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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. A current loop, having two circular arcs joined by two radial lines shown in the figure. It

carries a current of 10A. The magnetic field at point O will be close to:

1) 51.0 10 T 2) 51.5 10 T 3) 71.0 10 T 4) 71.0 10 T

Ans. 1

Sol.

0

1 2

i 1 1 ˆB k4 r r

2 2

1 2r 3cm 3 10 m,r 5cm 5 10 m

,i 10A4

7

2 2

4 10 1 1 ˆB 10 k16 3 10 5 10

5B 10 T3

51 10 T

2. A gas can be taken from A to B via two different processes ACB and ADB.

When the path ACB is used 60J if heat flows into the system and 30J of work is done by

the system. If path ADB is used work done by the system is 10J. The heat Flows into the

system in path ADB is:

1) 80J 2) 20J 3) 100J 4) 40J




Ans. 4

Sol.

Q W U

ACB ACB ACB

ACB ACB ADB ACB60J 30J U U 30J U U 30J

ACD ACB ADBQ U W 10J 30J 40J

3. A Plane electromagnetic wave of frequency 50MHz travels in free space along the

positive x direction. At a particular point in space and time, ˆE 6.3jV /m

. The

corresponding magnetic field B

, at that point will be:

1) 8 ˆ18.9 10 kT 2) 8 ˆ6.3 10 kT 3) 8 ˆ2.1 10 kT 4) 8 ˆ18.9 10 kT

Ans. 3

Sol. 8

8

E 6.3B 2.1 10 T

C 3 10

ˆˆ ˆandE B C

ˆ ˆˆj B i

ˆB̂ k

8 ˆˆB B B 2.1 10 kT

4. Two coherent sources produce waves of different intensities which interfere. After

interference, the ratio of the maximum intensity to the minimum intensity is 16. The

intensity of the waves are in the ratio:

1) 4 : 1 2) 25:9 3) 16:9 4) 5:3

Ans. 2

Sol. max

min

I16

I max

min

A4

A

1 2

1 2

A A 4

A A 1

Using componendo & dividendo.

2

1 1

2 2

A 5 I 5 25

A 3 I 3 9




5. An L-shaped object, made of thin rods of uniform mass density, is suspended with a

string as shown in figure. If AB = BC, and the angle made by AB with downward vertical

is , then:

1) 2

tan3

2) 1

tan3

3) 1

tan2

4) 1

tan2 3

Ans. 2

Sol. 1

LC P sin

2 2

Land C N cos Lsin

2

Let mass of one rod is m. Balancing torque about hinge point.

1 2mg C P mg C N

L Lmg sin mg cos Lsin

2 2

3 mgLmgLsin cos

2 2

1tan

3

6. A mixture of 2 moles of helium gas (atomic mass = 4 u), and 1 mole of argon gas (atomic

mass = 40 u) is kept at 300 K in a container. The ratio of their rms speeds

rms

rms

V helium

V arg on

, is close to:

1) 2.24 2) 0.45 3) 0.32 4) 3.16




Ans. 4

Sol.

rms Ar

rms He

V He M 403.16

V Ar M 4

7. When the switch S, in the circuit shown, is closed, then the value of current i will be:

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

Ans. 2

Sol. Let voltage at C = xv 1 2KCL : i i i

20 x 10 x x 0

2 4 2

x 10 and i = 5 amp.

8. A resistance is shown in the figure. Its value and tolerance are given respectively by:

1) 27K ,20% 2) 270K ,5% 3) 270K ,10% 4) 27K ,10%

Ans. 4

Sol. Color code:Red violet orange silver

3R 27 10 10% 27K 10%

9. A bar magnet is demagnetized by inserting it inside a solenoid of length 0.2 m, 100 turns,

and carrying a current of 5.2 A. The coercivitv of the bar magnet is:

1) 1200 A/m 2) 2600 A/m 3) 520 A/m 4) 285 A/m

Ans. 2

Sol. Coercivity = H = 0

B

N 100

ni i 5.20.2

= 2600 A/m




10. A rod, of length L at room temperature and uniform area of cross section A, is made of a

metal having coefficient of linear expansion / °C. It is observed that an external

compressive force F, is applied on each of its ends, prevents any change in the length of

the rod, when its temperature rises by T K. Young's modulus, Y, for this metal is:

1) F

2A T 2)

F

A T 273 3)

F

A T 4)

2F

A T

Ans. 3

Sol. Young’s modulus Stress

yStrain

F / A

/

Thermal strain T

l

l

F

A T

11. A block of mass m, lying on a smooth horizontal surface, is attached to a spring (of

negligible mass) of spring constant k. The other end of the spring is fixed, as shown in

the figure. The block is initially at rest in its equilibrium position. If now the block is

pulled with a constant force F, the maximum speed of the block is:

1) F

mk

2)

2F

mk 3)

F

mk 4)

F

mk

Ans. 3

Sol. Maximum speed is at mean position (equilibrium). F = kx F

xk

F spW W KE 2 21 1F x kx mv 0

2 2

2

2F 1 F 1F k mv

K 2 k 2

max

FV

mk

12. Three charges +Q, q, + Q are placed respectively, at distance, 0, d/2 and d from the

origin, on the x-axis. If the net force experienced by + Q, placed at x = 0, is zero, then

value of q is:

1) +Q/2 2) –Q/2 3) –Q/4 4) +Q/4




Ans. 3

Sol.

For equilibrium, a BF F 0

a BF F

22

kQQ kQq

d d / 2

Qq

4

13. A conducting circular loop made of a thin wire, has area 3.5 × 10–3

m2 and resistance l0

. It is placed perpendicular to a time dependent magnetic field B(t) = (0.4T) sin(50t).

The field is uniform in space. Then the net charge flowing through the loop during t = 0 s

and t = 10 ms is close to:

1) 140 c 2) 21mC 3) 6mC 4) 7 Mc

Ans. 1

Sol. 3

r i

1 1Q A B B 3.5 10 0.4sin 0

R 10 10 2

313.5 10 0.4 0

10

41.4 10 140 C

14. Two masses m and m

2 are connected at the two ends of a massless rigid rod of length l.

The rod is suspended by a thin wire of torsional constant k at the centre of mass of the

rod-mass system (see figure). Because of torsional constant k, the restoring torque is

=k for angular displacement 0. If the rod is rotated by 0 and released, the tension in it

when it passes through its mean position will be:

1) 2

03 k

l 2)

2

0

2

k

l 3)

2

02 k

l 4)

2

0k

l

Ans. 4

Sol. k

I

2

3k

m




0 average velocity 2

1T m r 2T m

3

2 2

0m3

2

02

3km

m 3

2

0k

2

2 2

m

2I3m

2

2m

3

15. A copper wire is stretched to make it 0.5% longer. The percentage change in its electrical

resistance if its volume remains unchanged is:

1) 2.5% 2) 0.5% 3) 1.0% 4) 2.0%

Ans. 3

Sol. R and volume V AA

ll

2

RV

l

R 21%

R

l

l V cons tan t

16. A parallel plate capacitor is made of two square plates of side 'a', separated by a distance

d (d<<a). The lower triangular portion is filled with a dielectric of dielectric constant K,

as shown in the figure. Capacitance of this capacitor is:

1) 2

01 k a

2 d

2)

2

0k aln K

d

3)

2

0k aln K

d K 1

4)

2

0k a

2d K 1

Ans. 3

Sol.

y d

x a

dy x

a d

dy dxa

0

d y1 y

dc KE.adx adx

0

1 1 yd y

dc adx k

0adx

dcy

d yk




d

0

0

a dyc a.

1dd y 1

k

d20

0

a 1n d y 1

1 k1 d

k

l

20

1d d 1

k a kn

1 k d d

l

2 20 0k a k a ln k1

n1 k d k k 1 d

l

17. Mobility of electrons in a semiconductor is defined as the ratio of their drift velocity to

the applied electric field. If, for an n-type semiconductor, the density of electrons is

1019

m–3

and their mobility is 1.6 m2/(V.s) then the resistivity of the semiconductor (since

it is an n-type semiconductor contribution of holes is ignored) is close to:

1) 2 m 2) 0.4 m 3) 4 m 4) 0.2 m

Ans. 2

Sol. dj E nev dvne ne

E

e e

1 1

n e

19 19

10.39 0.4 m

10 1.6 10 1.6

18. If the angular momentum of a planet of mass m, moving around the Sun in a circular

orbit is L, about the center of the Sun, its areal velocity is:

1) 4L

m 2)

L

m 3)

L

2m 4)

2L

m

Ans. 3

Sol. dA L

dt 2m

19. A block of mass 10 kg is kept on a rough inclined plane as shown in the figure. A force

of 3 N is applied on the block. The coefficient of static friction between the plane and the

block is 0.6. What should be the minimum value of force P, such that the block does not

move downward?




1) 32 N 2) 25 N 3) 23 N 4) 18 N

Ans. 1

Sol.

0 100mgsin 45 50 2

2

1

mgcos 0.6 mg 0.6 50 22

P 31.28 32N

3 mgsin

20. Temperature difference of 0120 C is maintained between two ends of a uniform rod AB

of length 2L. Another bent rod PQ, of same cross section as AB and length3L

2, is

connected across AB (See figure). In steady state, temperature difference between P and

Q will be close to:




1) 060 C 2) 0

75 C 3) 035 C 4) 0

45 C

Ans. 4

Sol.

eq.

120 5T 120 5I

R 8R 8R

0PQ

120 5 3 360T R 45 C

8R 5 8

21. A heavy ball of mass M is suspended from the ceiling of a car by a light string of mass m

(m<<M). When the car is at rest, the speed of transverse waves in the string is 160ms

.

When the car has acceleration a, the wave-speed increases to 160.5ms

. The value of a, in

terms of gravitational acceleration g, is closest to:

1) g

5 2)

g

20 3)

g

10 4)

g

30

Ans. 1

Sol. Mg

60

1/2

2 2 2 2

2

M g a 60.5 g a60.5

60 g

4 2 2

2

0.5 g a 21 1

60 60g

2 2 2 2 2g a g g

60

2 g g ga g

60 5.47 530




22. A sample of radioactive material A, that has an activity of 10 mCi ( 10lCi 3.7 10

decays/s), has twice the number of nuclei as another sample of a different radioactive

maternal B which has an activity of 20 mCi. The correct choices for half-lives of A and B

would then be respectively:

1) 20 days and 5 days 2) 20 days and 10 days

3) 5 days and 10 days 4) 10 days and 40 days

Ans. 1

Sol. Activity A = N

For A 0 A10 2N

For B 0 B20 N

B A 1/2 1/2A B4 T 4 T

23. Consider a tank made of glass (refractive index 1.5) with a thick bottom. It is filled with a

liquid of refractive index . A student finds that, irrespective of what the incident angle i

(see figure) is for a beam of light entering the liquid, the light reflected from the liquid

glass interface is never completely polarized. For this to

happen, the minimum value of is:

1) 3

5 2)

5

3 3)

5

3 4)

4

3

Ans. 1

Sol. bC i

Here bi is ‘brewester angle’

And ‘c’ is critical angle




sin c < sin bi since b rel

1.5tan i

b

2 22 2

1 1.5 1.5sin i

1.5 1.5

2

2 1.5 1.5

2 22 1.5 1.5

3

5

slab 1.5

24. An infinitely long current carrying wire and a small current carrying loop are in the plane

of the paper as shown. The radius of the loop is a and distance of its centre from the wire

is d (d>>a). If the loop applies a force F on the wire then:

1) 2

3

aF

d

2)

aF

d

3)

2a

Fd

4) F 0

Ans. 3

Sol.

For an infinite long wire

Equivalent dipole of given loop




dB

F m.dr

02

IdB d 1Now

dx dx 2 x x

2

2 2

M aSo F M NIA F

x d

25. Surface of certain metal is first illuminated with light of wavelength 1 350nm and

then, by light of wavelength 2 540 nm . It is found that the maximum speed of the

photo electrons in the two cases differ by a factor of 2. The work function of the metal (in

eV) is close to: (Energy of photon = 1240

eVin nm

)

1) 1.8 2) 1.4 3) 2.5 4) 5.6

Ans. 1

Sol. 2

1

hc 1m 2v

2

2

2

hc 1mv

2

1

1 2

2

hc

hc 4hc4 4

hc

2 1

4hc hc3

2 1

1 4 1hc

3

1 4 350 540

12403 350 540

1.8eV 1.88eV

26. A particle is moving with a velocity ˆ ˆv K yi xj , where K is a constant. The general

equation for its path is:

1) xy cons tan t 2) 2 2y x cons tan t

3) 2y x cons tan t 4) 2y x cons tan t

Ans.2

Sol. x yV ky, V kx

dx dy

ky, kxdt dt

dydy xdtNow,

dxdx y

dt

ydy xdx

Integrating both side 2 2y x c




27. A convex lens is put 10 cm from a light source and it makes a sharp image on a screen,

kept 10 cm from the lens. Now a glass block (refractive index 1.5) of 1.5 cm thickness is

placed in contact with the light source. To get the sharp image again, the screen is shifted

by a distance d. Then d is:

1) 0.55 cm away from the lens 2) 1.1 cm away from the lens

3) 0.55 cm towards the lens 4) 0

Ans.1

Sol.

1 1 1 1 1 1

f 5cmv u f 10 10 f

Shift due to slab = 1

t 1

in the direction of incident ray 2

1.5 1 0.53

Again, 1 1 1

v 9.5 5

1 1 2 9

u 5 19 95

95

v 10.55cm9

28. For a uniformly charged ring of radius R, the electric field on its axis has the largest

magnitude at a distance h from its centre. Then value of h is:

1) R

5 2) R 3)

R

2 4) R 2

Ans. 3

Sol. Electric field on axis of ring

3/22 2

kQxE

x R

For maximum electric field

dE

0dx

Rx

2




29. Three blocks A, B and C are lying on a smooth horizontal surface, as shown in the figure.

A and B have equal masses, m while C has mass M. Block A is given an brutal speed v

towards B due to which it collides with B perfectly in elastically. The combined mass

collides with C, also perfectly inelastically 5

th6

of the initial kinetic energy is lost in

whole process. What is value of M/m?

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

Ans. 1

Sol. 2

i 0

1k mv

2

Form linear momentum conservation

0 fmv 2m M v

0f

mvv

2m M

i

f

k6

k

2

0

0

1mv

2 61 mv

2m M2 2m M

2m M6

m

M4

m

30. Drift speed of electrons, when 1.5 A of current flows in a copper wire of cross section 5

mm2, is v. If the electron density in copper is 28 3

9 10 / m the value of v in mm/s is close

to (Take charge of electron to be = 191.6 10 C

)

1) 0.2 2) 3 3) 2 4) 0.02

Ans. 4

Sol. dI neAv

d 28 19 6

I 1.5v

neA 9 10 1.6 10 5 10

0.02mm / s




CHEMISTRY 31. Which one of the following statements regarding Henry’s law not correct ?

1) The value of HK increases with function of the nature of the gas

2) Higher the value of HK at a given pressure, higher is the solubility of the gas in the

liquids.

3) The partial of the gas in vapour phase is proportional to the mole fraction of the gas in

the solution.

4) Different gases have different HK (Henry’s law constant) values at the same

temperature.

Ans. 2

Sol. Liquid solution gas H gasP K X

More is HK less is solubility, lesser solubility is at higher temperature. So more is

temperature more is HK .

32. The correct decreasing order for acid strength is

1) 2 2 2 2 2NO CH COOH NCCH COOH FCH COOH C CH COOH l

2) 2 2 2 2FCH COOH NCCH COOH NO CHCOOH C CH COOH l

3) 2 2 2 2 2NO CH COOH FCH COOH NCCH COOH C CH COOH l

4) 2 2 2 2 2NCCH COOH O NCH COOH FCH COOH C CH COOH l

Ans. 1

Sol. EWG increase acidic strength

2 2 2 2 2NO CH COOH NCCH COOH FCH COOH CICH COOH

33. Two complexes 2 6 3Cr H O Cl (A) and 3 36Cr NH Cl (B) are violet and yellow

colored, respectively. The incorrect statement regarding them is

1) 0 value of (A) is less than that of (B)

2) 0 value of (A) and (B) are calculated from the energies of violet and yellow light,

respectively.

3) Both absorb energies corresponding to their complementary colors.

4) Bothe are paramagnetic with three unpaired electrons




Ans. 2

Sol. 0 order will be compared by spectro chemical series not by energies of violet and

yellow light 0 order is 2 3 3 36 6Cr H O Cl Cr NH Cl

34. Adsorption of a gas follows Freundlich adsorption isotherm. In the given plot, x is the

mass of the gas adsorbed on mass m of the adsorbent at pressure p. x

mis proportional to

2 unit4 unit

Log P

xLog

m

1)

1

4P 2) 2

P 3) P 4)

1

2P

Ans. 4

Sol. 1/nx

K Pm

x 1

log logK log Pm n

1 2 1

m n 2n 4 2

1/2x

So, K Pm

35. Correct statements among a to d regarding silicones are

a) They are polymers with hydrophobic character b) They are biocompatible

c) In general, they have high thermal stability and low dielectric strength

d) Usually, they are resistant to oxidation and used as greases

1) a, b and c only 2) a and b only 3) a, b,c and d 4) a, b and d only

Ans. 3

Sol. These are properties and uses of silicones




36. For emission line of atomic hydrogen from i in 8 to n = the plot of wave number v

against 2

1

n

will be (The Rydberg constant, HR is in wave number unit).

1) Linear with slope - HR 2) Linear with intercept- HR

3) None linear 4) Linear with sslope HR

Ans. 4

Sol. 2

H 2 21 2

1 1 1v R z

H 2 2

1

1 1v R

8

HH 2 2

1 Rv R

8

H

H 2

1 Rv R

64

Hm R

Linear with slope HR

37. The major product the following reaction is

1) 2) 3) 4)

Ans. 4

Sol.

38. The alkaline earth metal nitrate that does not crystallize with water molecules, is

1) 3 2Sr NO 2) 3 2

Mg NO 3) 3 2Ca NO 4) 3 2

Ba NO

Ans. 4




Sol. Smaller in size of center atoms more water molecules will crystallize hence 3 2Ba NO is

answer due to its largest size of +ve ion :

39. Major product of the following reaction is

1) 2)

3) 4)

Ans. 4

Sol.

2NH a will wact as nucleophile as (b) is having delocalized lone pair

40. The highest value of the calculated spin only magnetic moment (in BM) among all the

transition metal complexs is

1) 5.92 2)3.87 3) 6.93 4) 4.90

Ans. 1




Sol. n n 2 B.M.

n = Number of unpaired electrons

n = maximum number of unpaired electron = 5

Ex = 2

Mn

complex.

41. 20mL of 0.1 2 4MH SO solution is added to 30 mL of 0.2 M 4NH OH solution. The pH of

the resulatant mixture is: b 4pk of NH OH 4.7

1) 9.4 2)5.02 3) 9.0 4) 5.2

Ans. 3

Sol. 2 4 H20 ml 0.1M H SO 4

4 NH OH430 ml 0.2M NH OH 6

4 4 2NH OH H NH H O

6 4 0 0

2 0 4 4

Solution is basic buffer

4b

4

NHpOH pK log

NH OH

= 4.7 + log2 = 4.7 + 0.3 = 5

pH = 14 – 5 = 9

42. 0.5 moles of gas A and x moles of gas B exert a pressure of 200 Pa in a a container of

volume 10 3

m at 1000 K, given R is the gas constant in 1 1JK mol m

, x is

1) 2

4 12

R

2)

2R

4 R 3)

4 R

2R

4)

4 R

2R

Ans. 3

Sol. Tn 0.5 x PV n R T

200 10 0.5 x R 1000 2 0.5 x R 2 1

xR 2

4

1 2xR

4 Rx

2R




43. Consider the reversible isothermal expansion of an ideal gas in a closed system at two

different temperatures 1T and 2 1 2T T T . The correct graphical depiction of the

dependence of work done (w) on the final volume (V) is

1) 2)

3) 4)

Ans. 2

Sol. 2

1

Vw nRTln

V b

i

Vw nRTln

V

b

i

V| w | nRTln

V

b i| w | nRT ln V ln V

b i| w | nRT ln V nRTln V

Y m x C

So, slope of curve 2 is more than curve 1 and intercept of curve 2 is more negative then

curve 1.

44. The major product of following reaction is

1 AlH i Bu2

2 H O2R C N ?

1) RCHO 2) RCOOH 3) 2 2RCH NH 4) 2RCONH

Ans. 1

Sol. 2 2

AIH i Bu H OR C N R CH NH R CH O




45. In general, the properties that decrease and increase down a group in the periodic table

respectively, are

1) electro negativity and electron gain enthalpy.

2) electro negativity and atomic radius.

3) atomic radius and electro negativity.

4) electron gain enthalpy and electro negativity.

Ans. 2

Sol. Electronegativity decreases as we go down the group and atomic radius increases as we

go down the group.

46. A solution of sodium sulfate contains 92 g Na of ions per kilogram of water. The

molality of Na ions in that solution in mol 1kg is

1) 16 2) 8 3) 4 4) 12

Ans. 3

Sol. na

92n 4

23

So molality = 4

47. A water sample has ppm level concentration of the following metals : Fe = 0.2;

Mn = 5.0; Cu = 3.0; Zn = 5.0. The metal that makes the water sample unsuitable drinking

is

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

Ans. 3

Sol. i) zn 5.0 ppm ii) Fe 0.2 ppm iii) Mn 0.05 ppm iv) Cu 3.0 ppm

48. The increasing order of pKa of the following amino acids in aqueous solution is :

Gly Asp Lys Arg

1) Asp Gly Arg Lys 2) Arp Lys Gly Asp

3) Gly Asp Arg Lys 4) Asp Gly Lys Arg

Ans. 4

Sol. Asp 3 Gly 6 Lys 9.8 Arg 10.8




49. According to molecular orbital theory, which of the following is true with respect to

2 2Li and Li ?

1) Both are unstable 2) 2Li is unstable and 2Li is stable

3) 2Li is stable and 2Li is unstable 4) Both are stable

Ans. 4

Sol. Both 2 2Li and Li has 0.5 bond order and hence both are stable

50. The following results were obtained during kinetic studies of the reaction :

2A + B Products

The time (in minutes) required to consume half of A is :

1) 10 2) 5 3) 100 4) 1

Ans. 2

Sol. x y36.93 10 K 0.1 0.2

x y36.93 10 K 0.1 0.25

So y = 0

and x y21.386 10 K 0.2 0.30

x1 1

x 12 2

So 0r K 0.1 0.2

036.93 10 K 0.1 0.2

2K 6.93 10

1 1

2

0.693 0.693 10t 5

2K 20.693 10 2




51. The major product of the following reaction is :

1) 2)

3) 4)

Ans. 2

Sol.

During AES Br is o/p directing and major product will be formed on less hindrance ‘p’

position :




52. Arrange the following amines in the decreasing order of basicity :

1) I > II > III 2) III > II > I 3) I > III > II 4) III > I > II

Ans. 4

Sol. Order of basic strength :

53. Which amongst the following is the strongest acid?

1) 3CHI 2) 3CHCI 3) 3CHBr 4) 3

CH CN

Ans. 4

Sol. CN makes carbanion most stable so answer is CH(CN)3

54. The anodic half-cell of lead-acid battery is recharged unsing electricity of 0.05 Faraday.

The amount of PbSO4 electrolyzed in g during the process in : (Molar mass of PbSO4 =

303 g mol-1)

1) 22.8 2) 15.2 3) 7.6 4) 11.4

Ans. 3

Sol. (A) 4 2 2 4PbSO s 2OH PbO H SO 2e

0.05F

(C) 4 2 4 4PbSO 2e 2H Pb s H SO N PbSO 0.025 mole

pbSO4m 0.025 303 7.6 gm




55. The one that is extensively used as a piezoelectric material is :

1) Quartz 2) Amorphous silica 3) Mica 4) Tridymite

Ans. 1

Sol. Quartz

56. Aluminium is usually found in +3 oxidation state. In contarast, thallium exists in +1 and

+3 oxidation states. This is due to :

1) lanthanoid contraction 2) lattice effect

3) diagonal relationship 4) inert pair effect

Ans. 4

Sol. Due to inert pair effect Tl is stable is +1 oxidation state

57. The correct match between Item -I and Item II is :

1) A Q; B P; C S; D R 2) A R; B P; C S; D Q

3) A R; B S; C P; D Q 4) A Q; B S; C P; D R

Ans. 3

Sol.




58. The ore that contains both iron and copper is :

1) malachite 2) dolomite 3) azurite 4) copper pyrites

Ans. 4

Sol. Copper pyrites : 2Cu FeS

Malachite : 32Cu OH . CuCO

Azurite : 32Cu OH . 2CuCO

Dolomite : 3 3CaCO . MgCO

59. The compounds A and B in the following reaction are, respectively:

1) A = Benzyl alcohol, B = Benzyl isocyanide

2) A = Benzyl alcohol, B = Benzyl cyanide

3) A = Benzyl chloride, B = Benzyl cyanide

4) A = Benzyl chloride, B = Benzyl isocyanide

Ans. 4

Sol.

60. The isotopes of hydrogen are :

1) Tritium and protium only 2) Deuterium and tritium only

3) Protium and deuterum only 4) Protium, deuterium and tritium

Ans. 4

Sol. Isotopes of hydrogen is :

Proteium Deuterium Tritium




MATHEMATICS

61. The area (in sq. Units bounded by the parabola 2y x 1, the tangent at the point

(2, 3) to it and the y axis is:

1) 14

3 2)

56

3 3)

8

3 4)

32

3 Ans. 3

Sol. 2y x 1 1

Tangent at P 2,3 to 1 is y 3 4 x 2 as p

dy4

dx

Required area

=Area of PTN Area of curve A.P on y-axis

3

1

1NT NP y 1dy

2

3

3/2

1

1 2 168 2 y 1 8 8 / 3sq.units

2 3 3

62. The maximum volume (in cu. m) of the right circular cone having slant height 3m is:

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

3

Ans. 3

Sol. Volume of cone 2 2 2 21V r h & r h where 3

3l l

2V 9 h h3

2

2

dv d v0 h 3 & 6h 0

dh dh

V is max at h 3

Max V= 9 3 3 2 3 m.3




63. For 2x n 1,n N (the set natural numbers). The integral

2 2

2 2

2sin x 1 sin 2 x 1x dx

2sin x 1 sin 2 x 1

Is equal to:

(where c is a constant of integration)

1) 2

e

x 1log sec c

2

2) 2 2

e

1log sec x 1 c

2

3) 2

2

e

1 x 1log sec c

2 2

4) 2

e

1log sec x 1 c

2

Ans. 1

Sol. Given 2x n 1,n N and we have 22sin sin 2 1 cosTan / 2

2sin sin 2 1 cos

2 2 2

2 2

2sin x 1 sin 2 x 1 x 1x dx x tan dx

22sin x 1 sin 2 x 1

2 2x 1 x 1

tan .d2 2

2x 1

log sec c2

64. Let and be two roots of the equation 2x 2x 2 0, then 15 15 is equal to :

1) 512 2) -512 3) -256 4) 256

Ans. 3

Sol. , are roots of 2x 2x 2 0 2x 1 1 x 1 i

Let 1 i, 1 i

15 15 15 1515 151 i 1 i 1 i 1 i

1515

2cis 2cis4 4

15 15

2 . 2cos 2564

65. If y y x is the solution of the differential equation, 2dyx 2y x

dx satisfying

y 1 1, then 1

y2

is equal to:

1) 7

64 2)

13

16 3)

49

16 4)

1

4 Ans. 3




Sol. dy 2

y xdx x

is a Linear D.E

If 2/xdx

2e x 2 2

y x x x dx

4

2 xyx c & y 1 1 c 3 / 4

4

Put x=1/2 Then y 49 /16

66. Equation of a common tangent to the circle, 2 2x y 6x 0 and the parabola, 2y 4x,

is:

1) 2 3y 12x 1 2) 2 3y x 12 3) 3y x 3 4) 3y 3x 1

Ans. 3

Sol. Tangent to 2y 4x with slope m is 1

y mxm

2m x my 1 0 is tangent to the circle 2 2x y 6x 0

2

4 2

3m 1 1d r 3 m

3m m

For m 1/ 3, theCommon tangent isx

y 33

Ie; 3y x 3

67. Consider a class of 5girls and 7boys. The number of different teams consisting of 2 girls

and 3boys that can be formed from this class, if there are two specific boys A and B, who

refuse to be members of the same team, is:

1) 200 2) 300 3) 500 4) 350

Ans. 2

Sol. No of different teams consisting of 2 girls (out of 5)

And 3boys (out of 7) 2 3C C5 7 out of these,

No of teams consisting of 2 specific boys A and B 2 2 1C C C5 2 5

Answer 2 3 2 1 2 3 1C C C C C C C5 7 5 5 5 7 5

10 35 5 300

68. Three circles of radii a,b,c (a<b<c) touch each other externally. If they have x-axis as a

common tangent, then:

1) 1 1 1

a b c 2) a,b,c are in A.P

3) a , b, c are in A.P 4) 1 1 1

b a c

Ans. 1




Sol. a b c are radii of 3 circles, (say 1 2 3c ,c ,c ) touching each other externally

Also x-axis is their common tangent then

Sum of lengths of common tangents of 1 2 1 3c ,c andc ,c

=Lengths of the common tangent of 2 3c c

2 2 2 2 22a b a b a c a c b c b c

ab ac bc 1 1 1

c b a

69. If the fractional part of the number 4032

,15 15

kis then k is equal to:

1) 14 2) 6 3) 4 4) 8

Ans. 4

Sol.

100 100403 8 16 8 1 152 81 15

15 15 15 15

For some N

8

15

Fractional part= 8 k

k 815 15

70. Axis of a parabola lies along x-axis. If its vertex and focus are at distance 2and 4

respectively from the origin, on the positive x-axis then which of the following points

does not lie on it?

1) 4, 4 2) 5,2 6 3) 8,6 4) 6,4 2

Ans. 3

Sol. For the parabola vertex A 2,0 andfocus s 4,0

Equation of theparabola is

2 2y 0 4 2 x 2 .e; y 8 x 2 1 i

The point 8,6 is not satisfying 1

71. The plane through the intersection of the planes 1 2 3 4 0 x y z and x y z

and parallel to y-axis also passes through the point:

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

Ans. 3

Sol. Plane through the intersection of the given planes is of the form

x y z 1 2x 3y z 4 0




2 1 x 3 1 y 1 z 4 1 0, R

This is parallel to y-axis 3 1 0 1/ 3

The equation of the plane is x 4z 7 0

It is passing through 3,2,1

72. If a,b and c be three distinct real numbers in G.P and . a b c xb then x cannot be:

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

Ans. 4

Sol. Let r be the common ratio of the G.P then

2a b c xb a 1 r r x ar

1 1

x 1 r where r 2or 2r r

Range of x is , 1 3,

x can not be equal to 2

73. Consider the set of all lines 0 px qy r such that 3 2 4 0. p q r which one of

the following statements is true?

1) The lines are all parallel 2) Each line passes through the origin.

3) The lines are concurrent at the point 4) 3 1

,4 2

Ans. 4

Sol. 4px 4qy 4r 0 & 4r 3p 2q

p 4x 3 q 4y 2 0

This is equation of family of lines passing through the point of intersection of the lines

4x 3 0&4y 2 0 i.e; 3 / 4,1 / 2

74. The system of linear equations.

2

2

2 3 2 5

2 3 1 1

x y z

x y z

x y a z a

1) has infinitely many solutions for a=4

2) is inconsistent when 3a

3) is inconsistent when a=4

4) has a unique solution for 3a

Ans. 2

Sol. Augmented matrix A : B 2

1 1 1 2

2 3 2 5

2 2 1 1a a




By 2 1 3 2R 2R &R R

2

1 1 1 2

0 2 1 3

0 0 3 4a a

For a 3 clearly

Rank of A : B 3& rank of A=2

Which are unequal there exist no solution if a 3

75. Let , a i j b i j k and

c be a vector such that

2

0 . 4, a c b and a c then c

is equal to:

1) 19

2 2) 8 3)

17

2 4) 9

Ans. 1

Sol. a c b o a a c a b 0

2

a.c a a c a b 0anda.c 4(given)

2

a c 4a a b where

i j k

a b 1 1 0 i j 2k

1 1 1

2c 3i 5j 2k

2 19

2 c 9 25 4 38 c2

76. Let 1 2 30, ......a a a be an A.P,

30 15

52 1

1 1

.If a 27 2 75.

i i

i i

S a and a and S T then 10a is

equal to:

1) 57 2) 47 3) 42 4) 52

Ans. 4

Sol. 1 2 30a ,a ,.......,a . be the A.P let d be common difference ‘d’

30 15

i 2i 1

i 1 i 1

S a ,T a

5 1a 27 a 4d 27 1

S 2T 75

2 1 4 3 30 29a a a a ...... a a 75

15d 75

d 5

by 11 , a 7 10 1a a 9d 7 45 52.




77. 5 students of a class have an average height 150cm and variance 18 2.cm A new student,

whose height is 156 cm, joined them. The variance (in 2.cm ) of the height of these six

students is:

1) 22 2) 20 3) 16 4) 18

Ans. 2

Sol. Combine variance

22 2 1 2

1 21 21 2

1 2 1 2

1 n nn n x x

n n n n

Put 2 21 21 2

1 2n 5,n 1, 18, 0,x 150,x 156

1 5Combined var iance 90 0 36 20

6 6

78. Two cards are drawn successively with replacement from a well-shuffled deck of

52cards. Let X denote the random variable of number of aces obtained in the two drawn

cards. Then P X =1 P X = 2 equals:

1) 52/169 2) 25/169 3) 49/169 4) 24/169

Ans. 2

Sol. 4 1 12

P q52 13 13

k

x k k

Cn 2.p x k n .q .p

2

2 2

1 2

12 1 1 25p x 1 p x 2 C . C

13 13 13 169

79. For 1 2 3

1 10,1 ,let f ,f 1 f

1

x R x x x and x

x xbe three given functions.

If a function, J x is equal to:-

1) 3f x 2) 1f x 3) 2f x 4) 3

1f x

x Ans. 1

Sol. 1 3

1 1, 1 ,

1

2f x f x x f xx x

2 1 3f oJof x f x 2

1 1

1

f oJ

x x

2

1 1

1

f J

x x

1 1

11

J

x x

1 1

11

J

x x

3

1 1

11

J f x

x

x




80. Let 3 2 sin

, :2 1 2 sin

iA is purely imaginary

i

Then the sum of the element in A is

:

1) 5

6

2)

2

3

3)

3

4

4)

Ans. 2

Sol. ,2

3 2 sin

1 2 sin

i

i

is purely imaginary

23 4sin 0 2sin 3 / 4 3

n

2

, ,3 3 3

81. If denotes the acute angle between the curves, 2 210 2 y x and y x at a point of

their intersection. Then tan is equal to:

1) 4 / 9 2) 7/17 3) 8 /17 4) 8 /15 Ans. 4

Sol. 2 210 ,y 2 x y x

Point of intersections 2,6

1 22,6 , 4, 4

8 8tan

1 16 15

at m m

82. If cos sin

,sin cos

A

then the matrix 50

A when ,12

is equal to:

1)

3 1

2 2

1 3

2 2

2)

1 3

2 2

3 1

2 2

3)

1 3

2 2

3 1

2 2

4)

3 1

2 2

1 3

2 2

Ans. 1

Sol. cos sin

sin cos

A

50cos50 sin50

sin50 cos50

A

50cos50 sin50

sin50 cos50

A

25

50 50 412 6 6

50

3 1cos sin

6 6 2 2

1 3sin cos6 6 2 2

A




83. Let 0 .2

If the eccentricity of the hyperbola

2 2

2 21

cos sin

x y

is greater than 2,

then the length of its latus rectum lies in the interval:

1) 2,3 2) 3, 3) 3 / 2,2 4) 1,3 / 2

Ans. 2

Sol.

2 2

2

cos sin 12 0

cos cos 2

e and

2sin

. 2cos

L L

2tan .sin 3

84. The equation of the lines passing through 4,3,1 , parallel to the plane

2 5 0 x y z and intersecting the line 1 3 2

3 2 1

x y z

is:

1) 4 3 1

1 1 1

x y z

2) 4 3 1

3 1 1

x y z

3) 4 3 1

1 1 3

x y z 4)

4 3 1

2 1 4

x y z

Ans. 2

Sol. Required line 4 3 1

x y z

l m n 2 0 1 l m n and

3 0 1

3 2 1 0

l m n

3 0 2 l n

By 1 & 2 , 2 1 1 2

0 3 1 0

l m n

6 2 2

l m n

3 7 1

l m n

85. For any , ,4 2

the expression 4 2 63 sin cos 6 sin cos 4sin equals:

1) 613 4cos 2) 4 2 213 4cos 2sin cos

3) 2 413 4cos 6cos 4) 2 2 213 4cos 6sin cos Ans. 1

Sol. ,4 2




4 2 63 sin cos 6 sin cos 4sin

32 23 1 2sin cos 6 1 2sin cos 4 1 cos

2 2 2 4 613 12sin cos 4 3cos 3cos cos

2 2 2 2 613 12sin cos 12cos 1 cos 4cos

613 4cos

86. If 1 12 3 3cos cos

3 4 2 4

x

x x

then x is equal to:

1) 145

12 2)

145

10 3)

146

12 4)

145

11 Ans. 1

Sol. 1 12 3 3

cos cos3 4 2 4

x

x x

2

4 31

9 4

x x

2

4 91

9

x bx

2 145

144x

145

12x

87. The value of 3

0cos x dx

1) 2/3 2) 0 3) -4/3 4) 4/3

Ans. 4

Sol.

332

0 0cos 2 cos x dx xdx

2 4

2. %3 3

88. If the Boolean expression ~ q p q P is equivalent to q,P where

, , , then the ordered pair , is:

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

Ans. 1

Sol. p q p q ~ p q p p q q

F p q p q , ,

89.

4

40

1 1 2lim

y

y

y

1) exists and equals 1

4 2 2) does not exist

3) exists and equals 1

2 2 4) exists and equals

1

2 2 2 1

Ans. 1




Sol.

4

40

1 1 2

y

yLt

y

3

44

30

1 14

2 12 1 1

4

y

yyy

Lty

(L hospital rule)

1 1

.2 1 02 1 1 0

1

4 2

90. Let f : R R be a function defined as:

5, if 1

, if 1 3f x =

5 , if 3 5

30, if 5

x

a bx x

b x x

x

Then, f is:

1) continuous if 5 5 a and b 2) continuous if 5 10 a and b

3) continuous if 0 5 a and b 4) not continuous for any values of a and b

Ans. 4

Sol. Suppose f is Continuous

5

103 15

a bb

a b b

25 30 5 b b

So, f is not Continuous for any values of a and b.

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