2019 Jee-Main Q P K S · mirror ( )M2 parallel to the first mirror( )M1. The angle between the two...

2 0 1 9 _ Jee-Main Question Paper_ Key & Solutions


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

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PHYSICS 1. Two plane mirrors are inclined to each other such that a ray of light incident on the first

mirror 1( )M and parallel to the second mirror 2( )M is finally reflected from the second

mirror 2( )M parallel to the first mirror 1( )M . The angle between the two mirrors will be:

1) 090 2) 045 3) 075 4) 060

Ans. 4

Sol.

060

2. A particle having the same charge as of electron moves in a circular path of radius 0.5 cm

under the influence of a magnetic field of 0.5 T. If an electric field of 100 V/m makes it

to move in a straight path, then the mass of the particle is (Given charge of electron

–191.6 10 C )

1) –242.0 10 kg 2) –191.6 10 kg 3) –271.6 10 kg 4) –319.1 10 kg

Ans. 4

Sol. 2mvqvB

R

mv qBR

Path to be straight line, qE qvR

E v B

So 2

242 10qB R

m kgE

3. In a Young's double slit experiment, the slits are placed 0.320 mm apart. Light of

wavelength 500 nm is incident on the slits. The total number of bright fringes that are

observed in the angular range 0 030 30 is:

1) 320 2) 641 3) 321 4) 640




Ans. 2

Sol. 030d Sin n

320n

So, Number of maxima (Including central maxima) 2 1 641n

4. The pitch and the number of divisions, on the circular scale, for a given screw gauge

are 0.5 mm and 100 respectively. When the screw gauge is fully tightened without any

object, the zero of its circular scale lies

3 divisions below the mean line.

The readings of the main scale and the circular scale, for a thin sheet, are

5.5 mm and 48 respectively, the thickness of this sheet is

1) 5.755 mm 2) 5.725 mm 3) 5.740 mm 4) 5.950 mm

Ans. 2

Sol. 20.5 10

PitchLC mn

Noof Divisions

Positive zero error 2 23 0.5 10 1.5 10mm mm

Reading MSR CSR- ve zeroerror =5.725mm

5. At a given instant, say 0t , two radioactive substances A and B have equal activities. The

ratio B

A

R

R of their activities after time t itself decays with time t as –3t

e , If the half-life of

A is 2l n , the half-life of B is

1) 2

2

l n 2) 2 2l n 3)

2

4

l n 4) 4 2l n

Ans. 3

Sol. 30

0

B

B A

A

tt tB

t

A

R R ee e

R R e

So, 3B A

Hence n 2

n 2 1A A

A

lT l

n 2 n 2

4B

B

l lT

6. The top of a water tank is open to air and its water level is maintained. It is giving out

30.74 m water per minute through a circular opening of 2cm radius in its wall. The depth

of the centre of the opening from the level of water in the tank is close to:

1) 9.6m 2) 4.8m 3) 2.9m 4) 6.0m




Ans. 2

Sol. 40.744 10 2

60gh

4.8h m

7. Ge and Si diodes start conducting at 0.3 V and 0.7 V respectively. In the following figure

if Ge diode connection are reversed, the value of 0V changes by: (assume that the Ge

diode has large breakdown voltage)

1) 0.6 V 2) 0.8 V 3) 0.4 V 4) 0.2 V

Ans. 3

Sol. Initiall current will flow through Ge diode drop will be 0.3v.

In the second case, Si diode wll conduct and drop across the combination is 0.7v. so,

change in Voltage =0.4V

8. A power transmission line feeds input power at 2300 V to a step down transformer with

its primary windings having 4000 turns. The output power is delivered at 230 V by the

transformer. If the current in the primary of the transformer is 5A and its efficiency is 90%

, the output current would be:

1) 25 A 2) 50 A 3) 35 A 4) 45 A

Ans. 4

Sol. 0.9out s s

in p p

P V I

P V I

45sI A

9. A rod of mass 'M' and length '2L' is suspended at its middle by a wire. It exhibits

torsional oscillations; If two masses each of 'm' are attached at distance 'L/2' from its

centre on both sides, it reduces the oscillation frequency by 20% . The value of ratio m/M

is close to:

1) 0.17 2) 0.37 3) 0.57 4) 0.77




Ans. 2

Sol. 1

2

Kf

I

and 1 1

2

Kf

I

where K is torsigional constant

so, 1

1

20 11

100 5

f f I

f I

1

4

5

I

I

1

16

25

I

I

24

12

M LI

and

21

2

mLI I So, 0.37

m

M

10. In a car race on straight road, car A takes a time t less than car B at the finish and passes

finishing point with a speed 'v' more than that of car B. Both the cars start from rest and

travel with constant acceleration 1a and 2a respectively. Then 'v' is equal to:

1) 1 2

2

a at

2) 1 22a a t 3) 1 2

1 2

2a at

a a 4) 1 2a a t

Ans. 4

Sol. Given 1 2 1 2and t t t

Also, 1 1 1 2 2 2a t and v a t

So, 1 1 2 2v a t a t 1 2a a t 1 2

2 1

t a

t a

11. A 15 g mass of nitrogen gas is enclosed in a vessel at a temperature 027 C . Amount of heat

transferred to the gas, so that rms velocity of molecules is doubled, is about:

[Take 8.3 / R J K mole ]

1) 10 kJ 2) 0.9 kJ 3) 6 kJ 4) 14 kJ

Ans. 1

Sol. 15 54

28 2v

RnC T T T

10kJ

12. A mass of 10 kg is suspended vertically by a rope from the roof. When a horizontal force

is applied on the rope at some point, the rope deviated at an angle of 045 at the roof point.

If the suspended mass is at equilibrium, the magnitude of the force applied is –210 g ms

1) 200 N 2) 100 N 3) 140 N 4) 70 N




Ans. 2

Sol.

F

T

mg

2

Tmg and

2

TF So, 100f mg N

13. A particle is executing simple harmonic motion (SHM) of amplitude A, along the x-axis,

about 0x . When its potential Energy (PE) equals kinetic energy (KE), the position of

the particle will be:

1) 2

A 2)

2 2

A 3)

2

A 4) A

Ans. 3

Sol. 2 2 2 2 21 1

2 2m A x mx

2

Ax

14. In the given circuit the internal resistance of the 18 V cell is negligible. If 1 400R ,

3 100R and 4 500R and the reading of an ideal voltmeter across 4R is 5V, then the

value 2R will be:

1) 300 2) 230 3) 450 4)550




Ans. 1

Sol. As, 3R and 4R are in series, voltage across 3 1R V

So, voltage across 3 4,R R and 2R combination is 6V

Therefore , 2 3 4

1 1 1

200R R R

2 300R

15. A musician using an open flute of length 50 cm produces second harmonic sound waves.

A person runs towards the musician from another end of a hall at a speed of 10 km/h. If

the wave speed is 330 m/s, the frequency heard by the running person shall be close to:

1) 753 Hz 2) 500 Hz 3) 333 Hz 4) 666 Hz

Ans. 4

Sol. 2

nf

l

For 2nd harmonic, 2n and 660f Hz velocity of observers Vo=10Km/h=2.78m/s

Hence apparent frequency 0v vf f

v

=666Hz

16. The position co-ordinates of a particle moving in a 3 D coordinate system is given by

cosx a t

siny a t

and z a t

The speed of the particle is:

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

Ans. 3

Sol. 2 2 2 2x y z

v v v v a

17. In a communication system operating at wavelength 800 nm, only one percent of source

frequency is available as signal bandwidth. The number of channels accommodated for

transmitting TV signals of band width 6 MHz are (Take velocity of light

8 –343 10 / , 6.6 10c m s h J s )

1) 63.75 10 2) 54.87 10 3) 63.86 10

4) 56.25 10




Ans. 4

Sol. 143.75 10C

f hz

1% of 17 63.75 10 3.75 10f hz Mhz

Number of channels 63.75 10

6

56.25 10

18. A force acts on a 2 kg object so that its position is given as a function of time as 23 5x t .

What is the work done by this force in first 5 seconds?

1) 850J 2) 900J 3) 950J 4) 875J

Ans. 2

Sol. 6 /dz

v t m sdt

212 36 900

2w k t J

19. Two point charges 1 10q C and 2 (–25 )q C are placed on the x-axis at 1x m and 4x

m respectively. The electric field (in V/m) at a point 3y m on y-axis is,

9 2 2

0

19 10

4take Nm C

1) 263 27 10i j 2) 281 81 10i j 3) 263 27 10i j 4) 281 81 10i j

Ans. 3

Sol. 21 9 27 10

vE i j

m

and 22 72 54 10

vE i j

m

So, 21 2 63 27 10

vE E E i j

m

20. A carbon resistance has a following colour code. What is the value of the resistance ?

1) 1.64 5%M 2) 530 5%k 3) 64 10%k 4) 5.3 5%M

Ans. 2

Sol. 453 10 5%R 530 5%k




21. A parallel plate capacitor with square plates is filled with four dielectrics of dielectric

constants 1 2 3 4, , ,K K K K arranged as shown in the figure. The effective dielectric constant

K will be:

1)

1 2 3 4

1 2 3 42

K K K KK

K K K K

2)

1 2 3 4

1 2 3 4

K K K KK

K K K K

3)

1 4 2 3

1 2 3 42

K K K KK

K K K K

4)

1 3 2 4

1 2 3 4

K K K KK

K K K K

Ans. (Bonus)

Sol.

1K C 2K C

3K C 4K C

Kc

Hence

1 2 3 4

1 2 3 4

K K K KK

K K K K

22. Two Carrnot engines A and B are operated in series. The first one, A, receives heat at

1 600T K and rejects to a reservoir at temperature 2T . The second engine B receives heat

rejected by the first engine and, in turn, rejects to a heat reservoir at 3 400 T K . Calculate

the temperature 2T if the work outputs of the two engines are equal:

1) 400 K 2) 600 K 3) 500 K 4) 300 K




Ans. 3

Sol. 1 2 2 3 1 2 1 22 2 2

T TT

23. A rod of length 50cm is pivoted at one end. It is raised such that if makes an angle of 030

from the horizontal as shown and released from rest. Its angular speed when it passes

through the horizontal –1 in rad s will be –210g ms

1) 30 2) 30

2 3)

302

4) 20

3

Ans. 1

Sol. Applying energy conservation

2

21

4 2 3

l mLmg

36 /rad s

24. Charge is distributed within a sphere of radius R with a volume charge density

2 /

2,r aA

r er

where A and a are constants. If Q is the total charge of this charge

distribution, the radius R is:

1) log 12 2

a Q

aA

2) log 12

Qa

aA

3) 1

log1

2

aQ

aA

4) 1

log2 1

2

a

Q

aA

Ans. 4




Sol. 2

2

24

R R r

a

O O

tsdv e r dr

r

2

4 12

R

aa

A e

1log

2 12

aR

Aa

25. One of the two identical conducting wires of length L is bent in the form of a circular

loop and the other one into a circular coil of N identical turns. If the same current is

passed in both, the ratio of the magnetic field at the central of the loop LB to that at

the centre of the coil CB , i.e. L

C

B

Bwill be:

1) 1

N 2) 2

N 3) 2

1

N 4) N

Ans. 3

Sol. For loop 22

LL R R

0 0

2L

i iB

R L

2for coil L N r 2

Lr

20 0

2C

Ni iB N

r L

So,

2

1L

C

B

B N

26. The magnetic field associated with a light wave is given, at the origin, by

0B B 7 7sin 3.14 10 sin 6.28 10ct ct . If this light falls on a silver plate having a work

function of 4.7 eV, what will be the maximum kinetic energy of the photo electrons?

8 –1 –343 10 , 6.6 10c ms h J s

1)7.72eV 2)8.52eV 3)12.5eV 4)6.82eV

Ans. 1

Sol. To calculate maxU we must take the highest frequency

So 34 7 8

max 19

6.6 10 6.28 10 3 104.7

2 3.14 1.6 10U

7.675 7.7ev

27. The energy required to take a satellite to a height 'h' above Earth surface (radius of Earth

36.4 10 km ) is 1E and kinetic energy required for the satellite to be in a circular orbit at

this height is 2E . The value of h for which 1E and 2E are equal, is:




1) 41.28 10 km 2) 36.4 10 km 3) 33.2 10 km 4) 31.6 10 km

Ans. 3

Sol. 1

1 1E GMm

R R h

and

2 2

GMmE

R h

If 1 2E E , 3200

2

Rh km

28. Expression for time in terms of G (universal gravitational constant), h (Planck constant)

and c (speed of light) is proportional to:

1) 3

Gh

c 2)

5hc

G 3)

3c

Gh 4)

5

Gh

c

Ans. 4

Sol. 1 3 2G M LT

2 1h ML T

1C LT

x y zt G H C

1 1,x yz z

and 5z so 5

Ght

c

29. The energy associated with electric field is EU and with magnetic field is BU for an

electromagnetic wave in free space. Then:

1) 2B

E

UU 2) E BU U 3) E BU U 4) E BU U

Ans. 3

Sol. A BU U

30. A series AC circuit containing an inductor 20 mH , a capacitor (120 )F and a resistor

(60 ) is driven by an AC source of 24 V/50 Hz. The energy dissipated in the circuit in

60 s is:

1) 32.26 10 J 2) 33.39 10 J 3) 25.65 10 J 4) 25.17 10 J

Ans. 4

Sol. 6.28LX L

1

26.53cXc

So, 22 2 4010.5L Cz R X X

Energy dissipated 2

2517

vR t J

z




CHEMISTRY 31. Good reducing nature of H3PO2 attributed to the presence of

1) One P-OH bond 2) One P-H bond

3) Two P-H bonds 4) Two P-OH bonds

Ans. 3

Sol: Then is two P-H bonds in the structure.

32. The complex that has highest crystal field splitting energy (), is :

1) K3 [Co (CN)6] 2) [Co(NH3)5(H2O)]Cl3

3) K2 [CoCl4] 4) [Co (NH3)5Cl]Cl2

Ans. 4

Sol: The field strength of ligand “CN” is high.

33. The metal that forms nitride by reacting directly with N2 of air, is:

1) K 2) Cs 3) Li 4) Rb

Ans. 3

Sol: Among Alkali metals “Li” reacts with air forming both oxide and nitride.

34. In which of the following processes, the bond order has increased and paramagnetic

character has changed to diamagnetic?

1) 2 2N N 2)NO NO 3)

22 2O O 4) 2 2O O

Ans. 2

Sol: Bond order Magnetic behaviour

No 2.5 paramagnetic

NO 3 diamagnetic

35. The major product of the following reaction is:




1) 2) 3) 4)

OH

3CH

O

Ans: 4

Sol:

OH

OH 3CH

3CH

C

O2CH

2CH

2CH OH

O

OH

3CH

3AlCl

O

O

36. The transition element that has lowest enthalpy of atomisation, is :

1) Zn 2) Cu 3) V 4) Fe

Ans: 2

Sol:

TiVCr

Mn

Fe Co Ni

Cu

AtomisationH




37. Which of the following combination of statements is true regarding the

interpretation of the atomic orbitals ?

(a) An electron in an orbital of high angular momentum stays away from the

nucleus than an electron in the orbital of lower angular momentum.

(b) For a given value of the principal quantum number, the size of the orbit is

inversely proportional to the azimuthal quantum number.

(c) According to wave mechanics, the ground state angular momentum is equal to2

h

.

(d) The plot of Vs r for various azimuthal quantum numbers, shows peak

shifting towards higher r value.

1) (b), (c) 2) (a), (d) 3) (a), (b) 4) (a), (c)

Ans: 4

Sol: 2

hmvr n

-constant

If mvr is high n is high so electron stays away from the nucleus.

For ground state 1n .

38. The tests performed on compound X and their inferences are:

Test Inference

(a) 2,4- DNP test Coloured precipitate

(b) Iodoform test Yellow precipitate

(c) Azo-dye test No dye formation

Compound 'X' is:

1) 2) 3) 4)

Ans: 2

Sol: 2,4DNP test is given by aldehydes and ketones

Iodo form test given by C

O

3CH




Azo dye test not given by

3CH3CH

N

AR

39. The major product formed in the

following reaction is:

1) 3CH

O

H

OH

2)

3) 4)

Ans: (no option)

Sol:

CHO3 2OHCH CHO CH CHO

2CHOH CH CHO

40. For the reaction, 2A B products, when the concentrations of A and B both

wrere doubled, the rate of the reaction increased from –1 –10.3 mol L s to

– –12.4 lmolL s . When the concentration of A alone is doubled, the rate increased

from 1 –10.3 molL s to –1 –10.6 mol L s

Which one of the following statements is correct ?

1) Order of the reaction with respect to Bis2

2) Order of the reaction with respect to Ais2

3) Total order of the reaction is 4

4) Order of the reaction with respect to B is 1




Ans: 4

Sol: 1x y

R K A B

0.3 (1)

2.4 2 2 (2)

0.6 2 (3)

x y

x y

x Y

K A B

K A B

K A B

Order w.r.t B is (2)

From eq (1)&(3)

1 1

2 2

1

x

x

From eq (2)&(3)

4 (2)

2

y

y

41. The correct sequence of amino acids present in the tripeptide given below is :

1) Leu - Ser – Thr 2) Thr - Ser- Leu

3) Thr - Ser - Val 4) Val - Ser – Thr

Ans: 4

Sol:

3CH CH CH CHO

2NH

3CH 2NH

CH C OH

O

2CH OH2NH

OOH

3CH CH CH C OH

Valine serine Threonine




42. The correct statement regarding the given Ellingham diagram is:

1) At 800°C, Cu can be used for the extraction of Zn from ZnO

2) At 500 C, coke can be used for the extraction of Zn from ZnO

3) Coke cannot be used for the extraction of Cu from Ca2O.

4) At 1400°C, Al can be used for the extraction of Zn from ZnO

Ans: 4

Sol: At 01400 G for the formation of 2 3Al O is more negative compound to that of “zero”.

43. For the following reaction, the mass of water produced from 445 g of C57H110O6 is :

2C57H110O6(s) + 163O2(g)114CO2(g) + 110 H2OP(1)

1) 495 g 2) 490 g 3) 890 g 4) 445 g

Ans: 1

Sol: 57 110 6 2 2 2

16357 55

2C H O O CO H O

445

0.5890compoundn moles

Due to equation 1 mole of compound gives 55 moles of 2H O so 0.5 moles will give

0.5 55moles

2 0.5 55 18weight of H O formed g 495grms




44. The correct match between Item I and Item II is :

Item I Item II

(A) Benzaldehyde (P) Mobile phase

(B) Alumina (Q) Adsorbent

(C)Acetonitrile (R) Adsorbate

1) (A) (Q);(B) (R);(C) (P)

2) (A) (P); (B) (R); (C) (Q)

3) (A) (Q); (B) (P); (C) (R)

4) (A) (R); (B) (Q); (C) (P)

Ans: 4

Sol: Conceptual

45. The increasing basicity order of the following compounds is :

(A) 3 2 2CH CH NH (B) (C) (D)

1) (D)<(C)<(A)<(B) 2) (A)<(B)<(D)<(C)

3) (A)<(B)<(C)< (D) 4) (D)<(C)<(B)<(A)

Ans: 1

Sol: Donating groups will increase the basicity of amines, Ph N

3CH

H is least basic due to

delocalization of electron pair.

46. For coagulation of arscnious sulphide sol, which one of the following salt solution

will be most effective?

1) AlCl3 2) NaCl 3) BaCl2 4) Na3PO4

Ans: 1

Sol: To coagulate a negative colloid cation with higher charge is more effective.

47. At100°C, copper (Cu) has FCC unit cell structure with cell edge length of x

Å. What is the approximate density of Cu (in g cm–3) at this temperature?

[Atomic Mass of Cu = 63.55u]




1) 3

105

X 2)

3

211

X 3)

3

205

X 4)

3

422

X

Ans: 4

Sol: 3 23 8 3

4 63.5

(6 10 )( 10 )a

ZMd

N a X

= 3

3 3

4 63.5 421.71610 /

6gm cm

X X

48. The major product obtained in the following reaction is :

1) 2)

3) 4)

Ans: 3

Sol:

3 3CH C O C CH

2NH

O

OH OH

3NH C CH

OO

NH H

3CH C O

3O C CH

OOH




49. Which of the following conditions in drinking water causes

methemoglobinemia?

1) > 50ppm of lead 2) > 100 ppm of sulphate

3) > 50 ppm of chloride 4) > 50 ppm of nitrate

Ans: 4

Sol: Drinking water containing > 50 ppm of nitrate causes methemoglobinemia.

50. Homoleptic octahedral complexes of a metal ion 'M3+' with three monodentate ligands

L1, L2 and L3 absorb wavelengths in the region of green, blue and red respectively. The

increasing order of the ligand strength is:

1) L2 < L1 < L3 2) L3 < L2 < L1

3) L3 < L1 <L2 4) L1 < L2 < L3

Ans: 2

Sol: If ligand strength is high crystal field splitting is high so energy absorbed is high so wave

length is low So L3 < L2 < L1

51. The product formed in the reaction of cumene with O2 followed by treatment with dil. HCl

are:

1) 2)

3) 4)

Ans: 3

Sol:

3CH C 3CH3CH C 3CH

OO H

OH

3CH C 3CH

O

H

2O 3H O




52. The temporary hardness of water is due to :-

` 1) Ca (HCO3)2 2) NaCl 3) Na2SO4 4) CaCl2

Ans: 1

Sol: Hydrogen carbonates of calcium and magnesium causes temporary hardness

53. The entropy change associated with the conversion of 1 kg of ice at 273 K to water

vapours at 383 K is:

(Specific heat of water liquid and water vapour are 4.2 kJ K–1 kg–1 and 2.0 kJ K–1 kg–1 ;

heat of liquid fusion and vapourisation of water are 344 kJ kg–1 and 2491 kJ kg–1,

respectively). (log 273 = 2.436, log 373 = 2.572, log 383 = 2.583)

1) 7.90 kJ kg–1 K–1 2) 2.64 kJ kg–1 K–1

3) 8.49 kJ kg–1 K–1 4) 9.26 kJ kg–1 K–1

Ans: 3

Sol: 2 ( ) 2 ( )

(273 ) (273 )

S lH O H O

K K

0 33 1

1

344 101.26 10

273f

f

HS KJK

T

2 ( ) 2 ( )

(273 ) (383 )

l lH O H O

K K

33

2

10 383344 18 10 2.303log 0.677

18 273S KJ

2 ( ) 2 ( )

(383 ) (383 )

l gH O H O

K K

1

3

24916.5

383V

b

HS KJK

T

1.26 0.677 6.5 8.49S

54. The pH of rain water, is approximately :

1) 6.5 2) 7.5 3) 5.6 4) 7.0

Ans: 3

Sol: Rain water is acidic so-5.6

55. If the standard electrode potential for a cell is 2 V at 300 K, the equilibrium

constant (K) for the reaction 2 2( ) ( ) ( ) ( )Zn s cu aq Zn aq Cu s at 300 K

is approximately.

(R = 8 JK–1 mol–1, F = 96000 C mol–1)




1) e 160 2) e320 3) e–160 4) e–80

Ans: 1

Sol: 0 0 lnG nFE RT k 0 lnRT

E knf

1608 300 965002 ln ln

2 96500 600k k k e

56. A solution containing 62 g ethylene glycol in 250 g water is cooled to –10°C. If Kf for

water is 1.86 K kg mol–1, the amount of water (in g) separated as ice is:

1) 32 2) 48 3) 16 4) 64

Ans: 4 Sol: 01

1.86 7.440.25f f fT K m T C

New freezing point

0 110 1.86

water

molC

m

1.860.186 186

10250 186 64

waterm kg g

gm gm gm

57. When the first electron gain enthalpy (egH) of oxygen is –141 kJ/mol, its gain

enthalpy is:

1) almost the same as that of the first

2) negative, but less negative than the first

3) a positive value

4) a more negative value than the first

Ans: 3

Sol: Conceptual

58. The major product of the following reaction is :




1) 2) 3) 4)

Ans: 3

Sol:

2C NH

O

2 3CH CH

2 /i Br hv

ii KOH N-H

O

3CH

2C NH

O

3CH CH2

OH

H O

Br

O

3CH CH

Br

NH

59. Which of the following compounds is not aromatic?

1) 2) 3) 4) Ans: 3

Sol: not satisfying (4n+2) It is anti aromatic. 60. Consider the following reversible chemical reactions:

1

2

2 2

2 2

( ) ( ) 2 ( )......(1)

6 ( ) 3 ( ) 3 ( ).....(2)

K

K

A g B g AB g

AB g A g B g

The relation between K1 and K2 is:

1) 3

2 1K K 2) 3

2 1K K 3) K1K2 = 3 4) 1 2

1

3K K

Ans: 2

Sol: 3

2 1 31

1( )K K

K




MATHEMATICS 61. The number of all possible positive integral values of for which the roots of the

quadratic equation, 26x 11x 0 are rational numbers is:

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

Ans. 3

Sol. 11 121 24

x12

121-24 is a perfect square

= 3,4,5

62. Let a i j 2k,

1 2b b i b j 2k, and c 5i j 2k,

be three vectors such that

the projection vector of b

on a

is a

. If a b

is perpendicular to c

, then b

is equal to:

1) 6 2) 32 3)4 4) 22

Ans. 1

Sol. a b is to c a b .c 0 a.c b.c 0

1 2

1 2

5 1 2 5b b 2 0

5b b 10........ 1

Projection b on a

is 2

b.aa a

a

1 21 2

b b 21 b b 2...... 2

4

Solve eq. 1& 2

63. If

8 6

22 7

5x 7xf x dx, x 0 ,

x 1 2x

and f 0 0, then the value of f 1 is:

1)1

2 2)

1

2 3)

1

4 4)

1

4

Ans. 4




Sol.

8 6

214 5 7

5x 7xf x dx

x x x 2

6 8

25 7

5x 7xdx

x x 2

5 7x x 2 t 6 85x 7x dx dt

7

2 2 7

dt 1 xf x c c

t t x 1 2x

1 1f 0 0 c 0, f 1

1 1 2 4

64. If both the roots of the quadratic equation 2x mx 4 0 are real distinct and they lie in

the interval 1,5 then m lies in the interval:

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

Ans. 1

Sol. 51

f 1 .f 5 0

1 m 4 25 5m 4 0

29m 5 m 0

5

5

++

29

5 29m ,5 ,

5




And 20 m 16 0

++

44

m , 4 4,

44 5 29

5

29m , 4 4,5 ,

5

65. Let a,b and c be the th th7 ,11 and th13 terms respectively of a non- constant A.P. If these

are also the three consecutive terms of a G.P., then a

c is equal to:

1)2 2)4 3)7

13 4)

1

2

Ans. 2

Sol. A 10d A 12d

rA 6d A 10d

where a,b,c are A.P.also G.P.

A+10d=r (A+6d)……….(1)

A+12d=r (A+10d)……..(2)

Eq (2) - eq (1) 2d = r ( 4d)

1

r2

2 2

a a 14

c ar r

66. Let 0z be a root of the quadratic equation, 2 81 930 0x x 1 0, If z 3 6iz 3iz ,

Then arg z is equal to:

1)4

2)0 3)

6

4)

3




Ans. 1

Sol. Roots of 2x x 1 0 is 2,

0

81 93

z

z 3 6i 3i 3 6i 3i 3 3i

Arg 1 1z Tan

1 4

67. If 0 x2

, then the number of values of x for which sin x sin 2x sin3x 0, is:

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

Ans. 3

Sol. z 2sin 2x cos x sin 2x 0

1

sin 2x 0 cosx2

2x 0, ,2 ........ x3

x 0, , ......2

x 0,3

68. For each x R, let x be the greatest integer less than or equal to x. Then

x 0

x x x sin xlim

x

is equal to:

1)sin 1 2)-sin 1 3)1 4)0

Ans. 2

Sol. x 0 x 0 h,h 0 and x 1

x 0

h 1 h sin 1lim sin1

h




69. If

t t t

t t t t t

t t t

e e cos t e sin t

A e e cos t e sin t e sin t e cos t

e 2e sin t 2e cos t

, then A is:

1)not invertible for any t R 2) invertible only if t .

3)invertible for all t R 4) invertible only if t .2

Ans. 3

Sol. t t t

1 cos t sin t

A e e e 1 cos t sin t sin t cos t

1 2sin t 2cos t

t

1 cos t sin t

e 0 2cos t sin t 2sin t cos t

0 2sin t cos t 2cos t sin t

te 0

70. If /3

0

tan 1d 1 k 0 ,

2ksec 2

then the value of k is:

1)4 2)2 3)1

2 4)1

Ans. 4

Sol. /3

0

tan secd

2ksec .ssec

2sec t

2sec tan 2tdt

22

211

t dt 1 1

t2k.t.t 2k

1 1 1

1 12k 2 2

71. If x=3 tan t and y = 3sec t, then the value of 2

2

d y

dx at t

4

, is:

1)1

6 2)

1

6 2 3)

3

2 2 4)

1

3 2




Ans. 2

Sol. 2dx dy3sec t 3sec t tan t

dt dt

dy tan t

sin tdx sec t

2

2 2

d y dt 1cos t cos t.

dx dx 3sec t

72. The coefficient of 4t in the expansion of

361 t

1 t

is:

1)14 2)15 3)10 4)12

Ans. 2

Sol. 361 t 1 t

r

6 12 18 rc1 3t 3t t 3 r 1 t

2

6 12 18 rc1 3t 3t t r 2 t

2

4ct coefficient 6

73. Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and

the other two vertices lie on coordinate axes with integral coordinates. If each triangle in

S has area 50sq. units, then the number of elements in the set S is:

1)18 2)32 3)9 4)36

Ans. 3

Sol. 1

ab 50 ab 1002

Possible integer of a,b are

a:1 2 4 5 10

b:100 50 25 20 10




(0,0) (a,0)

(0,b)

No of points 9

74. The equation of the plane containing the straight line x y z

2 3 4 and perpendicular to the

plane containing the straight lines x y z

3 4 2 and

x y z

4 2 3 is:

1)3x 2y 3z 0 2) x 2y z 0 3)5x 2y 4z 0 4) x 2y 2z 0

Ans. 2

Sol.

x y z

2 3 4 0

8 1 10

x 30 4 y 20 32 z 2 24 0

x 2y z 0

75. An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the

drawn ball green, then a red ball is added to the urn and if the drawn ball is red, then a

green ball is added to the urn; the original ball is not returned to the urn. Now a second

ball is drawn at random from it. The probability that the second ball is red, is

1)32

49 2)

27

49 3)

21

49 4)

26

49

Ans. 1

Sol. A box contain 5R, 2G balls

1E a green ball in drawn no of balls in box 6R,1G

2E a red ball in drawn no of balls in box 4R,3G




A= Second ball is red

2 6 5 4p A

7 7 7 7

76. Let f : 0,1 R be such that f xy f x .f y , for all x,y 0,1 , and f 0 0. if

y y x satisfies the differential equation, dyf x

dx with y 0 1, then

1 3y y

4 4

is equal to:

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

Ans. 4

Sol. x 0 y f 0 f 0 f 0 1

Put y = 0 f 0 f x f 0 f x 1

dy1 y x c y 0 1

dx

1 0 c c 1 y x 1

1 3 1 3

y y 1 1 34 4 4 4

77. Let A 4, 4 and B 9,6 be points on the parabola, 2y 4x . Let C be chosen on the arc

AOB of the parabola, where O is the origin, such that the area of ACB is maximum.

Then, the area (in sq. units) of ACB , is:

1)1

302

2)1

314

3)32 4)3

314

Ans. 2

Sol. 2A 4, 11 B 9,6 C t ,2t

2

2

4 4 11

ACB 9 6 1 30 5t 5t 52

t 2t 1

d 1 1 1

0 t c ,1 area 31dt 2 4 4




78. If 1x sin sin10 and 1y cos cos10 , then y x is equal to:

1)10 2)0 3) 4) 7

Ans. 3

Sol. 1x sin sin10 3 10

1y cos cos10 4 10 y x

79. The area of the region A x, y : 0 y x x 1 and 1 x 1 in sq. units, is:

1)2 2)1

3 3)

4

3 4)

2

3

Ans. 3

Sol. 2

2

y x x 1

x 1, x 0

x 1 x 0

0 1

2 2

1 0

area x 1dx x 1dx

-1 1

80. If the system of linear equations

x 4y 7z g

3y 5z h

2x 5y 9z k

Is consistent, then:




1) g h k 0 2) 2g h k 0 3) g 2h k 0 4) g h 2k 0

Ans. 2

Sol. 3 3 1

1 4 7 g

0 3 5 h R R 2R

2 5 9 k

3 3 2

1 4 7 g

0 3 5 h R R R

0 3 5 k 2g

1 4 7 g

0 3 5 h

0 0 0 2g h k

system of eqs consistent 2g+h+k = 0

81. The sum of the following series

2 2 2 2 2 2 2 2 2 29 1 2 3 12 1 2 3 4 15 1 2 ....51 6 .....

7 9 11

up to 15 terms, is:

1)7830 2)7820 3)7510 4)7520

Ans. 3

Sol. 2 2 2

thn

3n 1 2 .....nn term t

2n 1

23n n n 1 n n 1

6 2

Sum of n terms 3 2n n

1S t n n

2

Put n = 15

82. The logical statement ~ ~ p q p r ~ q r is equivalent to:

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

Ans. 4

Sol. p ~ q p r ~ q r

p ~ q r ~ q r




p r ~ q

83. Let f be a differentiable function from R to R such that 3/2f x f y 2 x y , for all

x, y R, If f 0 1 then 1

2

0

f x dx is equal to:

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

2

Ans. 1

Sol. 1/2

x y x y

f x f ylim lim2 x y

x y

1 1f y 0 f y 0 f y k

Given f 0 1 f x 1

1 1

12

00 0

f x dx 1dx x 1

84. A hyperbola has its centre at the origin, passes through the point (4,2) and has transverse

axis of length 4 along the x- axis. Then the eccentricity of the hyperbola is:

1)2 2)2

3 3)

3

2 4) 3

Ans. 2

Sol. Given 2a = 4 a 2

Eq. of hyperbola 2 2

2 2

x y1

a b

Passing through (4,2)

2

2

16 4 41 b

4 b 3

Eccentricity 2 2

2

a be

a

85. The number of natural numbers less than 7,000 which can be formed by using the digits

0,1,3,7,9 (repetition of digits allowed) is equal to:




1)250 2)372 3)374 4)375

Ans. 3

Sol. Single digit no.s = =4

2 digit no.s = =4 5=20

3 digit no.s = =425=100

4 digit no.s which are less than 7000 is =2555 = 250

Total no 4+20+100+250=374

86. A data consists of n observation:

n

2

1 2 n ii 1

x ,x ,.....,x . x 1 9n

and n

2

ii 1

x 1 5n

, then the standard deviation of this

data is:

1) 5 2) 7 3)2 4)5

Ans. 2

Sol. 2 2

i ix 1 x 1 9n 5n

2i i2 x 1 14n x 6n

2 2

i ix 1 x 1 9n 5n

i4 x 4n xi n

S.D22

i ix xvar

n n

87. If the circles 2 2 2x y 16x 20y 164 r and 2 2x 4 y 7 36 intersect at two

distinct points, then:

1) r 11 2) r 11 3) 0 r 1 4)1 r 11

Ans. 4

Sol. 21 1c 8,10 r 64 100 164 r r

2 2c 4,7 r 6




Two circles intersect 1 2 1 2 1 2r r c c r r

88. Let A x R : xisnot positive int eger . Define a function 2xf : A R as f x

x 1

then f is:

1)not injective 2)neither injective nor surjective

3)injective but not surjective 4) surjective but not injective

Ans. 4

Sol. One – one onto

1 2f x f x f x y

1 2

1 2

2x 2x

x 1 x 1

2xy 2x xy y

x 1

1 2x x y

xy 2

one – one fn Range = R-{2}

Codamain =R

not onto function

89. Let the equations of two sides of a triangle be 3x 2y 6 0 and 4x 5y 20 0 . If the

orthocenter of this triangle is at (1,1), then the equation of its third side is:

1) 26x 61y 1675 0 2)122y 26x 1675 0

3) 26x 122y 1675 0 4)122y 26x 1675 0

Ans. 3

Sol. Line to 4x 5y 20 and

A

B

3x-2y+6=0 4x+5y-20=0

(1,1)

Passing through (1,1) is




5x 4y 3....... 1

And line to 3x 2y 6 0 and passes (1,1) in 2x 3y 5........... 2

Solve AC and eq. (2)35

, 102

Solve AB and eq. (1) 33

13,2

Then eq. of BC 26x 122y 1675 0

90. If the lines x ay b,z cy d and x a 'z b ', y c 'z d ' are perpendicular, then:

1) cc' a a ' 0 2) bb' cc' 1 0 3) aa ' c c' 0 4) ab' bc 1 0

Ans. 3

Sol. Eq. of lines x b y q d

a 1 c

1 1

1 1

x b y q d

a 1 c

are ler

aa ' 1 cc' 0

2019 Jee-Main Q P K S · mirror ( )M2 parallel to the first mirror( )M1. The angle between the two...

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