’ form an ideal solution. The vapour pressures of pure liquids ‘M’ and ‘N’ are 450 and 700mmHg, respectively, at the same temperature. Then correct statement is:




(xM = Mole fraction of ‘M’ is solution; xN = Mole fraction of ‘N’ in solution ; yM = Mole fraction of ‘M’ in vapour phase; yN = Mole fraction of ‘N’ in vapour phase)

1. M M

N N

x yx y

2. M M

N N

x yx y

3. M M

N N

x yx y

4. M M N Nx y x y

Ans: 3

Sol: 0 450PM

0 700PN 450700

M m

N n

y xy x

0

M mm

t

P XyP

. 1m N

n M

X YX Y

0

nN

PN XYPt

m m

n n

X YX Y




MATHS 61. Let 2S 2 ,2 : 2cos 3sin 0 . Then the sum of the elements of S is:

1. 53

2. 2 3. 13

6

4.

Ans: 2

Sol: solving the trigonometric equation, we get 1sin

2

2 ,2 so 5 7 11, , ,

6 6 6 6

so sum 2

62. If the standard deviation of the numbers 1,0,1,k is 5 where k 0 , then k is

equal to:

1. 1023

2. 2 6 3. 543

4. 6

Ans: 2

Sol:

22 2 2i i2 x x k 2 k5 k 2 6

n n 4 16

63. All the points in the set iS : Ri

i 1 lie on a:

1. circle whose radius is 2 2. circle whose radius is 1

3. straight line whose slope is -1 4. straight line whose slope is 1

Ans: 2

Sol: 2 2

2 2 2

i 1 2x iy x , y1 1 1

So 2 2x y 1 , is a circle of radius 1

64. For any two statements p and q, the negation of the expression p ~ p q is:

1. p q 2. ~ p ~ q 3. p q 4. ~ p ~ q

Ans: 2




Sol: p ~ p q p ~ p p q T p q p q

So ~ p ~ p q ~ p q ~ p ~ q (or) draw truth tables

65. If the function f : R 1, 1 A defined by 2

2

xf x1 x

, is surjective, then A is

equal to:

1. R 1,0 2. R 1,0 3. R 1 4. 0, Ans: 2

Sol: Let 2

2 22

xy y x y x1 x

2x 1 y y

2 yx 0 y 11 y

y y 1 0 y , 1 0,

So A R 1,0

66. If the fourth term in the Binomial expansion of g

6log x2 x x 0

x

is 720 8 , then

a value of x is:

1. 28 2. 8 3. 38 4. 28

Ans: 1

Sol: equate 7

4T 20 8

x8

3 3log6 7

32C . . x 20 8x

Take logaritham on both side to base 8, we get

2x x x8 8 8log 2 log log 2 (or) 21 x 8 (or)

18




67. Slope of a line passing through P 2,3 and intersecting the line, x y 7 at a

distance of 4 units from P, is:

1. 1 51 5

2. 1 71 7

3. 7 17 1

4. 5 15 1

Ans: 2

Sol:

x y 7

2

m,4

090

P 2,3

2 1 m 1 m 1tan

1 m 1 1 m14 7

m 1 11 m 7

simplify get 1 7m1 7

(or) 1 7m1 7

68. If f x is a non-zero polynomial of degree four, having local extreme points at

x 1,0,1 ; Then the set S x R : f x f 0 contains exactly:

1. four irrational numbers

2. two irrational and two rational numbers

3. two irrational and one rational number

4. four rational numbers

Ans: 3

Sol: Let 4 2

1 2 x xf x x x 1 f x c4 2

f 0 f x c 4 2x x c

4 2




2 2x x 1 0

2 2

x 0 (or) x 2

69. Let 3i j

and 2i j 3k

. If 21

, where 1

is parallel to

and 2

is

perpendicular to

, then 1 2

is equal to:

1. 1 3i 9 j 5k2 2. 1 3i 9 j 5k

2

3. 3i 9j 5k 4. 3i 9j 5k

Ans: 1

Sol: Let 1 1k , 2 2k i 3j k

Given 1 2 equating i, j,k coefficient we get 1 21 1k ,k2 2

find

1 21 3i 9 j 5k2

70. If the function f defined on ,6 3

by 2 cos x 1,xcot x 1 4f x

k, x4

is continuous,

then k is equal to:

1. 2 2. 12

3. 12

4. 1

Ans: 2

Sol: x

4

2 cos x 1 0k Lt formcot 1 0

by LH rule we get

1k2

71. Let and be the roots of the equation 2x x 1 0 . Then for y 0 in R,

y 1y 1

1 y

is equal to:

1. 2y y 1 2. 3y 3. 2y y 3 4. 3y 1




Ans: 2

Sol: 2 21, 1, 1

1 1 2 3R R R R & take common y 1

Given determinant 1 1 1

y 1 1 y 11 y

expanding simplify

2 3y y y

72. If the line y mx 7 3 is normal to the hyperbola 2 2x y 1

24 18 , then a value of m

is:

1. 52

2. 152

3. 25

4. 35

Ans: 3

Sol: use 22 22 2

2 2 2

a ba bm n

here 2 2a 24,b 18, m,m 1,n 7 3

Simplify we get 2m5

73. The value of /2 3

0

sin x dxsin x cos x

is:

1. 2

4

2. 2

8

3. 1

4

4. 1

2

Ans: 3

Sol: a a

0 0

use f x dx f a x dx

Simplify we get, given integral 1

4




74 If the line, x 1 y 1 z 2

2 3 4

meets the plane, x 2y 3z 15 at a point P, then

the distance of P from the origin is:

1. 52

2. 2 5 3. 92

4. 72

Ans: 3

Sol: Let P 2t 1,3t 1,4t 2 be any point on the line, P lies on plane also

1 1 9t ,P 2, ,4 ,OP2 2 2

75. Let 10

10

k 1

f a k 16 2 1

, where the function f satisfies f x y f x f y for

all natural numbers x, y and f 1 2 . Then the natural number ‘a’ is

1. 16 2. 2 3. 3 4. 4

Ans: 3

Sol: xf x 2 & simplify (using G.P formula) we get a 3

76. If one end of a focal chord of the parabola, 2y 16x is at 1,4 , then the length of

this focal chord is:

1. 24 2. 22 3. 20 4. 25

Ans: 4

Sol: use 1 2 1 21t t 1, t , t 22

So Q 16, 16 , P 1,4 given PQ 25

77. The solution of the differential equation 2dyx 2y x , x 0dx

with y 1 1 , is:

1. 32

4 1y x5 5x

2. 22

3 1y x4 4x

3. 2

2

x 3y4 4x

4. 3

2

x 1y5 5x

Ans: 3




Sol: LDE 2 dx 2xIF e x

4

2 2 x 3y.x x.x dx c,c4 4

2

2

x 3y4 4x

78. If the tangent to the curve, 3y x ax b at the point 1, 5 is perpendicular to

the line, x y 4 0 , then which one of the following points lies on the curve?

1. 2, 1 2. 2,2 3. 2, 2 4. 2,1 Ans: 3

Sol: f 1 5 1 a b 5 a b 6 ……(1)

1 2 1f x 3x a f 1 3 a 1 a 4

b a 6 2

3f x x 4x 2 check options 2, 2 satisfies

79. If 1 1 1 2 1 3 1 n 1 1 78

..............0 1 0 1 0 1 0 1 0 1

then the inverse of

1 n0 1

is

1. 1 120 1

2. 1 0

12 1

3. 1 0

13 1

4. 1 130 1

Ans: 4

Sol: n 1 n1 78 1 1 2 3... n 1 1

20 1 0 1 0 1

n 1 n 156 13 2 n 13 n N

So increase of 1 13 1 130 1 0 1




80. The value of is 2 0 0 0 2 0cos 10 cos10 cos50 cos 50 is :

1. 34

2. 03 1 cos 202

3. 03 cos 204 4.

32

Ans: 1

Sol: 2 2 3cos A cos B cos A cos B ,

4 if 0A B 60

81. Let the sum of the first n terms of a non-constant A.P., 1 2 3a ,a ,a ,... be

n n 750n A

2

, where A is a constant, If d is the common difference of this A.P.,

then the ordered pair 50d,a is equal to:

1. 50,50 45A 2. A,50 45A 3. 50,50 46A 4. A,50 46A Ans: 4

Sol: n n n n 1

n n 7S 50n A, t S S n 4 A 50

2

50d A,a 46A 50

82. A committee of 11 members is to be formed from 8 males and 5 females. If m is the

number of ways the committee is formed with at least 6 males and n is the number

of ways the committee is formed with at least 3 females, then:

1. m n 68 2. n m 8 3. m n 68 4. m n 78 Ans: 4

Sol: 8 5 8 5 8 5

6 5 7 4 8 3m C . C C . C C . C 28 40 10 78

8 5 8 5 8 56 3 7 4 6 5n C . C C . C C . C 78

83. Four persons can hit a target correctly with probabilities 1 1 1, ,2 3 4

and 18

respectively. If all hit at the target independently, then the probability that the

target would be hit, is:

1. 1

192 2.

2532

3. 25

192 4.

732




Ans: 2

Sol: 1 2 3 7 7 251 P no one hiting the t arg et 1 . . . 12 3 4 8 32 32

84. Let S be the set of all values of x for which the tangent to the curve

3 2y f x x x 2x at x, y is parallel to the line segment joining the points

1,f 1 ,and 1,f 1 , then S is equal to:

1. 1 , 13

2. 1 ,13

3. 1 ,13

4. 1 , 13

Ans: 2

Sol: 1,f 1 1, 2 , 1,f 1 1,0

2slope 1 3x 2x 2 1

23x 2x 1 0 x 1 3x 1 0

1x 1 or x3

85. The integral 2/3 4/3sec x cosec x dx is equal to (Here C is a constant of integration)

1. 1/33cot x C 2. 1/33 tan x C 3. 1/33tan x C 4. 4/33 tan x C4

Ans: 2

Sol: put 2 22 2 2 2 43 3 3 3 3

1 1dx dx put tan x t

cos x sin x cos x. tan x .cos x

1 1

3 34/3dt t . 3 c 3 tan x Ct

86. Letp,q R . If 2 3 is a root of the quadratic equation, 2x px q 0 , then :

1. 2q 4p 16 0 2. 2p 4q 12 0

3. 2p 4q 12 0 4. 2q 4p 14 0




Ans: 3

Sol: 2 3 also root, S.R 2 3 2 3 P p 4

P.R 4 3 q q 1

So 2p 4q 12 16 4 12 0

87. If a tangent to the circle 2 2x y 1 intersects the coordinate axes at distinct points

P and Q, then the locus of the mid-point of PQ is:

1. 2 2 2 2x y 16x y 0 2. 2 2 2 2x y 2x y 0

3. 2 2x y 2xy 0 4. 2 2 2 2x y 4x y 0 Ans: 4

Sol: Let P cos ,sin be any point on circle, tangent at P is x cos ysin 1

1 1A ,0 ,B 0,

cos sin

1 1P x , y be mid point of

1 10 0cos sinAB ,

2 2

1 11 12x ,2y

cos sin

1 1

1 1cos ,sin2x 2y

2 21 1

1 1 14x 4y

2 2 2 21 1 1x y 4x y

Locus of 1 1P x , y is 2 2 2 2x y 4x y 0

88. The area (in sq. units) of the region 2A x, y : x y x 2 is:

1. 92

2. 103

3. 136

4. 316

Ans: 1




Sol: 2x y x 2 2x x 2 0

Use

3 32 2 2

2

b 4ac 1 4.1. 2 27 96a 6.1 6 2

(or)

Draw diagram and find 2

2

1

9x 2 x dx2

89. Let f x 15 x 10 ;x R. Then the set of all values of x, at which the function,

g x f f x is not differentiable, is:

1. 5,10,15 2. 10 3. 10,15 4. 5,10,15,20 Ans: 1

Sol: f x 15 x 10 , x R

x 10 if x 520 x 5 x 10

g x f f xx 10 x 1530 x if x 15

90. A plane passing through the points 0, 1,0 and 0,0,1 and making an angle 4

with the plane y z 5 0 , also passes through the point:

1. 2, 1,4 2. 2, 1, 4 3. 2,1, 4 4. 2,1,4

Ans: 4

Sol: x y z 1a 1 1

is the equation of plane ……(1)

y z 0 ……(2)

2

1 .0 1.1 1 1acos

4 1 1 1 1 1a




2

2

21 14 2a2 12 2

a

21 12 aa 2

So plane equation is

2x y z 1 2x y z 1 , 2x y z 1

Check options 2,1,4 satisfies second equation

Prepared by

Sri Chaitanya Faculty

2019 apr jee-main q p k s - sri chaitanya · 2019-04-15 · 2019_jee-main question paper_key &...

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