aieee solution 2011

8/3/2019 Aieee Solution 2011

1/15


2/15

SOLUTION : AIEEE 2011 (Booklet Code : Q)

PART A (PHYSICS)1. The transverse displacementy(x,t) of a wave on a string is given by

2 2 2( )( , ) ax bt ab xty x t e This represents a :

(1) wave moving in x direction with speedb

a

(2) standing wave of frequency b

(3) standing wave of frequency1

b

(4) wave moving in+x direction with speeda

b

Ans. (1)

Sol.2( )( , )

a x bty x t e . This is a travelling wave in

x direction with speed ofb

a

2. A screw gauge gives the following reading when

used to measure the diameter of a wire.Main Scale reading : 0 mm.Circular Scale reading : 52 divisionGiven that 1mm on main scale corresponds to 100division of the circular scale.The diameter of wire from the above data is :(1) 0.052 cm (2) 0.026 cm(3) 0.005 cm (4) 0.52 cm

Ans. (1)

Sol. Least count of screw gauge =

10 01

100. mm mm

Diameter = Division on circular scale x leastcount + main scale reading.

152 0100

= 0.52 mm

Diameter = 0.052 cm3. A mass of m hangs with the help of a string

wrapped around a pulley on a frictionless bearing.The pulley has mass m and radius R. Assumingpulley to be a perfect uniform circular disc, theacceleration of the mass m, if the string does notslip on the pulley, is:

(1) g (2)2

3

(3)3

g(4)

3

2

Ans. (2)

Sol.

2

2

32

mg mg ga

I mm m

R

4. Work done in increasing the size of a soap bubble

from a radius of 3cm to 5cm is nearly (Surface

tension of soap solution = 0.03 Nm1) :

(1) 0.2 mJ (2) 2 mJ(3) 0.4 mJ (4) 4 mJ

Ans. (3)

Sol. W T A = 2 2 40 03 2 4 5 3 10. ( )

=

6

24 16 10( )

=3

0 384 10.

Joule =0 4. mJ

5. A thin horizontal circular disc is rotating about avertical axis passing through its centre. An insectis at rest at a point near the rim of the disc. Theinsect now moves along a diameter of the disc toreach its other end. During the journey of theinsect, the angular speed of the disc:(1) Continuously decreases(2) Continuously increases(3) First increases and then decreases(4) Remains unchanged

Ans. (3)

Sol. L I = ConstantIfIfirst decreases, then increases.

Therefore first increases and then decreases

6. Two particle are executing simple harmonicmotion of the same amplitude A and frequency along the x-axis. Their mean position isseparated by distance X0(X0 > A). If the maximumseparation between them is (X0 + A), the phasedifference between their motion is:

(1)3

(2)

4

(3)6

(4)

2

Ans. (1)

Sol. 1 1sin( ) x A t

2 2sin( ) x A t

1 2 1 21 2 2 2 2

sin sin x x A t

1 222

sinA A

1 2

2 6

1 3

7. Two bodies of masses m and 4 m are placed at adistance r. The gravitational potential at a pointon the line joining them where the gravitational

field is zero is:(1)

4Gm

r (2)

6Gm

r

(3)9Gm

r (4) zero

Ans. (3)

Sol.2 2

4( )

( )

Gm G m

x r x

1 2

x r x

, 2 r x x

3x r ,3

rx

m 4mr/3 2r/3


3/15


=4

3 2 3

( )

/ /

Gm G m

r r

=3 6 9Gm Gm Gm

r r r

8. Two identical charged spheres suspended from acommon point by two massless strings of length l

are initially a distance d(d < < l) apart because oftheir mutual repulsion. The charge begins to leakfrom both the spheres at a constant rate. As aresult the charges approach each other with avelocity v. Then as a function of distance xbetween them,(1) vx1 (2) v x1/2(3) vx (4) vx1/2

Ans. (4)Sol.

2

2sin

kq

d , cos mg

2

2tan .

k q

mg x ,

2

22.

k q

mg x

l

3 22kq

mg

l, q

2 x3, q x3/2

dq

dt 1 23

2/ dxx

dt(dq/dt is constant)

C x1/2 V, V x1/29. A boat is moving due east in a region where the

earths magnetic field is 51005 . NA-1m1 duenorth and horizontal. The boat carries a verticalaerial 2m long. If the speed of the boat is 1.50 ms1, the magnitude of the induced emf in the wire ofaerial is:(1) 0.75 mV (2) 0.50 mV(3) 0.15 mV (4) 1 mV

Ans. (3)

Sol. Bvl = 55 0 10 2 1 50. . = 515 10 = 0.15 mV

10. An object, moving with a speed of 6.25 m/s, isdecelerated at a rate given by:

2 5.dv

vdt

Where v is the instantaneous speed. The timetaken by the object, to come to rest, would be:(1) 2 s (2) 4 s(3) 8 s (d) 1 s

Ans. (1)

Sol.

0

6 25 0

2 5.

.

tdv

dtv

,0

6 252 2 5

..v t

0

6 252 2 5

..v t , 2 6 25 2 5. . .

t = 2 sec.11. A fully charged capacitor C with initial charge q0

is connected to a coil of self inductance L at t = 0.The time at which the energy is stored equallybetween the electric and the magnetic field is:

(1)4

LC (2) 2 LC

(3) LC (d) LC Ans. (1)

Sol. Since energy is halved when 0

2

qq

4t

1

4t

L

4LC

12. Let the x z plane be the boundary between twotransparent media. Medium 1 in 0z has a

refractive index of 2 and medium 2 with z < 0has a refractive index of 3 . A ray of light inmedium 1 given by the vector

6 3 8 3 10 A i j k is incident on the planeof separation. The angle of refraction in medium 2is:(1) 450 (2) 600(3) 750 (4) 300

Ans. (1)

Sol. 1 1 2sin sin

12 2

10 10 10

204006 3 8 3 100cos

( ) ( )

01 1

160

2cos ,

022 60 3sin sin , 2

32 3

2sin

2

1

2sin , 02 45

13. A current I flows in an infinitely long wire withcross section in the form of semicircular ring ofradius R. The magnitude of the magneticinduction along its axis is :

(1) 022

I

R

(2) 0

2

I

R

(3) 04

I

R

(4) 0

2

I

R

Ans. (4)

Sol.


4/15


5/15


(3)2 2 2

1 1 2 2 3 3

1 1 2 2 3 3

n T n T n T

n T n T n T

(4)

1 2 33

T T T

Ans. (1)Sol. Final Temperature

31 21 2 3

31 2

nn nT T T

N N N

T nn n

N N N

=1 1 2 2 3 3

1 1 2 2 3 3

n T n T n T

n T n T n T

19. A pulley of radius 2m is rotated about its axis by aforce F = (20t 5t2) Newton (wheret is measuredin seconds) applied tangentially. If the moment ofinertia of the pulley about its axis of rotation ofthe pulley about the axis of rotation is 10kgm2, thenumber of rotations made by the pulley before itsdirection of motion if reversed, is :(1) more than 3 but less than 6(2) more than 6 but less than 9(3) more than 9 (4) less than 3

Ans. (1)

Sol. To reverse the direction 0d (work done iszero)2 220 5 2 40 10( ) t t t

2240 10 4

10

t tt

I

32

02

3

t

dt t , is zero at3

22 03

tt , t = 6 sec.

dt 3

62

0

23

( )t

dt

63 4

0

236

3 12

t t

rad.

5 73 62

.n

20. A resistor R and 2F capacitor in series isconnected through a switch to 200V direct supply.Across the capacitor is a neon bulb that lights upat 120V. Calculate the value of R to make thatbulb light up 5s after the switch has been closed.(log102.5 = 0.4)

(1) 51071 . (2) 61072 . (3)

71033 . (4) 41031 .

Ans. (2)

Sol. Charge on capacitor q = q0(1-e-t/RC

)

qV

c , 0 1 /( ) t RC

qV e

c

, 200 1 /( ) t RCV e

120 200 1 /( ) t RCe , 0 4/ . t RCe ,

0 4( . )t

InRC

102 303 0 4

4. .

tIn

RC

6

5

2 10 2 303 0 4. .R

, = 62 7 10.

21. A Carnot engine operating between temperatures

T1 and T2 has efficiency61 . When T2 is lowered

by 62K, its efficiency increases to3

1. Then T1 and

T2 are, respectively:(1) 372 K and 330 K (2) 330 K and 268K(3) 310 K and 248 K (4) 372 K and 310K

Ans. (4)

Sol. 2

1

11

6

T

T 2

1 51

1 6 6

T

T

2

1

621 13

( )T

T 2

1

62 23

T

T

25 62 2

3

( )T

T

5T2 310 = 4T2

T2 = 310 and 16 310

5T

T1 = 372K.22. If a wire is stretched to make it 0.1% longer, its

resistance will:(1) increase by 0.2% (2) decrease by0.2%(3) decrease by 0.05% (4) increase by0.05%

Ans. (1)Sol. R l2

2R l

R l

23. Direction:The question has a paragraph followed by two

statements, Statement -1 andStatement -2. Of thegiven four alternatives after the statements,

choose the one that describes the statements

A thin air film is formed by putting the convexsurface of a plane-convex lens over a plane glassplate. With monochromatic light, this film givesan interference pattern due to light reflected fromthe top (convex) surface and the bottom (glassplate) surface of the film.Statement 1:

When light reflects from the air-glass plateinterface, the reflected wave suffers a phasechange of .Statement 2:The centre of the interference pattern is dark.


6/15


(1) Statement-1 is true, Statement 2 is true,Stateent-2 is the correct explanation ofstatement -1.

(2) Statement-1 is true, Statement -2 is true,Statement-2 is not the correct explanation ofStatement -1

(3) Statement -1 is false, Statement -2 is true.(4) Statement-1 is true, Statement -2 is false.Ans. (2)

Sol. The centre of interference pattern is dark,showing that the phase difference between two

interfering waves is . But this does not imply

statement-1.

24. A car is fitted with a convex side view mirror offocal length 20cm. A second car 2.8m behind thefirst car is overtaking the first car at a relativespeed of 15 m/s. The speed of the image of thesecond car as seen in the mirror of the first one is

(1)1

15/m s (2) 10 m/s

(3) 15 m/s (4) 110

/m s

Ans. (1)Sol. From mirror formula :

1 1 1

v u f ,

2 2

1 10

dv du

dt dtv u

22

2

dv v du f du

dt dt f u dtu

=

220

20 280

du

dt

=

21 1

15 15/

dum s

dt

25. Energy required for the electron excitation in Li++

from the first to the third Bohr orbit is :

(1) 36.3 eV (2) 108.8 eV(3) 122.4 eV (4)` 12.1 eV

Ans. (2)Sol. Every required

2

2 2

1 113 6 3

3 1. ( )E

=8

13 6 99

. = 108.8

eV26. The electrostatic potential inside a charged

spherical ball is given by bar 2 where r isthe distance from the centre; a, b are constant.Then the charge density inside the ball is:

(1) 06a r (2) 024 a (3) 06a (4) 024 a r

Ans. (3)

Sol. 2d

E ardt

,

0

.q

E dS

uuv

308q a r , 0

3

64

3

,q

a

r

27. A water fountain on the ground sprinkles water allaround it. If the speed of water coming out of thefountain is v, the total area around the fountainthat gets wet is:

(1)4

2

v

g (2)

4

22

v

g

(3)2

2v

g (4)

2

v

Ans. (1)

Sol.2 2

2max

sinv v

Rg g

area = 2R 2

v

28. 100g of water is heated from 300C to 500C.Ignoring the slight expansion of the water, thechange in its internal energy is (specific heat ofwater is 4184J / K g / K):(1) 8.4 kJ (2) 84 kJ(3) 2.1 kJ (4) 4.2 kJ

Ans. (1)

Sol. , ,Q M S T

= 3 3100 10 4 184 20 8 1 10. . 84 0,Q kJ W

8 4.Q kJ

29. The half life of a radioactive substance is 20minutes. The approximate time interval (t2 t1)

between the time t2 when3

2of its has decayed

and time t1 when3

1of it has decayed is:

(1) 14 min (2) 20 min(3) 28 min (4) 7 min

Ans. (2)

Sol. 10 02

3t

N N e , 20 0

1

3t

N N e , 2 12 ( )t te

2 1 2( )t n l 2 12

20( )n

t t

l

min.

30. This equation has Statement-1 and Statement2.Of the four choices given after the statements,chose the one that best describes the twostatements.Statement 1:

A metallic surface is irradiated by a

monochromatic light of frequency v > v0 (thethreshold frequency). The maximum kineticenergy and the stopping potential are Kmax and V0respectively. If the frequency incident on thesurface is doubled, both the Kmax and V0 are alsodoubled.Statement 2 :

The maximum kinetic energy and the stoppingpotential of photoelectrons emitted from a surfaceare linearly depended on the frequency of incidentlight.


7/15


(1) Statement-1 is true, Statement 2 is true,Stateent-2 is the correct explanation ofstatement -1.

(2) Statement-1 is true, Statement -2 is true,Statement-2 is not the correct explanation ofStatement -1

(3) Statement -1 is false, Statement -2 is true.(4)

Statement-1 is true, Statement -2 is false.Ans. (3)

Sol.max

k hf , 2max

' k hf

2max

'

maxk k 2max'

maxk k

PART- B (MATHEMATICS)31. The lines L1 : y x = 0 and L2 : 2x + y = 0

intersect the line L3 : y + 2 = 0 at P and Qrespectively. The bisector of the acute anglebetween L1 and L2 intersects L3 at R.

Statement 1 : The ratio PR : RQ equals 2 2 : 5 Statement 2 : In any triangle, bisector of an angle

divides the triangle into two similar triangles.(1) Statement-1 is true, Statement-2 is true;

Statement-2 is not a correct explanation forStatement -1.

(2) Statement-1 is true, Statement-2 is false.(3) Statement-1 is false, Statement-2 is true.(4) Statement-1 is true, Statement-2 is true;

Statement-2 is a correct explanation forStatement-1.

Ans. (2)Sol. The figure is self explanatory

2 2

5

PR OP

RQ OQ

Statement-2 is false

32. If 2 4sin cos A x x , then for all realx :

(1)13

116

A (2) 1 2A

(3)3 13

4 16A (4)

31

4A

Ans. (4)Sol. A = sin2x + cos4x = sin2x + (1- sin2x)2

= sin4x - sin2x + 1 =2

2 1 3sin2 4

x

3

14

A

33. The coefficient of x7 in the expansion of (1-x-x

2+x3)6 is:(1) -132 (2) -144 (3) 132 (4) 144

Ans. (2 )

Sol. Coeff. ofx7 in 62 31 x x x

Coeff. ofx7 in 66 21 1x x

= 6 6 6 6 6 61 3 3 2 5 1. . .C C C C C C = -144

34.

2

1 cos 2 2lim

2x

x

x

:

(1) equals 2 (2) equals - 2

(3) equals1

2(4) does not exist

Ans. (4)Sol.

2

1 cos2 2lim

2x

x

x

=

2

2 sin 2lim

2x

x

x

LHL = 2 ; RHL = 2 ; LHL RHL Limit doesnt Exist.

35. Statement-1 : The number of ways of distributing10 identical balls in 4 distinct boxes such that no

box is empty is 9 3C .Statement 2 : The number of ways of choosingany 3 places from 9 different places is 9 3C .(1) Statement -1 is true, Statement -2 is true;

Statement-2 is not a correct explanation forStatement-1.



Ans. (1)

Sol. Number of ways to distribute 10 identical balls infour distinct boxes such that no box remainsempty = 101C41 = 9C3Number of ways to select 3 different places from9 places = 9C3Clearly statement-2 is not a correct explanation ofstatement-1

36.2

2

d xequals

dy:

(1)

1 32

2

d y dy

dxdx

(2)

22

2

d y dy

dxdx

(3)32

2

d y dy

dxdx

(4)

12

2

d y

dx

Ans. (1)

Sol.2

2

1 1d x d dx d d dxdy dydy dy dy dy dx dy

dx dx

32 2 2

2 32 2 2

1 1 1. . . .d y d y dy d y

dydx dx dx dxdy dy

dxdx dx


8/15


37. If 3 0dy

ydx

and 0 2y , then ln 2y =

(1) 5 (2) 13 (3) -2 (4) 7Ans. ( 4 )

3 log 33

dy dy y dx y x c

dx y

3 x y c e

0, 2 5 x y c log2log 2 5 3 7y e

38. Let R be the set of real numbers.Statement-1 :

A , x : integerx y R R y x is an is anequivalence relation on R.Statement-2 :

B , x : x y R R x y for for some rationalnumber is an equivalence relation on R.

(1) Statement-1 is true, Statement-2 is true;Statement-2 is not a correct explanation forStatement-1.



Ans. (1)

Sol. Statement 1 is true but Statement-2 is falsebecause it is not symmetric.

39. The value of 12

0

8log 1

1

xdx

x

is :

(1) log28

(2) log2

2

(3) log2 (4) log2

Ans. (4)

Sol. 1

20

8log 1

1

x I dx

x

, Put tanx

/ 4 22

0

log 1 tan8 sec

secI d

/ 4

0

8 log 1 tan d

/ 4

0

1 tan8 log 1

1 tand

=/ 4

0

28 log

1 tand

8 log 2 .4

I

2 2 log2 log2I I

40. Let , be real and z be a complex number. If2 0z z has two distinct roots on the lineRe z = 1, then it is necessary that :

(1) 1,0 (2) 1

(3) 1, (4) 0,1 Ans. ( 3)

Sol. Let roots are p iq and p iq ; where ,p q R .As root lie on line Re(z) = 1 1p . Now product

of roots = 2 2 21 p q q ( 1 ) 1,

41. Consider 5 independent Bernoullis trials eachwith probability of success p. If the probability of

at least one failure is greater than or equal to31

32,

then p lies in the interval :

(1)3 11

,4 12

(b)1

0,2

(3)11

,112

(4)1 3

,2 4

Ans. ( 2)

Sol. Probability of atleast one failure= 1 prob of no failure.

5 531 1 1132 32 2

p p p

As probability 0; Hence1

0,2

p

42. A man saves Rs. 200 in each of the first threemonths of his service. In each of the subsequentmonths his saving increases by Rs. 40 more thanthe saving of immediately previous month. Histotal saving from the start of service will be Rs.11040 after :(1) 19 months (2) 20 months(3) 21 months (4) 18 months

Ans. (3)Sol. Total saving = 200 + 200 + 200 + 240 + 280 +

to n months = 11040

2

400 400 3 .40 110402

nn

2 5 546 0 21 26n n n as n

43. The domain of the function 1

f xx x

is :

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

Ans. (2)

Sol. For f x to be defined 0 0 x x x

44. If the angle between the line1 3

2

y zx

and

the plane 2 3 4 x y z is 15

cos14

, then

equals :

(1)3

2(b)

2

5(3)

5

3(4)

2

3

Ans. (4 )

Sol. The angle is 13

sin14

2 2

2

1 4 3 314 3 5 9 x 14 5

141 4 1 4 9

2 2 29 30 25 9 453


9/15


45. If 1 310

a i k r

and 1 2 3 67

b i j k r

, then

the value of 2 x x 2a b a b a b r r rr r r

is :

(1) -3 (2) 5 (3) 3 (4) -5Ans. ( 4)

Sol.

2

2 . x x 2 2 . 2 2a b a b a b a b b a a b r r r r r rr r r r r r

= 5

46. Equation of the ellipse whose axes are the axes ofcoordinates and which passes through the point (-

3, 1) and has eccentricity2

5is :

(1) 2 25 3 48 0x y (2) 2 23 5 15 0x y

(3) 2 25 3 32 0x y (4) 2 23 5 32 0x y Ans. (1,4)

Sol.2 2

2 2 2 2

9 11 1..................(1)

x y

a b a b

Case-1 (when a > b)

2 2 2 2 2 2 21 1 2/5 5 3 ................( )b a e b a b a ii

From (i) and (ii) : 2 232 32

&5 3

b a

Hence : equation of ellipse is 2 23 5 32 0x y

Case 2 : (when b > a)

Equation of ellipse is 2 25 3 48 0x y 47. Let I be the purchase value of an equipment and

V(t) be the value after it has been used fort years.The value V(t) depreciates at a rate given by

differential equation

dV t

k T t

dt

, where k

> 0 is a constant and T is the total life in years ofthe equipment. Then the scrap value V(T) of theequipment is :

(1)2

I2

kT (2)

2

I2

k T t

(3) kTe (4) 2I

Tk

Ans. (1 )

Sol.

2

0 02

TV T T

I t

T tdV t k T t dt V T I k

2 2

2 2

T kTV T I k V T I

48. The vectors a and br

are not perpendicular and

c and dr

are two vectors satisfying :

x xb c b d r

and 0a dr

g . Then the vector d

is equal to :

(1)a c

c ba b

r r rgrrr

g(2)

b cb c

a b

rr g rrr

g

(3)a c

c ba b

r r rgrrr

g(4)

b cb c

a b

rr g rrr

g

Ans. (1)

Sol. x x x x x xb c b d a b c a b d r r r r r rr r r r

. . . .a c b a b c a d b a b d r r r r r rr r r r r r

.. . .

.

a ca b d a c b a b c d b c

a b

r rr r r r r rr r r r r rrr

49. The two circles 2 2 x y ax and

2 2 2 0 x y c c touch each other if :

(1) a c (2) 2a c

(3) 2a c (4) 2 a c

Ans. (1)

Sol. 2 2 ...... 1 x y ax

1 1,02 2

a acentre c and radius r

2 2 2............. 2 x y c

2 20,0centre c and radius r c

Now 1 2 1 2c c r r a c

50. If C and D are two events such that C D and

0P D , then the correct statement among thefollowing is :(1) |P C D P C (2) |P C D P C

(3)

|P D

P C DP C

(4) |P C D P C

Ans. (1)

Sol. We have C D C D C

( ) ( )

( )( ) ( )

C P C D P C P P C

D P D P D

As 0 1( )P D 51. The number of values of k for which the linear

equations : 4 2 0 x ky z , 4 0kx y z ,2 2 0 x y z . Possess a non-zero solution is :(1) 2 (2) 1 (3) zero (4) 3

Ans. (1)Sol. For non-trivial solution of given system of linear

equations

4 2

4 1 0

2 2 1

k

k 8 2 2 2 8 0( ) ( ) k k k

2 6 8 0k k 2 6 8 0k k 2 4,k 52. Consider the following statements :

P : Suman is brilliant; Q : Suman is richR : Suman is honestThe negation of the statement Suman is brilliant anddishonest if and only if Suman is rich can beexpressed as

(1) Q P R : : (2) Q P R : :


10/15


11/15


Ans. (2)Sol. The given functionfis continuous atx = 0 if

0 00 0 0

lim ( ) ( ) lim ( )

h h f h f f h

12

2p q

3 1

2 2,p q

59. The area of the region enclosed by the curves1, ,y x x e y

x and the positive x axis is :

(1) 1 square units (2)3

2square units

(3)5

2square units (4)

1

2square units

Ans. (2)Sol. Required area = OAB + ACDB

1

1 11

2

e

dxx

11

2( )

enx l

3

2 square

units.

60. For 50, 2x , define

0sin dt

x

f x t t . Then f x has :

(1) local minimum at and 2 .(2) local minimum at and local maximum at2 .(3) local maximum at and local minimum at2 .(4) local maximum at and 2.

Ans. (3)

Sol.0

( ) sin

x

f x t tdt '( ) sin f x x x

local maximum at and local minimum at 2.54. If the mean deviation about the median of the

numbersa, 2a, .,50a is 50, then a equals :

(1) 3 (2) 4 (3) 5 (4) 2Ans. (2)

Sol. A = Rs.200 d = Rs.40Savings in first two monts = Rs.400Remained savings = 200 + 240 + 280 + .. upto nterms

400 1 40 11040 4002

( )n

n

200n + 20n2

20n 10640 = 0

20n2

+ 180n 10640 = 0

n2

+ 9n 532 = 0, (n+28 )(n-19) =0

n = 1955. Statement 1 : The point A(1, 0, 7) is the mirror

image of the point B(1, 6, 3) in the line1 2

1 2 3

x y z .

Statement 2 : The line :1 2

1 2 3

x y z

bisects the line segment joining A 1,0,7 and

B 1, 6, 3 .

(5) Statement 1 is true, Statement-2 is true;Statement-2 is not a correct explanation forStatement-1.


Statement-2 is a correct explanation for

Statement-1.Ans. (1)Sol. The mid point of A(1,0,7) and B(1,6,3) is (1,3,5)

which lies on the line1 2

1 2 3

x y z

Also the line passing through the points A and Bis perpendicular to the given line, hence B is themirror image of A, consequently the statement -1is true.Statement -2 is also true but it is not a correctexplanation of statement -1 as there are infinitelymany lines passing through the midpoint of theline segment and one of the lines is perpendicularbisector.

56. Let A and B be two symmetric matrices of order3.Statement-1 : A(BA) and (AB)A are symmetricmatrices.Statement-2 : AB is symmetric matrix if matrixmultiplication of A with B is commutative.(1) Statement-1 is true, Statement-2 is true;

Statement-2 is not a correct explanation forStatement-1.

(2) Statement-1 is true, Statement-2 is false.(3) Statement-1 is false; Statement-2 is true.(4) Statement-1 is true, Statement-2 is true;


Ans. ( 1 )Sol. if AB = BA (AB)T= ATBT AB issymmetric

Statement-2 is true (ABA)T= A

TB

TAT

Take A = I and B = some non

symmetric ABA always

A(BA) and (AB)A are symmetricStatement-1 is true nut does not

depend on Statement-2

57. If 1 is a cube root of unity, and

71 A B . Then (A, B) equals :

(1) (1, 1) (2) (1, 0)(3) (-1, 1) (4) (0, 1)

Ans. (1)

Sol. 71( ) A B 2 7( ) A B 2 A B 1 A B 1 1,A B 1 1( , ) ( , )A B

58. The values of p and q for which the function


12/15


2

3/ 2

sin 1 sin, 0

, 0

, 0

p x xx

x

f x q x

x x xx

x

Is continuous for allx is R, are :(1)

5 1,

2 2p q (2)

3 1,

2 2p q

(3)1 3

,2 2

p q (4)1 3

,2 2

p q

Ans. (2)

Sol. The given functionfis continuous atx = 0 if

0 00 0 0lim ( ) ( ) lim ( )

h h f h f f h

12

2p q

3 1

2 2,p q

59. The area of the region enclosed by the curves1

, ,y x x e yx

and the positive x axis is :

(1) 1 square units (2)3

2square units

(3)5

2square units (4)

1

2square units

Ans. ( 2)Sol. Required area = OAB + ACDB

1

1 11

2

e

dxx

11

2( )enx l

2

2 square

unit.

60. For5

0,2

x

, define 0

sin dtx

f x t t . Then

f has :(1) local minimum at and 2 .(2) local minimum at and local maximum at2 .(3) local maximum at and local minimum at2 .(4) local maximum at and 2.

Ans. (3)

Sol.0

( ) sin

x

f x t tdt ; '( ) sin f x x x

local maximum at and local minimum at 2.

PART- C (CHEMISTRY)61. Among the following the maximum covalentcharacter is shown by the compound :(1) SnCl2 (2) AlCl3 (3) MgCl2 (4) FeCl2

Ans. ( 2 )

Sol. According to Fajans rule, cation with greatercharge and smaller size favours covalency.

62. The presence or absence of hydroxyl group onwhich carbon atom of sugar differentiates RNA andDNA ?(1) 2nd (2) 3rd (3) 4th (4) 1st

Ans. ( 1 )Sol. RNA contains D ribose while DNA

contains D2deoxyribose.63. Trichloroacetaldehyde was subjected to

Cannizzaros reaction by using NaOH. The mixtureof the products contains sodium trichloroacetate andanother compound. The other compound is :

(1) Trichloromethanol (2) 2,2,2-Trichloropropanol(3) Chloroform (4) 2,2,2-Trichloroethanol

Ans. (4)

Sol. In connizzaros reaction CCl3CHO undergoesdisproportionation reaction as :

64. Sodium ethoxide has reacted with ethanoyl chloride.The compound that is produced in the abovereaction is :(1) 2-Butanone (2) Ethyl chloride(2) Ethyl ethanoate (4) Diethyl ether

Ans. ( 3 )

Sol.

65. The reduction potential of hydrogen half-cell will benegative if :(1) p(H2) = 1 atm and [H+] = 1.0 M(2) p(H2) = 2 atm and [H+] = 1.0 M(3) p(H2) = 2 atm and [H+] = 2.0 M(4) p(H2) = 1 atm and [H+] = 2.0 M

Ans. ( 2 )Sol.


13/15


Therefore E is (-)ve66. The strongest acid amongst the following

compounds is :(1) HCOOH (2)

CH3CH2CH(Cl)CO2H(3) ClCH2CH2CH2COOH (4) CH3COOH

Ans. ( 2)

Sol.

Presence of electron withdrawing group nearest to thecarboxylic group increase the acidic strength to maximumextent

67. The degree of dissociation of a weakelectrolyte, A B

x yis related to Vant Hoff factor (i)

by the expression:

(1)1

1

i

x y

(2)

1

1

x y

i

(3)1

1

x y

i

(4)

1

1

i

x y

Ans. (4)Sol. Vant Hoff factor (i)

Observed colligative property

Normalcolligative property

1

y x

x yx y

A B xA yB

Total moles after dissociation = 1 x y

1 11

x yi

1

1

i

x y

68. a and b are van der Waals constants for gases.Chlorine is more easily liquefied than ethanebecause(1) a and b for Cl2 < a and b for C2H6(2) a for Cl2 < a and C2H6 but b for Cl2 > b for C2H6(3) a for Cl2 > a and C2H6 but b for Cl2 < b for

C2H6(4) a and b for Cl2 > a and b for C2H6

Ans. (3)

Sol. a is a measure of attraction between the molecules

and b is the size of the molecules.69. A vessel at 1000K contains CO2 with a pressure of

0.5 atm. Some of the CO2 is converted into CO onthe addition of graphite. If the total pressure atequilibrium is 0.8atm, the value of K is :(1) 3 atm (2) 0.3 atm

(3) 0.18 atm (4) 1.8 atmAns. (4)

Sol. 20.5

2atm

CO g C s CO g

Total pressure = 0.5-P+2P=0.8; P = 0.3

2

2 22 2 0.61.8

0.5 0.5 0.3CO

p

CO

PPK

P P

70. Boron cannot form which one of the followinganions ?

(1) BH (2) B OH

(3) 2BO (4) 36BF

Ans. (4)

Sol. Due to lack of vacant d orbital in B. sp3d2 hybridization

is not possible hence BF63- will not form.

71. Which of the following facts about the complex[Cr(NH3)6]Cl3 is wrong ?(1) The complex is paramagnetic(2) The complex is an outer orbital complex(3) The complex gives white precipitate with silver

nitrate solution(4) The complex involves d2sp3 hybridisation and isoctahedral in shape.

Ans. (2)Sol. Cr+3 in octahedral geometry always form inner orbital

complex72. Ethylene glycol is used as an antifreeze in a cold

climate. Mass of ethylene glycol which should beadded to 4 kg of water to prevent it from freezing at-60C will be : (Kf for water = 1.86 K kg mol

-1, andmolar mass of ethylene glycol = 62g mol-1).(1) 204.30 g (2) 400.00g(3) 304.60g (4) 804.32g

Ans. (4 )Sol.

1000 1.86 x x 10006

. . . 62 x 4000f fw w

T K xmol wt wt of solvent

800w g ; So, weight of solute should be more than800g.

73. Which one of the following orders presents thecorrect sequence of the increasing basic nature of thegiven oxides ?(1) MgO < K2O < Al2O3 < Na2O(2) Na2O < K2O < MgO < Al2O3(3) K2O < Na2O < Al2O3 < MgO(4) Al2O3


14/15


(3) 64 times (4) 10 timesAns. (2)

Sol. 150

510Rate at 50C

2 2 2 32Rate at T C

T

T times

75. The magnetic moment (spin only) of 24NiCl is :(1) 5.46 BM (2) 2.82 BM

(3) 1.41 BM (4) 1.82 BMAns. (2)Sol. Hybridisation of Ni is sp3

Unpaired e in 3d8 is 2

So, 2n n BM 2 x 4 8 2.82BM

76. The hybridization of orbitals of N atom in3 2 4NO , NO NHand are respectively :

(1) 2 3, ,sp sp sp (2) 3 2, ,sp sp sp

(3) 2 3, ,sp sp sp (4) 2 3, ,sp sp sp

Ans. (1)

Sol. 2 33 2 4; ; NO sp NO sp NH sp

77. In context of the lanthanoids, which of the followingstatements is not correct ?(1) All the members exhibit +3 oxidation state(2) Because of similar properties the separation of

lanthanoids is not easy.(3) Availability of 4f electrons results in the

formation of compounds in +4 state for all themembers of the series.

(4) There is a gradual decrease in the radii of themembers with increasing atomic number in theseries

Ans. (3)

Sol. All the lanthanoids does not exhibit +4 oxidation

state.78. A 5.2 molal aqueous solution of methyl alcohol,

CH3OH, is supplied. What is the mole fraction ofmethyl alcohol in the solution ?(1) 0.190 (2) 0.086 (3) 0.050 (4) 0.100

Ans. (2)

Sol. Mole fraction of methanol5.2

0.0861000

5.218

Mole of methanol

total moles

79. Which of the following statement is wrong ?(1) Nitrogen can not form d p bond.(2) Single N-N bond is weaker than the single P P

bond(3) N2O4 has two resonance structures(4) The stability of hydrides increases from NH3 to

BiH3 in group 15 of the periodic tableAns. (4)

Sol. Stability of hydrides from NH3 to BiH3 decreases dueto decreasing ionic character.

80. The outer electron configuration of Gd (Atomic no :64) is:(1) 4f8 5d0 6s2 (2) 4f4 5d4 6s2(3) 4f75d1 6s2 (4) 4f3 5d5 6s2

Ans. (3)

Sol. 7 1 24 5 6Gd f d s

81. Which of the following statements regarding sulphuris incorrect ?(1) The vapour at 2000C consists mostly of S8

rings.(2) At 6000C the gas mainly consists of S2

molecules.(3) The oxidation state of sulphur is never less than

+4 in its compounds.(4) S2 molecule is paramagnetic

Ans. (3)Sol. Sulphur exhibit + 2, + 4, + 6 oxidation states but + 4

and + 6 are more common.82. The structure of IF7 is :

(1) trigonal bipyramid (2) octahedral(3) pentagonal bipyramid (4) square pyramid

Ans. (3 )

Sol. Hybridisation of iodine is sp3d3 So, structure ispentagonal bipyramid

83. Ozonolysis of an organic compound givesformaldehyde as one of the products. This confirmsthe presence of :

(1) a vinyl group (2) an isopropylgroup(3) an acetylenic triple bond (4) two ethylenic doublebonds

Ans. ( 1)

Sol.

84. A gas absorbs a photon of 355nm and emits at twowavelengths. If one of the emissions is at 680nm,the other is at :

(1) 325 nm (2) 743 nm(3) 518 nm (4) 1035 nm

Ans. (2 )

Sol.1 2 2

1 1 1 1 1 1

355 680

2 742.76nm

85. Silver Mirror test is given by which one of thefollowing compounds ?(1) Acetone (2) Formaldehyde(3) Benzophenone (4) Acetaldehyde

Ans. (2 ,4)

Sol. :

86. Which of the following reagents may be used todistinguish between phenol and benzoic acid ?(1) Tollens reagent (2) Molisch reagent(3) Neutral FeCl3 (4) Aqueous NaOH

Ans. ( 3 )Sol. Neural FeCl3 gives violet coloured complex


15/15


87. Phenol is heated with a solution of mixture of KBrand KBrO3. The major product obtained in the abovereaction is :(1) 3-Bromophenol (2) 4-Bromophenol(3) 2, 4,6- Tribromophenol (4) 2-Bromophenol

Ans. ( 3)

Sol.

88. In a face centred cubic lattice, atom A occupies thecorner positions and atom B occupies the face centrepositions. If one atom of B is missing from one ofthe face centred points, the formula of the compoundis :(1) AB2 (2) A2B3(3) A2B5 (4) A2B

Ans. (3 )

Sol.8

8AZ ;

5

2BZ ,

So, formula of compound is AB5/2; i.e., A2B589. The entropy charge involved in the isothermal

reversible expansion of 2 moles of an ideal gas froma volume of 10dm3 to a volume of 100 dm3 at 270Cis :(1) 35.8 J mol1K1 (2) 32.3J mol1K1(3) 42.3 J mol1K1 (4) 38.3 J mol1K1

Ans. (4 )

Sol. 2

1

lnV

S nRV

;100

2.303 x 2 x 8.314log10

S

38.3 / / S J mole K 90 Identify the compound that exhibits tautomerism :

(1) Lactic acid (2) 2-Pentanone(3) Phenol (4) 2-Butene

Ans. ( 2)

Sol.O

2-Pentanone has -hydrogen & hence it will exhibit

tautomerism

aieee solution 2011

Documents