Aieee 2007 Part i

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Aakash Institute-Corporate Office:Plot No. 4, Sector-11, MLU, Dwarka, New Delhi-75 Ph. : 25084107 / 108 Fax : 011-25084119

(1)

1. If (2, 3, 5) is one end of a diameter of the sphere 0202126222 =+++ zyxzyx , then the coordinates

of the other end of the diameter are

(1) (4, 3, 3) (2) (4, 9, 3)

(3) (4, 3, 3) (4) (4, 3, 5)

Sol. Answer (2)

The center of the sphere

x2+ y2+ z2 6x 2y 27 + 20 = 0

is (3, 6, 1), which is the mid-point of the diameter with ends (2, 3, 5) and (, , ), (say), then

62

3,3

2

2=

+=

+and 1

2

5=

+,

yielding = 4, = 9, = 3, i.e. the point (4, 9, 3)

2. Let kjibkjia 2, +=++= and kjxixc )2( += . If the vector c lies in the plane of a and b , then

x equals

(1) 2 (2) 0

(3) 1 (4) 4

Sol. Answer (1)

cba ,, being coplanar, we have

0

1

2

1

2

1

1

1

1

=

xx

x = 2

3. Let A(h, k), B(1, 1) and C(2, 1) be the vertices of a right angled triangle with AC as its hypotenuse. If the area

of the triangle is 1, then the set of values which k can take is given by

(1) {3, 2} (2) {1, 3}

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

Aakash InstituteCorporate Office : Aakash Tower, Plot No. 4, Sector-11, MLU, Dwarka, New Delhi-110075

Solutions to AIEEE - 2007

MATHEMATICS

PAPER - I (Code - N)

Time : 3 hours M. Marks : 360

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Sol. Answer (4)

A

B C(2, 1)(1, 1)

Area of the triangle is 1, length AB is 2.

Clearly h is 1, hence the possible value of k are given by

|k 1| = 2, k = 1, 3

4. Let P = (1, 0), Q = (0, 0) and R = )33,3( be three points. The equation of the bisector of the angle PQR

is

(1) 03 =+ yx (2) 03 =+yx

(3) 02

3=+ yx (4) 0

2

3=+yx

Sol. Answer (2)

N

P

y

xQ

60

R

(0, 0)(1, 0)

(3, 3 3)

Slope of 6033

33===QR

PQR = 120

QN, the bisecer makes 1202

12060 =+ with the positive x-axis and passed through origin. So its equation is

xy 3=

03 =+yx

5. If one of the lines of my2+ (1m2)xy mx2= 0 is a bisector of the angle between the lines xy= 0, then mis

(1) 2 (2)2

1

(3) 2 (4) 1

Sol. Answer (4)

Bisector of the lines given by

xy = 0 are y = x or y = x.

Given equation is

my2+ (1 m2) xy mx2= 0

in the form

01 22 =

+

2

xxym

my

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We that coeff. of xy should be zero

1 m2= 0 m= 1

Remark : The question should have read, then one of possible values of m instead of value of m. But given by bestchoice, answe is (4)

6. Let

+= xfxfxF

1

)()( , where +=

x

dtt

t

xf1

1

log

)( . Then F(e) equals

(1) 2 (2)2

1

(3) 0 (4) 1

Sol. Answer (2)

+++=

+=

xx

dtt

tdt

l

t

xfxfxF

/1

111

log

1

log1)()(

In the 2ndintegral setu

t1

= , so that dt= du/u2, which gives

+++=xx

duuu

udt

t

txf

11)1(

log

1

log)(

+++=xx

dttt

tdt

t

t

11)1(

log

1

log

dttt

t

t

tx

++

+= )1(

log

1

log

1

dtt

tx

=1

log

Now let t= ev

Then

dvee

v

eF

v

v

=1

0)(

2

1

2

1

0

1

0

=

==

vvdv

7. Let f: RRbe a function defined by f(x) = Min {x+ 1, |x| + 1}. Then which of the following is true?

(1) f(x) is not differentiable at x = 0 (2) 1)( xf for all Rx

(3) f(x) is not differentiable at x = 1 (4) f(x) is differentiable everywhere

Sol. Answer (4)

From the graph F(x) is seen to be F(x) = x+ 1

Which is diffrentable xR.

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8. The function f : R\{0} Rgiven by

1

21)(

2 =

xexxf

can be made continuous at x= 0 by defining f(0) as

(1) 1 (2) 2

(3) 1 (4) 0

Sol. Answer (1)

For fto be continuous at x= 0

f(0) = )(lim0

xfx

which can be calculated by expassion of ex. f(0) truns out be 1.

9. The solution for xof the equation

=

x

tt

dt

222 21

is

(1) 22 (2) 2

(3) (4)2

3

Sol. Answer (No choice is available)

2112

=

x

tt

dt

2

)(sec 11 = xt , which gives 2=x .

10. + xxdx

sin3cosequals

(1) Cx

+

122tanlog (2) C

x+

+

122tanlog

2

1

(3) Cx

+

122tanlog

2

1(4) C

x+

+

122tanlog

Sol. Answer (2)

Express xx sin3cos + as 2 sin (x+ /6) and then integrate.

11. The area enclosed between the curvesy2= xandy= |x| is

(1)3

1(2)

3

2

(3) 1 (4)6

1

Sol. Answer (4)

The expression 1.1.211

0

dxx gives the required area which is 61

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12. If the difference between the roots of the equation x2+ ax+ 1 = 0 is less than 5 , then the set of possible

values of ais

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

(3) (3, ) (4) (3, )

Sol. Answer (2)

5||

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Sol. Answer (1)

,055544

4 = baCbaC nnnn gives

5

4=n

b

a

16. The set S = {1, 2, 3, ......, 12} is to be partitioned into three sets A, B, C of equal size. Thus, ,SCBA =

=== CACBBA . The number of ways to partition Sis

(1) 4)!3(

!12(2) 3)!4(!3

!12

(3) 4)!3(!3

!12(4) 3)!4(

!12

Sol. Answer (4)

The number of curves in the same as distributive 12 things into 3 ordered sets of equal sizes.

The required number = 3

)!4(

!12

17. The largest interval lying in

2,

2for which the function

+

+= )log(cos1

2cos4)( 1

2

xx

xf x is defined,

is

(1)

2,0 (2) [0, ]

(3)

2,

2(4)

2,

4

Sol. Answer (1)

For the domain of admissible values

>

Rx

x

x

0cos

112

1

x[0, /2)

This is the largest interval in

2,

2

18. A body weighing 13 kg is suspended by two strings 5 m and 12 m long, their other ends being fastened to

the extremities of a rod 13 m long. If the rod be so held that the body hangs immediately below the middle

point. The tensions in the strings are

(1) 5 kg and 13 kg (2) 12 kg and 13 kg

(3) 5 kg and 5 kg (4) 5 kg and 12 kg

Sol. Answer (4)

B

C

A

M13

5

13 kg

T2

T1

12

( + = ) 2

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The tensions are given by

)sin()sin(90sin

13 21

=

= TT

=

=cossin

13 21 TT

As135sin = , T1and T2are 5 kg and 12 kg.

19. A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is

(1)243

8(2)

729

1

(3)9

8(4)

729

8

Sol. Answer (1)

The required probability = 243

8

9

8

9

1 21

3 =

C

20. Consider a family of circles which are passing through the point (1, 1) and are tangent to x-axis. If (h, k)

are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval

(1)2

1k (2)

2

10

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22. The differential equation of all circles passing through the origin and having their centres on the x-axis is

(1)dx

dyxyxy 222 = (2)

dx

dyxyyx += 22

(3)dx

dyxyyx 322 += (4)

dx

dyxyxy 222 +=

Sol. Answer (4)

x2+ y2+ 3gx= 0 ...(1)

By eliminating g between (1) & x+dx

dyy + g= 0

We get y2= x2+ 2xydx

dy

23. If pand qare positive real numbers such that p2+ q2= 1, then the maximum value of (p + q) is

(1) 2 (2) 2

(3)2

1(4)

2

1

Sol. Answer (1)

Let p= cos qa= sin

max. valuve of p+ q= (cos + sin )max.= 2 .

24. A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that

AB (=a) subtends an angle of 60 at the foot of the tower, and the angle of elevation of the top of the tower

from A or B is 30. The height of the tower is

(1) 3a (2)3

2a

(3) 32a (4) 3

a

Sol. Answer (4)

Obviously h= atan 30

25. The sum of the series

10

20

3

20

2

20

1

20

0

20

........... CCCCC +++ is

(1) 20C10 (2) 20C10

(3) 1020

2

1C (4) 0

Sol. Answer (3)20C0

20C1+20C2

20C3.....20C9+

20C1020C11..... +

20C20= 0.

Using nCr= nC

n r

We have

2(20C020C1+

20C2.......20C9) +

20C10 = 0

20C020C1+

20C2.......20C9+

20C10 = 2

120C10

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26. The normal to a curve at P(x, y) meets the x-axis at G. If the distance of Gfrom the origin is twice the abscissa

of P, then the curve is

(1) Hyperbola (2) Ellipse

(3) Parabola (4) Circle

Sol. Answer (1)

From the given condition, we have

xdx

dyyx 2=+

x2 y2= a2, a hyperbola.

27. If 3|4| +z , then the maximum value of |z+ 1| is

(1) 0 (2) 4

(3) 10 (4) 6

Sol. Answer (4)

|z+ 1| = |z + 4 3|,|z + 4| + 3

= 6

28. The resultant of two forces P N and 3 N is a force of 7 N. If the direction of the 3 N force were reversed, the

resultant would be N19 . The value of Pis

(1) 4 N (2) 5 N

(3) 6 N (4) 3 N

Sol. Answer (2)

)9(2)19(7 222 +=+ P

P= 5N

29. Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are

0.3 and 0.2 respectively. The second plane will bomb only if the first misses the target. The probability that

the target is hit by the second plane is

(1) 0.7 (2) 0.06

(3) 0.14 (4) 0.2

Sol. Answer (3)

Required probability = P(ICII)

= P(IC

) P(II)= (0.7) (0.2)= 0.14

30. If

y

xD

+

+=

111

111

111

for 0,0 yx then Dis

(1) Divisible by ybut not x (2) Divisible by neither xnor y

(3) Divisible by both xand y (4) Divisible by xbut not y

Sol. Answer (3)Using C2C2 C1and C3C3 C1D = xy

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31. For the hyperbola 1sincos 2

2

2

2

=

yx, which of the following remains constant when varies?

(1) Abscissae of foci (2) Eccentricity

(3) Directrix (4) Abscissae of vertices

Sol. Answer (1)

1sin

y

cos

x2

2

2

2

=

Eccentricity of hyperbola is sec .

x coordinate of foci are = 1( ae), which is independent of .

32. If a line makes an angle of4

with the positive directions of each of x-axis and y-axis, then the angle that

the line makes with the positive direction of the z-axis is

(1)2 (2) 6

(3)3

(4)

4

Sol. Answer (1)

l2+ m2+ n2= 1

1n2

1

2

1 2 =++

n = 0

angle with z-axis is2

33. A value of C for which the conclusion of Mean Value Theorem holds for the function f(x) = logexon the interval

[1, 3] is

(1) loge3 (2) 2 log3e

(3) 3log2

1e (4) log3e

Sol. Answer (2)

13

)1(f)3(f)c(f=

2

3log

c

1= c= 2 log3e

34. The function f(x) = tan1(sin x+ cos x) is an increasing function in

(1)

2,

2(2)

2,

4

(3)

4,2 (4)

2,0

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Sol. Answer (3)

f(x) = tan1(sin x + cos x) = tan1

+

4xsin2

which is increasing when 24x

2

+

4

x4

3

4,

4

3

4,

2

35. Let A=

500

50

55

If |A2| = 25, then || equals

(1) 5 (2) 52

(3) 1 (4)5

1

Sol. Answer (4)

Matrix being upper triangular, the det. = product of diagonal elements = 25= |A| and |A2| = |A|2= (25)2= 25

5

1=

5

1|| =

36. The sum of the series ........!4

1

!3

1

!2

1+ upto infinity is

(1) 21

+e

(2) 2e

(3) 1e (4) 21

e

Sol. Answer (3)

Put x = 1 in expansion of ex.

37. If u and v are unit vectors and is the acute angle between them, then vu 32 is a unit vector for

(1) Exactly one value of (2) Exactly two values of

(3) More than two values of (4) No value of

Sol. Answer (2)

nsin6v3u2 = which will be a unit vector for values of given by6

1sin = , acute angle between two vectors

is possible only for two values of .

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38. A particle just clears a wall of height bat a distanceaand strikes the ground at a distance cfrom the point

of projection. The angle of projection is

(1)a

bc1tan (2)ac

b1tan

(3) 45 (4))(

tan 1aca

bc

Sol. Answer (4)

Trajectory is

= tan

R

x1xy as (a, b) lies on it.

= tan

c

a1ab

Hence tanis

)ac(a

bc

39. The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys and girls

combined is 50. The percentage of boys in the class is

(1) 60 (2) 40

(3) 20 (4) 80

Sol. Answer (4)

The combined average is 80gb

g42b52=

+

+, where b & g denotes number of boys and girls respectively.

b = 4gHence % of boys in the class = 80%

40. The equation of a tangent to the parabola y2= 8x is y = x+ 2. The point on this line from which the other

tangent to the parabola is perpendicular to the given tangent is

(1) (2, 0) (2) (1, 1)

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

Sol. Answer (1)

Perpendicular tangents to the parabola intersect at directrix which is x= 2.

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41. A circular disc of radius Ris removed from a bigger circular disc of radius 2Rsuch that the circumferencesof the discs coincide. The centre of mass of the new disc is R from the centre of the bigger disc. The valueof is

(1)4

1(2)

3

1

(3)2

1(4)

6

1

Sol. Answer (2)

123

1OOOO =

R

O1

OO2[Q Mremaining disc= 3 Mcut]

ROO3

12=

3

1=

42. A round uniform body of radius R, mass Mand moment of inertia I, rolls down (without slipping) an inclinedplane making an angle with the horizontal. Then its acceleration is

(1)/IMRl

gsin2

(2) 2I/MRl

gsin

+

(3)/IMRl

gsin2+

(4) 2I/MRl

gsin

Sol. Answer (2)

2

2

1

sin

R

K

ga

+

=

43. Angular momentum of the particle rotating with a central force is constant due to

(1) Constant Torque (2) Constant Force(3) Constant linear momentum (4) Zero Torque

Sol. Answer (4)

44. A 2 kg block slides on a horizontal floor with a speed of 4 m/s. It strikes a uncompressed spring, andcompresses it till the block is motionless. The kinetic friction force is 15 N and spring constant is 10,000 N/m.The spring compresses by

(1) 8.5 cm (2) 5.5 cm

(3) 2.5 cm (4) 11.0 cm

Sol. Answer (2)

22

2

10

2

1mvxFKx r = (by W.E. theorem)

PHYSICS

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45. A particle is projected at 60 to the horizontal with a kinetic energy K. The kinetic energy at the highest pointis

(1) K/2 (2) K

(3) Zero (4) K/4

Sol. Answer (4)

= 2Highest cosKK

46. In a Youngs double slit experiment the intensity at a point where the path difference is6

(being the

wavelength of the light used) is I. If I0denotes the maximum intensity,0I

Iis equal to

(1)4

3(2)

2

1

(3)2

3(4)

2

1

Sol. Answer (1)

=

=

2

6,

2cos20II

47. Two springs, of force constants k1and k2, are connected to a mass mas shown. The frequency of oscillationof the mass is f. If both k1 and k2 are made four times their original values, the frequency of oscillationbecomes

CLk1 k2

m

(1) 2f (2) f/2

(3) f/4 (4) 4f

Sol. Answer (1)

m

KKf 21

2

1 +

= ,

m

KKf

)(4

2

1 21+

=

48. When a system is taken from state i to state falong the path iaf, it is found that Q = 50 cal and W= 20cal. Along the path ibfQ = 36 cal. Walong the path ibf is

a

i

f

b

(1) 14 cal (2) 6 cal

(3) 16 cal (4) 66 cal

Sol. Answer (2)

Q1 w1= Q2 w2= U

49. A particle of mass mexecutes simple harmonic motion with amplitude aand frequency v. The average kineticenergy during its motion from the position of equilibrium to the end is

(1) 22ma2v2 (2) 2ma2v2

(3)2

1ma2v2 (4) 42ma2v2

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Sol. Answer (2)

2222 )2(2

1

2

1amamKav v==

50. The displacement of an object attached to a spring and executing simple harmonic motion is given byx = 2 102cos t metres. The time at which the maximum speed first occurs is

(1) 0.25 s (2) 0.5 s

(3) 0.75 s (4) 0.125 s

Sol. Answer (2)

tdt

dxv == sin102 2

This because maximum at second5.04

==T

t

51. In an a.c. circuit, the voltage applied is E= E0sin t. The resulting current in the circuit is

=

2

sin0 tII .

The power consumption in the circuit is given by

(1) 002 IEP= (2)2

00IEP=

(3) P= zero (4)2

00IEP=

Sol. Answer (3)

,2

= 0cos

== vvav IP

=

2

Q

52. An electric charge 103C is placed at the origin (0, 0) of X Yco-ordinate system. Two points A and B

are situated at )2,2( and (2, 0) respectively. The potential difference between the points A and B will be

(1) 4.5 volt (2) 9 volt

(3) Zero (4) 2 volt

Sol. Answer (3)

)2(4

1

0

qVA

= , )2(4

1

0

qVB

=

VA= VB53. A battery is used to charge a parallel plate capacitor till the potential difference between the plates becomes

equal to the electromotive force of the battery. The ratio of the energy stored in the capacitor and the workdone by the battery will be

(1)2

1(2) 1

(3) 2 (4)4

1

Sol. Answer (1)

Energy stored =

2

2

1

CV

Work done = QV = CV2

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54. An ideal coil of 10 H is connected in series with a resistance of 5 and a battery of 5 V. 2 seconds afterthe connection is made, the current flowing in amperes in the circuit is

(1) (1 e1) (2) (1 e)

(3) e (4) e1

Sol. Answer (2)

)1( LRt

eR

EI

=

55. A long straight wire of radius acarries a steady current i. The current is uniformly distributed across its cross

section. The ratio of the magnetic field at2

aand 2ais

(1)2

1(2)

4

1

(3) 4 (4) 1

Sol. Answer (4)

,2

22

0

1a

ai

B

=)2(2

02

a

iB

=

56. A current Iflows along the length of an infinitely long, straight, thin walled pipe. Then

(1) The magnetic field at all points inside the pipe is the same, but not zero

(2) The magnetic field is zero only on the axis of the pipe

(3) The magnetic field is different at different points inside the pipe(4) The magnetic field at any point inside the pipe is zero

Sol. Answer (4)

57. If MOis the mass of an oxygen isotope 8O17.MPand MNare the masses of a proton and a neutron respectively,

the nuclear binding energy of the isotope is

(1) (MO 17 MN)C2

(2) (MO 8 MP)C2

(3) (MO 8 MP 8 MN)C2

(4) MO

C2

Sol. Answer (3)

20

2 )88( CMMMmcBE NP +==

58. In gamma ray emission from a nucleus

(1) Only the proton number changes

(2) Both the neutron number and the proton number change

(3) There is no change in the proton number and the neutron number

(4) Only the neutron number changes

Sol. Answer (3)

In -decay, only energy is released configuration does not change.

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59. If in a p-njunction diode, a square input signal of 10V is applied as shown

5V

5V

RL

Then the output signal across RLwill be

(1)

+5V

(2)

10V

(3)

10V

(4)5V

Sol. Answer (1)

Only Positive half of the input will be there in the output

60. Photon of frequency v has a momentum associated with it. If c is the velocity of light, the momentum is

(1) h v/c (2) v/c

(3) hvc (4) h v/c2

Sol. Answer (1)

c

h

c

E

p

v

==

61. The velocity of a particle is v = v0+ gt + ft2. If its position is x = 0 at t = 0, then its displacement after unit

time (t = 1) is

(1) v0+ g/2 + f (2) v0+ 2g + 3f

(3) v0+ g/2 + f/3 (4) v0+ g + f

Sol. Answer (3)

++==1

0

20 )( dtgtgtvdtvx

320

gtgv ++=

62. For the given uniform square lamina ABCD, whose centre is O

D F

C

A BE

O

(1) EFAC II 2= (2) EFAC II =2

(3) IAD= 3IEF (4) IAC= IEF

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Sol. Answer (4)

I1

I2I4

I3

I0

For a square lamina,

43210 IIIII +=+=

(By axis theorem)

By symmetry, I1= I2= I3 = I463. A point mass oscillates along the x-axis according to the law x = x

0

cos (t /4). If the acceleration of theparticle is written as a= A cos (t + ), then

(1) A = x02 , = 3/4 (2) A = x0

, = /4

(3) A = x0

2 , = /4 (4) A = x02 , = /4

Sol. Answer (1)

As xa 2=

a

=

4cos0

2 tx

+=

4cos02 tx

+=

4

3cos0

2 tx

64. Charges are placed on the vertices of a square as shown. LetE be the electric field and V the potential at

the centre. If the charges on A and B are interchanged with those on D and C respectively, then

A B

D Cq q

q q

(1)E changes, V remains unchanged (2)

E remains unchanged, V changes

(3) BothE and V change (4)

E and V remain unchanged

Sol. Answer (1)

Vremains zero

E reverses its direction

65. The half-life period of a radio-active element X is same as the mean life time of another radio-active elementY. Initially they have the same number of atoms. Then

(1) X and Y decay at same rate always (2) X will decay faster than Y

(3) Y will decay faster than X (4) X and Y have same decay rate initially

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Sol. Answer (3)

In one half life, Xbecause2

X, where as in the same time Ybecomes

e

Y

Ydecays faster.

66. A Carnot engine, having an efficiency of = 1/10 as heat engine, is used as a refrigerator. If the work doneon the system is 10 J, the amount of energy absorbed from the reservoir at lower temperature is

(1) 100 J (2) 99 J

(3) 90 J (4) 1 J

Sol. Answer (3)

1Q

W=

Q1= 100 J

Q2= 100 10 = 90 J

67. Carbon, silicon and germanium have four valence electrons each. At room temperature which one of the followingstatements is most appropriate

(1) The number of free elecrons for conduction is significant only in Si and Ge but small in C

(2) The number of free conduction electrons is significant in C but small in Si and Ge

(3) The number of free conduction electrons is negligibly small in all the three

(4) The number of free electrons for conduction is significant in all the three

Sol. Answer (1)

68. A charged particle with charge q enters a region of constant uniform and mutually orthogonal fieldsE and

B with a velocity

v perpendicular to both

E and

and comes out without any change in magnitude or

direction ofv . Then

(1) 2/EEBv

= (2)2/BBEv

=

(3) 2/BEBv

= (4) 2/EBEv

=

Sol. Answer (2)

B

v

E

The magnitude of vfor the given condition is B

E

v= and it is directed along BErr

.

69. The potential at a point x (measured in m) due to some charges situated on the x-axis is given by

V (x) = 20/(x2 4) Volts

The electric field E at x = 4 m is given by

(1) 10/9 Volt m and in the +ve x direction (2) 5/3 Volt m and in the ve x direction

(3) 5/3 Volt m and in the +ve x direction (4) 10/9 Volt m and in the +ve x direction

Sol. Answer (1)

dx

dVE =

22 )4(

220

=

x

xE

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70. Which of the following transitions in hydrogen atoms emit photons of highest frequency?

(1) n = 1 to n = 2 (2) n = 2 to n = 6

(3) n = 6 to n = 2 (4) n = 2 to n = 1

Sol. Answer (4)

The energy difference E2 E1is maximum.

71. A block of mass m is connected to another block of mass Mby a spring (massless) of spring constant k.The blocks are kept on a smooth horizontal plane. Initially the blocks are at rest and the spring is unstretched.Then a constant force Fstarts acting on the block of mass Mto pull it. Find the force on the block of massm

(1))( Mm

MF

+(2)

M

mF

(3)m

FmM )( +(4)

)( Mm

mF

+

Sol. Answer (4)

mMFa+

= m m Ff

a

mM

mFamf

+==

72. Two lenses of power 15D and +5D are in contact with each other. The focal length of the combination is

(1) + 10 cm (2) 20 cm

(3) 10 cm (4) + 20 cm

Sol. Answer (3)

P = P1+ P2 or 21

1

PPf +=

73. One end of a thermally insulated rod is kept at a temperature T1and the other at T2. The rod is composedof two sections of lengths l1and l2and thermal conductivities k1and k2respectively. The temperature at theinterface of the two section is

T1

l1

l2

T2

k1

k2

(1) (k1l1T1+ k2l2T2) / (k1 l1+ k2l2) (2) (k2l2T1+ k1l1T2) / (k1 l1+ k2l2)

(3) (k2l

1T

1+ k

1l

2T

2) / (k

2 l

1+ k

1l

2) (4) (k

1l

2T

1+ k

2l

1T

2) / (k

1 l

2+ k

2l

1)

Sol. Answer (4)

2

22

1

11 )()(

ll

TTAKTTAKQ

=

=

74. A sound absorber attenuates the sound level by 20 dB. The intensity decreases by a factor of

(1) 100 (2) 1000

(3) 10000 (4) 10

Sol. Answer (1)

=

1

210log1020

I

I

=

1

212 log10

I

ILL

100

12

II =

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75. If Cpand Cvdenote the specific heats of nitrogen per unit mass at constant pressure and constant volumerespectively, then

(1) Cp Cv= 28 R (2) Cp Cv= R/28

(3) Cp C

v= R/14 (4) C

p C

v= R

Sol. Answer (2)

Cmolar= Mol. mass cspecificCp CV= R

M(cp cV) = R

cp cV= 28

R

M

R=

76. A charged particle moves through a magnetic field perpendicular to its direction. Then

(1) Kinetic energy changes but the momentum is constant

(2) The momentum changes but the kinetic energy is constant

(3) Both momentum and kinetic energy of the particle are not constant

(4) Both momentum and kinetic energy of the particle are constantSol. Answer (2)

As )( BVqFrrr

= Vr

KE= 0

77. Two identical conducting wires AOB and COD are placed at right angles to each other. The wire AOB carriesan electric current I1and COD carries a current I2. The magnetic field on a point lying at a distance dfromO, in a direction perpendicular to the plane of the wires AOB and COD, will be given by

(1) )I(I2

22

21

0 +d

(2)

2/1210 II

2

+

dd

(3)1/22

22

10 )I(I

2

+

d(4) )I(I

2

21

0 +d

Sol. Answer (3)

B2B1B

AC

O

I2

D

d

I1

22

21 BBB +=

d

IB

=

2

101 ,

d

IB

=

2

202

78. The resistance of a wire is 5 ohm at 50C and 6 ohm at 100C. The resistance of the wire at 0C will be

(1) 3 ohm (2) 2 ohm

(3) 1 ohm (4) 4 ohm

Sol. Answer (4)

0100

50100

6

56

0

=

R

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79. A parallel plate condenser with a dielectric of dielectric constant K between the plates has a capacity C andis charged to a potential V volts. The dielectric slab is slowly removed from between the plates and thenre-inserted. The net work done by the system in this process is

(1) Zero (2)2

1(K 1) CV2

(3) CV2(K 1)/K (4) (K 1) CV2

Sol. Answer (1)

No change in initial and final energy.

80. If gEand gmare the accelerations due to gravity on the surfaces of the earth and the moon respectively andif Millikans oil drop experiment could be performed on the two surfaces, one will find the ratio

electronic charge on the moon

electronic charge on the earthto be

(1) gM/gE (2) 1

(3) 0 (4) gE/g

M

Sol. Answer (2)

Charge of an electron is a universal constant.

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CHEMISTRY

81. The equivalent conductance of two strong electrolytes at infinite dilution in H2O (where ions move freely through

a solution) at 25C are given below:

equiv/cmS0.91 2COONaCH3 =

equiv/cmS2.426 2HCl=

What additional information/quantity one needs to calculate of an aqueous solution of acetic acid?

(1) of chloroacetic acid (ClCH2COOH) (2) of NaCl

(3) of CH3COOK (4) The limiting equivalent conductance )(H

H++

Sol. Answer (2)

NaClHClCOONaCHCOOHCH 33+=

82. Which one of the following is the strongest base in aqueous solution?

(1) Methylamine (2) Trimethylamine

(3) Aniline (4) Dimethylamine

Sol. Answer (4)

83. The compound formed as a result of oxidation of ethyl benezene by KMnO4is

(1) Benzyl alcohol (2) Benzophenone

(3) Acetophenone (4) Benzoic acid

Sol. Answer (4)

KMnO4

H+

COOH

84. The IUPAC name of is

(1) 3-ethyl-4, 4-dimethylheptane (2) 1, 1-diethyl-2, 2-dimethylpentane

(3) 4, 4-dimethyl-5, 5-diethylpentane (4) 5, 5-diethyl-4, 4-dimethylpentane

Sol. Answer (1)

1

24

35

6

7

3-ethyl -4, 4-dimethylheptane

85. Which of the following species exhibits the diamagnetic behaviour?

(1) NO (2) 22O

(3) +2O (4) O2

Sol. Answer (2)

22O total electrons = 18

Electronic configuration :

22

22

22

22

22

22

22

22

21

21 **,** yxzyxz ppppppssss ==

86. The stability of dihalides of Si, Ge, Sn and Pb increases steadily in the sequence

(1) Pb X2

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87. Identify the incorrect statement among the following :

(1) Br2reacts with hot and strong NaOH solution to give NaBr, NaBrO

4and H

2O

(2) Ozone reacts with SO2to give SO

3

(3) Silicon reacts with NaOH(aq) in the presence of air to give Na2SIO

3and H

2O

(4) Cl2reacts with excess of NH

3to give N

2and HCl

Sol. Answer (1)

88. The charge/size ratio of a cation determines its polarizing power. Which one of the following sequencesrepresents the increasing order of the polarizing power of the cationic species, K+, Ca2+, Mg2+, Be2+?

(1) Ca2+< Mg2+< Be2+< K+ (2) Mg2+ Be2+< K+< Ca2+

(3) Be2+< K+Ca2+< Mg2+ (4) K+< Ca2+< Mg2+< Be2+

Sol. Answer (4)

89. The density (in g mL

1

) of a 3.60 M sulphuric acid solution that is 29% H2SO4(Molar mass = 98 g mol

1

) bymass will be

(1) 1.45 (2) 1.64

(3) 1.88 (4) 1.22

Sol. Answer (4)

1L of soln. contains = 3.6 mole

= 3.6 98 gram of H2SO

4

soln.SOHofweightlitre)1(in

42 =29

100983.6

mL/g100029

100986.3d

=

1.22 g/mL

90. The first and second dissociation constants of an acid H2A and 1.0 105and 5.0 1010respectively. The

overall dissociation constant of the acid will be

(1) 0.2 105 (2) 5.0 105

(3) 5.0 1015 (4) 5.0 1015

Sol. Answer (4)

91. A mixture of ethyl alcohol and propyl alcohol has a vapour pressure of 290 mm at 300 K. The vapour pressureof propyl alcohol is 200 mm. If the mole fraction of ethyl alcohol is 0.6, its vapour pressure (in mm) at thesame temperature will be

(1) 360 (2) 350

(3) 300 (4) 700

Sol. Answer (2)

Totalalcohol

opylPralcohol

opylPralcoholEthyl

alcoholEthyl PPP =+

4.02006.0PE + = 290

EP = 350 mm of Hg

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92. In conversion of lime-stone to lime,

CaCO3(s) CaO(s) + CO

2(g)

the values of H and S are +179.1 kJ mol1and 160.2 J/K respectively at 298 K and 1 bar. Assuming thatH and S do not change with temperature, temperature above which conversion of limestone to lime willbe spontaneous is

(1) 1118 K (2) 1008 K

(3) 1200 K (4) 845 K

Sol. Answer (1)

For the reaction to be spontaneous, i.e., G should be negative.

TS > H

T 160.2 > 179.1 1000

T > 1118 K

93. The energies of activation for forward and reverse reactions for A2 + B

2 2AB are 180 kJ mol1 and

200 kJ mol1respectively. The presence of a catalyst lowers the activation energy of both (forward and reverse)reactions by 100 kJ mol1. The enthalpy change of the reaction (A

2+ B

22AB) in the presence of catalyst

will be (in kJ mol1)

(1) 20 (2) 300

(3) 120 (4) 280

Sol. Answer (1)

H = Eaf E

ab

= 180 200 = 20 kJ/mol

Reaction is exothermic and heat evolved is 20 kJ/ mol. (No effect on H, in pressure of catalyst)

94. The cell, Zn | Zn2+(1M) || Cu2+(1M) | Cu )V10.1E( cell = , was allowed to be completely discharged at 298 K. The

relative concentration of Zn2+to Cu2+

+

+

]Cu[]Zn[

2

2

is

(1) 9.65 104 (2) Antilog (24.08)

(3) 37.3 (4) 1037.3

Sol. Answer (4)

Ecell

= 0

+

+

=2

2

cellCu

Znlog

n

0591.E

= +

+

2

2

CuZnlog

20591.1.1

=

=

+

+

2

2

Cu

Znlog

591.0

21.1

3.37

2

2

10Cu

Zn=

+

+

95. The pKa of a weak acid (HA) is 4.5. The pOH of an aqueous buffered solution of HA in which 50% of the acidis ionized is

(1) 7.0 (2) 4.5

(3) 2.5 (4) 9.5

Sol. Answer (4)

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96. Consider the reaction,

2A + B Products

When concentration of B alone was doubled, the half-life did not change. When the concentration of A alonewas doubled, the rate increased by two times. The unit of rate constant for this reaction is

(1) s1 (2) L mol1s1

(3) No unit (4) mol L1

s1

Sol. Answer (1)

Rate = K[A] [B]0

Q Reaction is first order reaction.

Unit of K = sec1

97. Identify the incorrect statement among the following

(1) 4fand 5forbitals are equally shielded

(2) d-Block elements show irregular and erratic chemical properties among themselves

(3) La and Lu have partially filled d-orbitals and no other partially filled orbitals

(4) The chemistry of various lanthanoids is very similar

Sol. Answer (1)

98. Which one of the following has a square planar geometry?

(1) [PtCl4]2 (2) [CoCl

4]2

(3) [FeCl4]2 (4) [NiCl

4]2

(At. Nos. Co = 27, Ni = 28, Fe = 26, Pt = 78)

Sol. Answer (1)

Crystal field splitting increases on moving from 3d to 4d to 5d

In case of Pt, even in case of weak ligand 0> P.

99. Which of the following molecules is expected to rotate the plane of plane-polarised light?

(1)

COOH

H

HH N2 (2)

CHO

CH OH2

HHO

(3)SH

(4) H H

H N2

Ph

NH3

Ph

Sol. Answer (2)

100. The secondary structure of a protein refers to

(1) Fixed configuration of the polypeptide backbone

(2) -helical backbone

(3) Hydrophobic interactions

(4) Sequence of -amino acids

Sol. Answer (2)

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101. Which of the following reactions will yield 2, 2-dibromopropane?

(1) CH3CH = CH

2+ HBr (2) CH

3C CH + 2HBr

(3) CH3CH = CHBr + HBr (4) CH CH + 2HBr

Sol. Answer (2)

CH C CH + 2HBr3

CH C3 3

CH

Br

|

|Br

(2, 2-dibromopropane)

102. In the chemical reaction,

CH3CH

2NH

2+ CHCl

3+ 3KOH (A) + (B) + 3H

2O, the compounds (A) and (B) are respectively :

(1) C2H

5NC and 3KCl (2) C

2H

5CN and 3KCl

(3) CH3CH

2CONH

2and 3KCl (4) C

2H

5NC and K

2CO

3

Sol. Answer (1)

CH3CH

2NH

2+ CHCl

3+ 3KOH

Reaction)ne(Carbylami252 O3H3KClNCHC ++

103. The reaction of toluene with Cl2and presence of FeCl

3gives predominantly

(1) m-Chlorotoluene (2) Benzoyl chloride

(3) Benzyl chloride (4) o - and p-chlorotoluene

Sol. Answer (4)

CH3

+ Cl2FeCl

3

CH3

Cl+

CH3

ClO-chloro Toluene P-chloro toluene

104. Presence of a nitro group in a benzene ring

(1) Deactivates the ring towards electrophilic substitution

(2) Activates the ring towards electrophilic substitution

(3) Renders the ring basic

(4) Deactivates the ring towards nucleophilic substitution

Sol. Answer (1)

NO2is a deactivating group for electrophilic aromatic substituion.

105. In which of the following ionization processes, the bond order has increased and the magnetic behaviour haschanged?

(1) N2N

2+ (2) C

2C

2+

(3) NO NO+ (4) O2O

2+

Sol. Answer (3)

Bond order of NO is 2.5, No. of unpaired electron = 1 (paramagnetic)

Bond order of NO+is 3, No. of unpaired electron = 0 (diamagnetic)

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106. The actinoids exhibit more number of oxidation states in general than the lanthanoids. This is because

(1) The 5forbitals extend further from the nucleus than the 4forbitals

(2) The 5f orbitals are more buried than the 4forbitals

(3) There is a similarity between 4fand 5forbitals in their angular part of the wave function

(4) The actinoids are more reactive than the lanthanoids

Sol. Answer (1)The 5f orbitals extend into space beyond the 6s and 6p-orbitals and participate in bonding where as in lanthanides4f-orbitals are buried deep inside and thus unable to participate in bonding.

107. Equal masses of methane and oxygen are mixed in an empty container at 25C. The fraction of the totalpressure exerted by oxygen is

(1)2

1(2)

3

2

(3)298

273

3

1 (4)

3

1

Sol. Answer (4)

24

2

2

oCH

o

Onn

nP

+=

totalP

32

w

16

w32

w

+

=

totalPw3

w

fraction of total pressure is31

108. A 5.25% solution of a substance is isotonic with a 1.5% solution of urea (molar mass = 60 g mol1) in the samesolvent. If the densities of both the solutions are assumed to be equal to 1.0 g cm3, molar mass of the substancewill be

(1) 210.0 g mol1 (2) 90.0 g mol1

(3) 115.0 g mol1 (4) 105.0 g mol1

Sol. Answer (1)

For isotonic solution

2

2

1

1

v

n

v

n

=

Q v1= v

2

60

5.1

M

25.5=

M = 210

109. Assuming that water vapour is an ideal gas the internal energy change (U) when 1 mol of water is vapourised at 1bar pressure and 100C, (Given : Molar enthalpy of vapourisation of water at 1 bar and 373 K = 41 kJ mol1and R =8.3 J mol1K1) will be

(1) 41.00 kJ mol1 (2) 4.100 kJ mol1

(3) 3.7904 kJ mol1 (4) 37.904 kJ mol1

Sol. Answer (4)

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110. In a saturated solution of the sparingly soluble strong electrolyte AgIO3(Molecular mass = 283) the equilibrium

which sets in is

AgIO3(s)

Ag+(aq)

+ IO3(aq)

If the solubility product constant Ksp

of AgIO3at a given temperature is 1.0 108, what is the mass of AgIO

3

contained in 100 ml of its saturated solution ?

(1) 1.0 104

g (2) 28.3 102

g(3) 2.83 103g (4) 1.0 107g

Sol. Answer (3)

AgIO3(s) Ag+(aq) + IO

3(aq)

The solubility of AgIO3is S mol/L

Ksp

= [Ag+] [IO3

]

1 108= S2

S = 104mol/L

moles of AgIO3is 100 mL = 105

molmass of AglO

3in 100 mL = 105 283

= 2.873 103gm

111. A radioactive element gets spilled over the floor of a room. Its half-life period is 30 days. If the initial activity is tentimes the permissible value, after how many days will it be safe to enter the room?

(1) 100 days (2) 1000 days

(3) 300 days (4) 10 days

Sol. Answer (1)

At

Alog

t

303.2 0= A0= 10 At

10logt

303.2

30

6932.0=

days100~3010.0

30t =

112. Which one of the following conformations of cyclohexane is chiral?

(1) Boat (2) Twist boat

(3) Rigid (4) Chair

Sol. Answer (2)

113. Which of the following is the correct order of decreasing SN2 reactivity? (X = a halogen)

(1) R2CHX > R

3CX > RCH

2X (2) RCH

2X > R

3CX > R

2CHX

(3) RCH2X > R

2CHX > R

3CX (4) R

3CX > R

2CHX > RCH

2X

Sol. Answer (3)

SN2 mechanism proceed through transition state. for Alkyl halide which is less hindered, transition state will be

more stable.

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114. In the following sequence of reactions,

BAOHCHCHether

MgIP

232 + DC

OHHCHO 2

the compound D is

(1) Propanal (2) Butanal

(3) n-butyl alcohol (4) n-propyl alcohol

Sol. Answer (4)

+ Mg23IP

23 ICHCHOHCHCH2 HCHO23 MgICHCH CH CH CH3 2

OMgI|

|

H

alcoholpropyln223

H/OH OHCHCHCH2

+

115. Which of the following sets of quantum number represents the highest energy of an atom?

(1) n = 3, l= 0, m = 0, s = + 2

1

(2) n = 3, l= 1, m = 1, s = + 2

1

(3) n = 3, l= 2, m = 1, s = +2

1(4) n = 4, l= 0, m = 0, s = +

2

1

Sol. Answer (3)

For n = 3, l= 2 value of (n + l) is highest.

Its energy will be highest.

116. Which of the following hydrogen bonds is the strongest?

(1) O H ...... F (2) O H ...... N

(3) F H ...... F (4) O H ...... O

Sol. Answer (3)

Electronegativity of fluorine is highest.

It forms strongest H-bonds.

117. In the reaction,

2Al(s)

+ 6HCl(aq)

2Al3(aq)

+ 6Cl (aq)

+ 3H2(g)

,

(1) 11.2 L H2(g) at STP is produced for every mole HCl

(aq)consumed

(2) 6 L HCl(aq)

is consumed for every 3 L H2(g) produced

(3) 33.6 L H2(g) is produced regardless of temperature and pressure for every mole Al that reacts

(4) 67.2 L H2(g) at STP is produced for every mole Althat reacts

Sol. Answer (1)

2Al(s) + 6HCl(aq) 2Al3+(aq) + 6Cl(aq) + 3H2(g)

Q 6 moles HCl reacts with Al and evolves = 3 22.4 litre H2at STP.

1 mole of HCl will evolve 11.2 litre of H2(g) at STP.

118. Regular use of which of the following fertilizers increases the acidity of soil?

(1) Ammonium sulphate (2) Potassium nitrate

(3) Urea (4) Superphosphate of-lime

Sol. Answer (1)

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119. Identify the correct statement regarding a spontaneous process

(1) Lowering of energy in the reaction process is the only criterion for spontaneity

(2) For a spontaneous process in an isolated system, the change in entropy is positive

(3) Endothermic processes are never spontaneous

(4) Exothermic processes are always spontaneous

Sol. Answer (2)

For spontaneous process is an isolated system the change in entropy is positive S > 0.

120. Which of the following nuclear reactions will generate an isotope?

(1) -particle emission (2) Neutron particle emission

(3) Positron emission (4) -particle emission

Sol. Answer (2)

1o

1mz

mz nYX +

Here X and Y isotopes.

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