farewel song

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Quest Tutorials

H-108, New Asiatic Building, Connaught place, New Delhi 110001

CHEMISTRY PAPER-I

PART I CHEMISTRY

SECTION I (TOTAL MARKS: 21)

(SINGLE CORRECT CHOICE TYPE)

This section contains 7 multiple choice questions. Each question has four choices (a),(b), (c) and (d) outof

which ONLY ONE is correct.

1. Dissolving 120g of urea (mol. wt 60) in 1000g of water gave a solution of density 1.15g/mL. The

molarity of the solution is

(a) 1.78 M (b) 2.00 M (c) 2.05 M (d) 2.22 M]

1. (c)

2. AgNO3

(aq.) was added to an aqueous KCl solution gradually and the conductivity of the solution

was measured. The plot of conductance versus the volume of AgNO3 is

(a) (P) (b) (Q) (c) (R) (d) (S)

2. (d) AgNO KCl AgCl KNO 3

3

Conductivity will remain same for some time then gradually increase due to excess of

KCl.

3. Among the following compounds, the most acidic is

(a) p-nitrophenol (b) p-hydroxybenzoic zcid

(c) ohydroxybenzoic acid (d) p-toluic acid

3. (c) Due to ortho effect 2


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respectively are

(a) Octahedral, tetrahedral and square planar

(b) Tetrahedral, square planar and octahedral

(c) Square planar, tetrahedral and octahedral

(d) Octahedral, square planar and octahedral

6. (b)

2

3

NiCl sp

4

2

2

Ni CN dsp

4

2

3 2

2 6

Ni H O sp d


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3

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7. Bombardment of aluminium by


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(b)

3 2

An NO (c)

3 3

Al NO (d)

3 2

Pb NO

15. (a)

16. The final solution contains

(a)

2 2

3 4 4

Pb NH and CoCl

(b)

3 2

3 3 4 4

Al NH and Cu NH


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(c)

2

3 3 2

Ag NH and Cu NH 4

(d)

2

3 3 2 6

Ag NH and Ni NH

16. (c)7

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SECTION IV (TOTAL MARKS: 28)

(INTEGER ANSWER TYPE)

This section contains 7 questions. The answer to each of the questions is a single-digit integer, ranging

from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS.

17. The work functionof some metals is listed below. The number of metals which will show

photoelectric effect when light of 300nm was wavelength falls on the metals is

Metal Li Na K Mg Cu Ag Fe Pt W

eV 2.4 2.3 2.2 3.7 4.8 4.3 4.7 6.3 4.75

17. (4)

o

3000A 4.1 eV.


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All below this value will eject photo electron

18. To an evacuated vessel with movable piston under external pressure of 1atm, 0.1mol of He and 1.0

mol of an unknown compound (vapour pressure 0.68 atm. At 0

o

C) are introduced. Considering the

ideal gas behaviour, the total volume (in lite) of the gases at 0

o

C is close to.

18. (7)

1 273

0.1 7

12 0.32

19. Reaction of Br2 with Na2CO3 in aqueous solution gives sodium bromide and sodium bromated

with evolution of CO2

gas. The number o sodium bromide molecules involved in the balanced

chemical equation is

19. (5)

2 2 3 3 2

3Br 3Na CO 5NaBr NaBrO 3CO

20. The difference in the oxidation numbers of the two types of sulphur atoms in Na S O is

2 4 6

20. (5)

5 0 = 5


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rotated on a horizontal circular path about vertical axis. The maximum tension that the string can

bear is 324 N. The maximum possible value of angular velocity of ball (in radian/s) is:

(a) 9 (b) 18 (c) 27 (d) 36

25. (d) T

2 2

sin m r m Lsin

2

T m L

T 324

18 / 0.5 36

mL 0.5 0.5

10

Quest Tutorials


26. A meter bridge is set up as shown, to determine an unknown resistance X using a standard 10

ohm resistor. The galvanometer shows null point when tapping key is at 52 cm. mark. The end

corrections are 1 cm and 2 cm respectively for the ends A and B. The determined value of X is

(a) 10.2 ohm (b) 10.6 ohm (c) 10.8 ohm (d) 11.1 ohm

26. (b) x /10 52 1 / 48 2 x 10.6

27. A 2 -F capacitor is charged as shown in figure. The percentage of its stored energy dissipated

after the switch S is turned to position 2 is

(a) 0% (b) 20% (c) 75% (d) 80%


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% of its stored energy dissipated =

2

2

Q 4

100 80%

5 Q

28. A police car with a siren of frequency 8 kHz is moving with uniform velocity 36 km/hr towards a

tall building which reflects the sound waves. The speed of sound in air is 320 ms

-1

. The frequency

of the siren heard by the car driver is:

(a) 8.50 kHz (b) 8.25 kHz (c) 7.75 kHz (d) 7.50 kHz

28. (a) Velocity of sound, C = 320 ms

-1

frequency of the siren heared by the car driver

f =

0

f C V / C V 8 320 10 / 320 10 8.50kHz


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This section contains 4 multiple choice questions. Each question has four choices (a), (b), (c) and (d) out

of which ONE OR MORE may be correct.

31. A metal rod of length L and mass m is pivoted at one end. A thin disk of mass M and radius

R ( Angular frequency for case B

(d) Angular frequency for case A < Angular frequency for case B

31. (a, d) Torque due to weight is same in both case (A)

T I 1 / I

sys sys

Case A: When disc is not rotating then M.I of system about point of suspension = IA

Case B : When disc is rotating then M.I. of system about point of suspension = IB

2

pivot pivot Pivot

B rod disc A rod disc

MR

I I I I I I

2


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inside

E 0 from Gauss law A A

2

A B A A B B

A B 2 2

A B B B A A

Q Q Q R R

, C

4 R 4 R Q R R

$

$ $

# # $

on surface on surface

E & E A A B B $ $

on surface on surface E < E D A B14

Quest Tutorials


SECTION-III (TOTAL MARKS: 15)

(PARAGRAPH TYPE)

This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions andbased

on the paragraph 3 multiple choice questions have to be answered. Each of these questions has four

choices (a), (b), (c) and (d) out of which ONLY ONE is correct.

Paragraph for questions 35 to 36

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids


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containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let N bethe

number density of free electrons, each of mass m. When the elec trons are subjected to an electricfield,

they are displace relatively away from the heavy positive ions. If the electric field becomes zero, the

electrons begin to oscillate about the positive ions with a natural angular frequency p

, which is called

the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied thathas

an angular frequency , where a part of the energy is absorbed and a part of it is reflected. As

appropriate p

, all the free electrons are set to resonance together and all the energy is reflected. This is the

explanation of high reflectivity of metals.

35. Taking the electronic charge as e and the permittivity as

0

% , use dimensional analysis to

determine the correct expression for p

.

(a)

0

Ne

m%

(b)

m 0

Ne

%

(c)


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2

1/2 1/2

p 0 Ne m

%

2

p

0

Ne

m

%15

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36. Estimate the wavelength at which plasma reflection will occur for a metal having the density of

electrons

27 3

N 4 10 m


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, where these quantities are in proper SI

units.

(a) 800 nm (b) 600 nm (c) 300 nm (d) 200 nm

36. (b) f / 2 #

2

27 19

30 11

4 10 1.6 10

10 10

2

19 68 15 15

1.6 10 4 10 2 1.6 10 3.2 10

8

7

15

3 10 6

c / f 10 m

3.2 10 3.2


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1 2

E 4E 1 2

39. Consider the spring mass system, with the mass submerged in water, as shown in the figure. The

phase space diagram for one cycle of this system is:

39. (b) The mass has zero momentum at highest position. From here, it moves down, during

which position decreases but momentum increases in negative value . After one cycle,

its position is slightly less than initial value (damped oscillation). 17

Quest Tutorials




This section contains 7 questions. The answer to each question is a single digit integer ranging from 0 to9.

The bubble corresponding to the correct answer is to be darkened in the ORS.

40. Steel wire of length L at 40

o

C is suspended from the ceiling and then a mass m us hung from

its free end. The wire is cooled down from 40

o

C to 30

o

C to regain its original length L. The

coefficient of linear thermal expansion of the steel is 10

-5

/

o


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6

m 10 kg 1 mg

42. A block is moving on an inclined plane making an angle 45

o

with the horizontal and the

coefficient of friction is - . The force required to just push it up the inclined plane is 3 times the

force required to just prevent it from sliding down. If we define N = 10 -, then N is

42. (5)

u

d u d

F mg sin cos

F mg sin cos & F 3F

1 / 2 tan 1 / 2

N 10 5

-

-

-

- 18

Quest Tutorials


43. A boy is pushing a ring of mass 2 kg and radius 0.5m with a sticks as shown in the figure. The


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2

0 0

2

A / LR 300 I sin 300t

N 0.6 6

- -

#

46. Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap fil of

side a. The surface tension of the soap film is . The system of charges and planar film asre in

equilibrium, and

1/ N

2

q

a k

) *

, where k is a constant . then N is

46. (3) Consider the upper half of the setup


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For equilibrium of the system

2 2

2 2

0 0

1 q 1 q 1

2. a 2 . . .

4 4 a 2

2a

) *

#% #%

3 2

0

a q 1 / 4 . 1 1/ 2 2 #%

) *

a = constant

1/3

2

q / N 3


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" 5 . If

n n

n

a


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6 2 0

2


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2 6

and

2

5 5 6 2 0

2 5 5 2 6

10 10 8 8 8 2 8 2

8 8

10 8

9 9 9 9 9 9

9

a 2a 2 2 2 6 6

3

2a 2 2 2


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5


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5


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5 5


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5 5


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521

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50. A straight line L through the point (3, -2) is inclined at an angle 60

o

to the line 3x y 1 . If L

also intersects the x axis, then the equation of L is

(a) y 3x 2 3 3 0 (b) y - 3x 2 3 3 0

(c) 3y x 3 2 3 0 (d) 3y x 3 2 3 0

50. (b) If required line has slope = m

m 3 o

tan 60 m 0, 3

1 3m

Since, the line intersects the x axis

equations is, y 2 3 x 3 y 3x 2 3 3 0

51. Let

0 0

x , y be the solution of the following equations

ln 2 ln 3

2x 3y

ln x ln y

3 2


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Then x0 is

(a) 1/6 (b) 1/ 3 (c) (d) 6

51. (c) Since ln 2 ln 2 ln x ln 3 ln 3 ln y and lnx . ln 3 ln y. ln 2 = 0

2 2

ln 2 ln x ln y ln 3 ln 3 ln 2 .(i)

and ln x.ln 3 ln y.ln 2 0 (ii)

Using (i) ln 2 (ii) ln 3, we get

2 2 2 2

ln x. ln 2 ln 3 ln 2 ln 3 ln 2

ln x ln 2 x 1 / 2

52. The value of

ln 3 2

2 2

ln 2

x sin x

dx

sin x sin ln 6 x

4

is

(a) 1 / 4 ln 3 / 2 (b) 1 / 2 ln 3 / 2 (c) ln 3 / 2 (d) 1 / 6 ln 3 / 2

52. (a)

ln 3 2

2 2

ln 2

x sin x


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I dx

sin x sin ln 6 x

4

2

x t 2xdx=dt

ln 3

ln 2

1 sin t

I dt

2 sin t sin ln 6 t

4

Using

b b

a a

1

f x dx f x f a b x dx

2

4 4

ln 3

ln 2


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1 sin t sin ln 6 t

I dt

4 sin t sin ln 6 t sin ln 6 t sin t

4

[since ln 3 + ln 2 = ln 6]

ln 3

ln 3

ln 2

ln 2

1 1 1 3

I dt t ln

4 4 4 2

422

Quest Tutorials


53. Let

a i j k, b = i - j + k


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and

c i j k

be three vectors. A vector v in the plane of a

and

b

, whose projection on c

is 1/ 3 is given by

(a)

i 3j 3k (b) -

3i 3j k (c)

3i j 3k (d)

i 3j 3k

53. (c)

v i j k i j k -

v.c 1


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1

c 3

- - -

- 1

v i 2 1 j k 2 1

Only option (c) satisfies this one. 23

Quest Tutorials


SECTION-II (TOTAL MARKS: 16)

(Multiple Correct Choice Type)

This section contains 4 multiple choice questions. Each question has four choices (a), (b), (c) and (d) outof

which ONE OR MORE may be correct.

54. Let f: R R be a function such that f ( x + y ) = f x f y , x, y R 7 8

If f(x) is differentiable at x = 0, then

(a) f(x) is differentiable only in a finite interval containing zero.

(b) f(x) is continuous 7 8x R

(c) f ' x is constant 7 8x R

(d) f(x) is differentiable except at finitely many points

54. (b) (c)

h 0 h 0 h 0

f x h f x f x f h f x f h f 0

f ' x Lim Lim Lim

h h h


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Put x y 0, f 0 0

f ' x f ' 0

f x f ' 0 x c

f x kx c, c = 0 as f 0 0

Function is continuous. f(x) is constant.

55. Let the eccentricity of the hyperbola

2 2

2 2

x y

1

a b

be reciprocal to that of the ellipse

2 2

x 4y 4 .

If the hyperbola passes through a focus of the ellipse, then

(a) the equation of the hyperbola is

2 2

x y

1

3 2

(b) a focus of the hyperbola is (2, 0)

(c) the eccentricity of the hyperbola is


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5

3

(d) the equation of the hyperbola is

2 2

x 3y 3

55. (b) (d)

2 2

x y

1

4 1

,

2 2 2 2 2

a 4, b 1,e 1 b / a e 3 / 2

eccentricity of hyperbola = 2 / 3

Focus of ellipse = 9 9 ae,0 3,0

2 2

2 2

x y

1

a b

passes through focus 9 3,0

2

a 3

2 2 2

b a e 1 1


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Equation of hyperbola:

2 2 2 2

2 2

x y x y

1 1

a b 3 1

2 2

x 3y 3

Focus of hyperbola : 9 9 ae,0 2,024

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56. Let M and N be two 3 3 non singular skew symmetric matrices such that MN = NM. If P

T

denotes the transpose of P, then

1

2 2 T 1

M N M N MN T

is equal to

(a)

2

M (b)

2


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N (c)

2

M (d) MN

56. (c) Since

T T

M M, N N

1 T

2 2 T 1

M N M N MN

1 T 1 1

2 2 1 2 2 1 T

M N MN MN M N N M N M

1 1 1

M N M N M M MN MN M M 2 2

57. The vector (s) which is/are coplanar with vectors

i j 2k and

i 2 j k , and perpendicular to


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the vector

i j k is /are

(a)

j k (b) -

i j (c)

i j (d)

j k

57. (a) (d) Let

a i j 2k i 2 j k -

- - - i 2 j 2 k

a i j k 0

- a= j k

Taking


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1, a = -j + k

1, a= j - k

25

Quest Tutorials



(Paragraph Type)

This section contains 2 paragraphs. Based upon one of the paragraphs 2 multiple choice questions and

based on other paragraph 3 multiple choice questions have to be answered. Each of these has fourchoices

(a), (b), (c) and (d) out of which ONLY ONE is correct.

Paragraph for question Nos 58 and 59

Let U1

and U2

be two urns such that U1

contains 3 white and 2 red balls, and U2

contains only 1 white ball.

A fair coin is tossed. If head appears then 1 ball is drawn at random from U1

and put into U2

. However, if

tail appears then 2 balls are drawn at random from U1 and put into U2.Now 1 ball is drawn at randomfrom

U2.

58. The probability of the drawn ball from U2 being white is


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(a) 13/30 (b) 23/30 (c) 19/ 30 (d) 11/30

58. (b)

3 2 3 2

1 1 2 2

5 5 5

2 2 2

For appearing head For appearing tail

1 3 2 1 1 2 1 C C C C

P 1 + 1 23 / 30

2 5 5 2 2 3 3 C C C

) * ) *

59. Given that the drawn ball from U2 is white, the probability that head appeared on the coin is

(a) 17/23 (b) 11/23 (c) 15/23 (d) 12/23

59. (d)

Using Bayes theorem

P =

3 2 3 2

1 1 2 2


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5 5 5

2 2 2

1 3 2 1

1

2 5 5 2

12 / 23

1 3 2 1 1 2 1 C C C C

1 1

2 5 5 2 2 3 3 C C C

) *

) * ) *

Paragraph for question Nos 60 and 62

Let a, b and c be three real numbers satisfying

1 9 7


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a b c 8 2 7 0 0 0

7 3 7

) *

(E)

60. If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a +

b + c is

(a) 0 (b) 12 (c) 7 (d) 6

60. (d) D =

1 8 7

9 2 3 0

1 1 1

system has non trivial solution.

If c = k, b = -6/7 k, a = -k/7

2a + b+ c = 1

2k / 7 6k / 7 k 1 k 726

Quest Tutorials


: P a,b,c 1,6, 7

7a b c 7 6 7 6

61. Let be a solution of


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62. Let b = 6, with a and c satisfying (E). If


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and 5 are the roots of the quadratic equation

2

ax bx c 0 , then

n

n 0

1 1

;


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5

< is

(a) 6 (b) 7 (c) 6/7 (d) ;

62. (b) b = 6, k = -7, c = - 7, a = 1

Q.E. becomes

2

x 6x 7 0; a + b = -6 , ab = -7

n n 2

n 0 n 0

6 6 6 1

1 ...... 7

7 7 7 6

1

7

; ;


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5


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2

f x x cx with c = 1.

2

f x x x . f 2 6

64. If z is any complex number satisfying z 3 2i 2 3 , then the minimum value of 2z 6 5i is

64. (5) Expression

5

2 Z 3 i

2

which is equivalent to double the distance between complex

number z and 3, 5 / 2 which lies on one the diameter x = 3 of circle given. Hence

minimum value of expression =

5

2 5

2

65. Let

1 2 100

a ,a ,.......a be an arithmetic progression with

1

a 3 and

p

p i

i 1

S a , 1 p 100


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5 6 5nd d

6 nd d

m

n

S

S

) *

independent of n when d = 6

2 1 a a d 928

Quest Tutorials


66. Consider the parabola

2

y 8x . Let 1

be the area of the triangle formed by the end points of its

latus rectum and the point P

1

,2

2


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on the parabola, and 2

be the area of the triangle formed by

drawing tangents at P and at the end points of the latus rectum. Then

1

2

is

66. (2) Extremities of L.R. are (2,4) & (2, -4)

1 6

Points of intersection of tangents at these points are (-2, 0), (1, 3) & (-1, -1)

2 3

67. Let f

1 sin

sin tan

cos 2

, where

4 4


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# #

! ! . Then the value of

d

f

d tan

is .

67. (1) Put k =

1 sin sin

tan tan k

cos 2 cos 2

2

2 2

2 2

sin 1

tan k sec k 1

1 2sin cos ec 2

2


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2 2 2

2 2

cos ec 1 1

sec k sin k tan

cos ec 2 cot

sin k tan f sin k tan Hence.

d f

1

d tan

68. The minimum value of the sum of real numbers

5 4 3 8

a ,a ,3a ,1,a

and

10

a with a > 0 is

68. (8) Using A.M 6 G.M

5 4 3 3 3 8 10

a a a a a 1 a a


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(d) III in haematite and II, III in magnetite

3. (d) Hamatite is Fe2O3 where iron is in III oxidation state.

Magentite is Fe3O4 which is a mixed oxide where iron exhibits both II and III oxidation

state.

4. Among the following complexes (K-P)

K Fe CN K , CO NH Cl L , Na Co oxalate M , Ni H O ]Cl N ,K Pt

3

6 6 3 3 3 3 6 4 2 2 2 CN O

and

2 3 6 2

Zn H O NO P

the diamagnetic complexes are

(a) K,L,M,N (b) K,M,O,P (c) L,M,O,P (d) L,M,N,O

4. (c) K has one unpaired e

N all octahedral complexes are outer orbital complex 2

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5. Passing H2S gas into a mixture of

2 2 2

Mn , Ni ,Cu

and

2

Hg

ions in an acidified aqueous

solution precipitates

(a) CuS and HgS (b) MnS and CuS

(c) MnS and NiS (d) NiS and HgS

5. (a) CuS and HgS get precipitated

6. Consider the following cell reaction:

2 o

2Fe O 4H 2Fe 2H O E 1.67V s 2 g aq aq 2

At

2 3

Fe 10 M, P O 0.1 atm 2


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and pH = 3, the cell potential at

o

25 C is

(a) 1.47 V (b) 1.77 V (c) 1.87 V (d) 1.57 V

6. (d)

4

2 2

2

2

2

O H

.06 .06

1.67 log Fe log 1.55V

2 4 H O

7. The freezing point (in

o

C ) of a solution containing 0.1g of

2

6


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K Fe CN Mol.Wt.329

in 100g

of water

1

K 1.86K kg mol

f

is

(a)

2

2.3 10

(b)

2

5.7 10

(c)

3

5.7 10

(d)


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2

1.2 10

7. (a)

2

T 1.86 0.1 20 4 / 329 2.3 10

f

8. Amongst the compound given, the one that would form a brilliant colored dye on treatment with

NaNO2 in dil. HCl followed by addition to an alkaline solution of


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naphthol is

8. (c) 3

Quest Tutorials


SECTION II (TOTAL MARKS: 16)

(MULTIPLE CORRECT ANSWER(S) TYPE)

This section contains 4 multiple choice questions. Each question has four choices (a),(b),(c) and (d) out of

which ONE OR MORE may be correct.

9. For the first reaction

2 5 2 2

2N O g 4NO g O g

(a) The concentration of the reactant decreases exponentially with time.

(b) The half-life of the reaction decreases with increasing temperature.

(c) The half-life of the reaction depends on the initial concentration of the reactant.

(d) The reaction proceeds to 99.6% completion in eight half-half duration.

9. (abc)

1/2 1/2 t

693 2.3 100

log

t 8 t A

t


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100

2.4 log

A

t

A 100 / 250 0.4

99.6%

10. The correct functional group X and the reagent / reaction conditions Y in the following scheme are

(a) X COOCH ,Y H / Ni / heat 3 2

(b) X CONH ,Y H / Ni / heat 2 2

(c) X CONH ,Y Br / NaOH 2 2

(d) X CN,Y H / Ni / heat 2

10. (abcd) Alcohol gives polyster

NH2 group gives polyamide

11. The equilibrium

I o II

2Cu Cu Cu


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In aqueous medium at

o

25 C shifts towards the left in the presence of

(a) NO3

(b) Cl

(c) SCN

(d) CN

11. (bcd) Cl ,SCN and CN

form insoluble compound of CuCl,CuSCN and CuCN which

drags the equilibrium reaction in backward direction.

12. Reduction of the metal centre in aqueous permanganate ion involves

(a) 3 electrons in the neutral medium (b) 5 electrons in neutral medium

(c) 3 electrons in alkaline medium (d) 5 electrons in acidic medium

12. (ad)4

Quest Tutorials




This section contains 6 questions. The answer to each of the questions is a single-digit integer, ranging


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101/163


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Cl 1.6 10 10 10

Cl 10

Ag 1.6 10 x 7

15. The number of hexagonal faces that are present in truncated octahedron is

15. (8)

16. The total number of contributing structures showing hyperconjugation (involving C-H bonds) for

the following carbocation is

16. (6) There are six hydrogens. 5

Quest Tutorials



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B, darken the bubbles corresponding to q and r in the ORS.

19. Match the transformations in Column I with appropriate options in Column II

Column I Column II

(a) CO s CO g

2 2 (p) Phase transition

(b) CaCO s CaO s CO g

3 2 (q) Allotropic change

(c)

2

2H H g

(r)H is positive

(d) (s)

(t)

S is positive

S is negetive

19. a (p,r,s) b (r,s) c (t) d (q,r)

20. Match the reactions in Column I with appropriate types of steps/ reactive intermediate involved in

these reactions as given in Column II.

Column- I Column- II

(a) (p) Nucleopphilic substitution

(b) (q) Electrophilic substitution

(c) (r) Dehydration

(d) (s)

(t)


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Nucleophilic addition

Carbanion

20. a (r,s, t) b (s,p) c (s,r) d q,r 7

Quest Tutorials


PHYSICS PAPER-II

PART II-PHYSICS


(SINGLE CORRECT CHOICE TYPE)

This Section contains 8 multiple choice questions. Each question ha four choice (a),(b),(c) and (d) out of


21. The density of a solid ball is to be determined in an experiment. The diameter of the ball is

measured with a screw gauge, whose pitch is 0.5mm and there are 50divisions on the circular

sacle. The reading on the main scale is 2.5mm and that on the circular scale is 20 divisions. If the

measured mass of the ball has relative error of 2%, the relative percentage error in the density is

(a) 0.9% (b) 2.4% (c) 3.1% (d) 4.2%

21. (c) LC = (0.5/50)=10

2

mm

Reading = 2.5 + 20

10

2

= 2.7 mm

3

m


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/ m / m 3. d / d

4 d

3 2

2

2 3 10 / 2.7 100 3.1%

22. A wooden block performs SHM on a frictionless surface with frequency, 0

. The block carries a

charge +Q on its surface. If now a uniform electric filed E

is switched-on as shown, the SHM of

the block will be:

(a) of the same frequency and with shifted mean position


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(b) of the same frequency and with the same mean position

(c) of changed frequency and with shifted mean position

(d) of changed frequency and with the same mean position.

22. (a) Restoring force remains same, so frequency does not change.

Only mean position shifted rightward by (EQ/K) 8

Quest Tutorials


23. A light ray travelling in glass medium is incident on glass-air interface at an angle of incidence .

The reflected (R) and transmitted (T) intensities, both as function of , are plotted. The correct

sketch is

23. (c) for

C

0 T R

&

C

- T 0,R 100%

Also 0 T 100%,R 0%

24. A satellite is moving with a constant speed V in a circular orbit about the earth. An object of

mass m is ejected from the satellite such that it just escapes from the gravitational pull of the

earth. At the time of its ejection, the kinetic energy of the object is

(a)

2

(1 / 2)mV (b)

2

mV (c)


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2

3 / 2 mV (d)

2

2mV

24. (b)

2 2 2

mV / r GmM / r r GM / V e e

Now using energy conservation for particle

2

e

1/ 2 Mu GM m / r 0

2 e

e

2

1 GM

mu m 0

2 GM

V

2 2 1 / 2 Mu mV K.E9


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Quest Tutorials


25. A long insulated copped wire is closely wound as a spiral of N turns. The spiral has inner radius

a and out radius b. The spiral lies in the X -Y plane and a steady current l flows through the

wire. The Z-component of the magnetic filed at the centre of the spiral is:

(a)

0Nl b

ln

2 b a a

(b)

0Nl b a

ln

2 b a b a

(c)

0Nl b

ln

2b a


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111/163

0Nl / 2 b a ln b / a

26. A point mass is subjected to two simultaneous sinusoidal displacement in x-direction,

1

x t A sin t ! and

2

x t A sin t 2 / 3 !

. Adding a third sinusoidal displacement

3

x t Bsin t ! " brings the mass to a complete rest. The values of B and " are

(a) 2 A, 3 / 4 (b) A, 4 / 3

(c) 3 A, 5 / 6 (d) A, / 3

26. (b)

1 2

x x A sin t A sin t 2 / 3 ! !

A 1 / 2 sin t 3 / 2 cos t ! !

! A sin t / 3

Now

1 2 3

x x x 0


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! ! " Asin t / 3 Bsin t

" B A & 4 / 310

Quest Tutorials


27. Which of the filed patterns given below is valid for electric field as well as for magnetic field?

27. (c) Electric field produced by the time varying magnetic field is circular.

For a long straight wire, the magnetic field lines are circles with their centres on the wire.

28. A ball of mass 0.2kg rests on a vertical post of height 5m. A bullet of mass 0.01kg, travelling with

a velocity

1

Vms

in a horizontal direction, hits the centre of the ball. After the collision, the ball

and bullet travel independently. The ball hits the ground at a distance of 20 m and the bullet ay a

distance of 100m from the foot of the post. The initial velocity V of the bullet is :

(a)

1

250ms

(b)

1

250 2 ms

(c)


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1

400ms

(d)

1

500ms

11

Quest Tutorials


28. (d) Time of flight after collision for both are

1

bullet

ball

t 2h / g 1sec

V 100 /1 100ms

V 20 /1 20ms 1

From conservation of Linear momentum

m u m v m v

bullet bullet bullet bullet ball ball


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, then the Gauss law will

still be valid

(b) The Gauss law can be used to calculate the field distribution around an electric dipole

(c) If the electric field between two point charges is zero somewhere, then the sign of the two

charges is the same.

(d) The work done by the external force in moving a unit positive charge from point A at potential

VA to point B at potential VB is V V B A

29. (cd) For point charge, flux for a sphere if point charge is placed at the centre

2

2.5

0

1

4 r

4 r

#

[According to question]

0 $ # q /

So Gauss law is not valid.

Electric field due to a dipole is not radially symmetric.

So Gauss law application does not give field distribution.

For electric field to be zero, directions of electric field must be opposite and equal in


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magnitude. This occurs only if sign of both the charges is same.

ext.agent elc.field B A

A B B A

W U U

W V V

q 1 q 1

30. A thin ring of mass 2kg and radius 0.5m is rolling without slipping on a horizontal plane with

velocity

1

1ms

. A small ball of mass 0.1kg moving with velocity

1

20ms

in the opposite direction,

hits the ring at a height of 0.75m and goes vertically up with velocity

1

10ms

. Immediately after


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the collision.

(a) the ring has pure rotation about its stationary CM

(b) the ring comes to a complete stop

(c) friction between the ring and the ground is to the left

(d) there is no friction between the ring and the ground 13

Quest Tutorials


30. (ac) Applying CLM horizontal direction:

CoM ring

0.1 20 2 1 2V

V 0

CoM

Using CAM about contact point:

From given data, initial angular momentum is anticlockwise.

Lring finally

is anticlockwise

Hence friction on the ring acts leftward to oppose rotation.

31. A series R-C circuit is connected to AC voltage source. Consider two cases; (a) when C is without

a dielectric medium and b when C is filled with dielectric of constant 4. The current

R

I of the


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following is/are true ?

(a)

A B

R R

I I (b)

A B

R R

I I (c)

A B

V V C C

(d)

A B

V V C C

31. (bc) Z for 1

st

case,

2

2

A

1

Z R

C


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!

Z for 2

nd

case,

2

B 2

1

Z R

k c

!

So

B A

Z Z So, I I

B A R R

Since, V is same

source

V increases for B

Resistance

V decreases for B

Capacitor

32. Two solid spheres A and B of equal volumes but of different densities

A B

d and d are connected by


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a string. They are fully immersed in a fluid of density dF. They get arranged into an equilibrium

state as shown in the figure with a tension in the string. The arrangement is possible only if

(a)

A F

d d (b)

B F

d d (c)

A F

d d (d)

A B F

d d 2d

32. (abd) Considering the F.B.D

T = Vg d d

F A and T Vg d d B F

For both conditions, T > 0

F A d d and

B F

d d

Equating tension,

F B A

2d d d 14

Quest Tutorials


SECTION-III (TOTAL MARKS 24)



121/163


122/163

train f f

o

S ut 10cos 60 t S S 1.15

ball f f ball train

2

3u 5 3 3u 3 / 2 a 1.15 a 5 ms

34. Two batteries of different emfs and different internal resistances are connected as shown. The

voltage across AB in volts is

34. (5) If V is p.d. across A & B , i1 is current through 1% & i2 is current through 2% , then

V 6 i 3 2i

1 2

..(i)

1 2 i V 6 & i V 3 / 2

Also

1 2

i i 0 ..(ii)

V = 5V 15

Quest Tutorials


35. Water (with refractive index = (4/3)) in a tank is 18 cm deep. Oil of refractive index (7/4) lies on

water making a convex surface of radius of curvature R = 6cm as shown. Consider oil to act as

thin lens. An object S is place 24 cm above water surface. The location of its image is at x cm


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max incident

& Maximum potential acquired (when maximum no. of photoelectrons emitted)

V K ne / r

max

S / unit & KE eV

max

max

7

n 15 /14.4 10

37. A series R C combination is connected to an AC voltage of angular frequency

1

500 radian s

! . If the impedance of the R C circuit is R 1.25 , the time constant (in

millisecond) of the circuit is

37. (4)

2

2 2 2 2 2

Z R X R 1.25 R 1 / C C

!

2 2


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R / 4 1 / 500C

2 2

RC 4 / 500

RC 1 / 250 s 4 ms16

Quest Tutorials


38. A block of mass 0.18 kg is attached to a spring of force constant 2 Nm

-1

. The coefficient of

friction between the block and the floor is 0.1. Initially the block is at rest and the spring is un

stretched. An impulse is given to the block as shown in the figure. The block slides a distance of

0.06 m and comes to rest for the first time. The initial velocity of the block in ms

-1

is V = N/10.

Then N is

38. (4)

2 2

1 / 2 mV 1 / 2 Kx fx

2

2 2 2 2 1 2

1/ 2 18 10 V 1 / 2 2 6 10 18 10 10 10 ) 6 10


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2 V 144 / 900 N /10 12 / 30 N 417

Quest Tutorials



(MATRIX MATCH TYPE)

This section contains 2 questions. Each question four statements (A, B, C and D) given in Column I and

five statements (p, q, r, s and t) in Column II. Any given statement in Column I can have correct matching

with ONE or MORE statement(s) given in Column II. For example, if for a given question, statement B

matches with the statements given in q and r, then for the particular question, against statement B,darken

the bubbles corresponding to q and r in the ORS.

39. One mole of a monatomic ideal gas is taken through a cycle ABCDA as shown in the P V

diagram. Column II gives the characteristics involved in the cycle. Match them with each of the

processes given in Column I.

Column I Column II

(a) Process A B (p) Internal energy decreases


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128/163

Q W U;

Q ve 0 ve

r, t is correct 18

Quest Tutorials


40. Column I shows four system, each of the same length L, for producing standing waves. The

lowest possible natural frequency of a system is called its fundamental frequency, whose

wavelength is denoted as

1

) . Match each system with statements given in Column II describing

the nature and wavelength of the standing waves.

Column I Column II

(A) Pipe closed at one end (p) Longitudinal waves

(B) Pipe open at both ends (q) Transverse waves

(C) Stretched wire clamped at

both ends

(r) )

f L

(D) Stretched wire clamped at

both ends and at mid point

(s)

f ) 2L

(t)

f ) 4L

40. A (p,t), B (p,s), C (q,s), D (q,r)


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A Longitudinal standing wave also,

f f

) ) / 4 L 4L p, t is correct

B Longitudinal standing wave also,

f f

) ) / 2 L 2L p,s is correct

C Longitudinal standing wave also,

f f ) ) / 2 L 2L q,s is correct

D Longitudinal standing wave also,

f f f

) ) ) / 2 / 2 L L q, r19

Quest Tutorials


MATHEMATICS PAPER II

PART III MATHEMATICS


(SINGLE CORRECT ANSWER TYPE)



41. Let f(x) = x

2

and g(x) = sin x for all x *R. Then the set of all x satisfying ( f o g o g o g ) (x) = (g

o g o f) (x), where (f o g) (x) = f (g (x)) , is

(a) + n , n * 0,1,2,....... (b) + n , n* 1,2,........

(c) 2 2n , n * ......, 2, 1,0,1,2........ (d) 2n , n ...., 2, 1,0,1,2...... *


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41. (a) (f o g o g o f ) (x) = ( g o g o f) (x)

2

2 2

sin sin x sin sin x

2 sin sin x 0 or 1

+ * x n , n 0,1,2......

42. Let (x, y) be any point on the parabola

2

y 4x . Let P be the point that divides the line segment

from (0, 0) to (x, y) in the ratio 1 : 3. Then the locus of P is

(a)

2

x y (b)

2

y 2x (c)

2

y x (d)

2

x 2y

42. (c) y = 2 x

Let P be (h, k) x = 4h, x 2k

2 k h


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132/163

44. A value of b for which the equations

2

x bx 1 0

2

x x b 0

have one root in common is

(a) - 2 (b) -i 3 (c) i 5 (d) 2

45. (b) be the common root

2

b 1 0

2

b 0

2

2 b 1

1 b

,

1 b

b 1


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3 2

b 3b 0 b 3b i 3

45. Let ! $1 be a cube root of unity and S be the set of all non singular matrices of the form

2

1 a b

1 c

1

, -

!

! !

, where each of a, b and c is either ! or

2

! . Then the number of distinct matrices in

the set S is

(a) 2 (b) 6 (c) 4 (d) 8

45. (a)

2

1 a b

1 c 0

1

! $

! !

2 ! ! $ 1 a c ac 0a c 1 $ ac 1 $


134/163

Clearly a = ! or

2

! or c =

2

! ! or

but

2

c $ ! as it makes the determinant zero.

also

2

a $ !

2

! ! a , b or ! ! ,c

No of such matrices = 2

46. The circle passing through the point (-1, 0) and touching the y axis at (0, 2) also passes through

the point

(a) (-3/2, 0) (b) (-5/2,2) (c) (-3/2, 5/2) (d) (-4, 0)

46. (d) Let C be (h, 2)

Equation of circle:

circle passes through (-1, 0) , h = -5/2

Equation of circle will be (x + (5/2))

2

+ (y 2)

2


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= 25/4

Only (-4, 0) satisfies

47. If

1 x

2 2

x 0

Lim 1 x ln 1 b 2bsin

, b > 0 and * , then the value of is

(a) + 4 (b) + 3 (c) + / 6 (d) + / 2

47. (d)

2 2 2 2

1 b 2bsin b 2bsin 1 0

2 4

2sin 4sin 4

b

2

+

=

2 2 2

sin cos 1 sin +

For b to be real, cos 0 + 221


136/163

Quest Tutorials


48. Let f: . 1,2 0, be a continuous function such that f(x) = f( 1 x) for all x * 1,2 . Let

2

1

1

R xf x dx

, and R2 be the area of the region bounded by y = f(x), x = -1, x = 2, and the x

axis. Then

(a) R 2R 1 2 (b) R 3R 1 2 (c)

1 2

2R R (d)

1 2

3R R

48. (c)

2

1 2 1 1 2

1

R 1 x f 1 x dx R R 2R R

22


137/163

Quest Tutorials



MULTIPLE CORRECT CHOICE TYPE


which ONE or MORE may be correct.

49. Let L be a normal to the parabola

2

y 4x . If L passes through the point (9, 6), the L is given by

(a) y x + 3 = 0 (b) y + 3x 33 = 0

(c) y + x 15 (d) y 2x + 12 = 0

49. (a,b,d) Equation of normal y = mx 2am am

3

i.e y = mx 2m m

3

Passing through (9, 6)

3

m 7m 6 0 m 3,2,1

Equation of normals are y + 3x 33 = 0, y 2x 12 0 , y x 3 0

50. Let f: 0,1 R be defined by f(x) =

b x

1 bx


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139/163

f ' x 0 x 0,1 as b 0,1

1 bx

/ * *

Also, f(x) is continuous for all x 0,1 * as domain of definition is R 1 / b where 1/b

> 1. Hence, f(x) is strictly decreasing, so invertible.

b x y b b x 1

f x y x f x f x

1 bx yb 1 1 bx

2

1

f ' b

b 1

and

2

f ' 0 b 1

Since , f ' x exist for all x * 0,1

1

f


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141/163


142/163

1

2

2

4

3

2

5

Clearly, f(x) is not differentiable at x = 0, f(x) is differentiable at x = 1 and f(x) is

differentiable x = -3/2 24

Quest Tutorials


SECTION III (TOTAL MARKS: 24)


This section contains 6 questions. The answer to each of the questions is a single digit integer, ranging

from 0 to 9. The bubble corresponding to the correct answer is to be darkened in the ORS.

53. Let

i

3

e

! , and a, b, c, x, y, z be non zero complex numbers such that

2

2

a b c X


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a b c y

a b c z

! !

! !

Then the value of

2 2 2

2 2 2

x y z

a b c

is

53. (3)

2 2 2

c c c

x y z xx yy zz

= (a + b+ c)

c c c 2 c c 2 c 2 c c c 2

a b c a b c a b c a b c a b c ! ! ! ! ! ! ! !

=

2 2 2

2 2 2

2 2 2

x y z


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3 a b c 3

a b c

54. Let y ' x y x g ' x g x g ' x , y 0 0, x R * , where f(x) denotes

df x

dx

and g(x) is a

given non constant differentiable function on R with g(0) = g(2) = 0. Then the value of y(2) is

54. (0) y ' x y x g ' x g x g ' x

Linear differential equation with integrating factor

g x

e

g x g x g x g x

' ' ' ' y x e g x g ' x e dx y x e e g x 1 c

Since, y 0 0 and g 0 0 c = 1

g x g 2

y x g x 1 e y 2 g 2 1 e 0

55. Let M be a 3 3

matrix satisfying

0 1


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M 1 2 ,

0 3

, - , -

1 1

M 1 1

0 1

, - , -

and

1 1

M 1 1

1 12

, - , -


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147/163

M 1 1

0 1

1 2 4 5 7 8

x x 1, x x 1, x x 1

1 4 7 x 0, x 3, x 2

1 2 3 4 5 6 7 8 9

1 0

M 1 M 0 x x x 0; x x x 0, x x x 12

1 12

3 6 9 x 1, x 5, x 7

0 1 1

M 3 2 5

2 3 7


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Sum of diagonal entries = 9

56. The straight line 2x 3y = 1 divides the circular region

2 2

x y 6 2 into two parts. If

3 5 3 1 1 1 1

S 2, , , , , , , *

4 2 4 4 4 8 4

1 6

4 7

5 8

Then the number of point (s) in S lying inside the smaller part is

56. (2) 2x 3y = 1 meets co ordinate axes at

1 1 5 3

,0 & 0, ,

2 3 2 4

'

lies outside the circle

hence ruled out.

For,

3

x 2, y


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4

: 2x 3y 1 0

For,

1 1

x , y

4 4

: 2x 3y 1 0

For,

1 1

x , y

8 4

: 2x 3y 1 < 0

3 1 1

2, & ,

4 4 4

are the only two points

57. Let

a i k, b = -i + j


150/163

and

c i 2 j 3k

be three given vectors. If r

is a vector such that

r b c b

and r a 0 '

, then the value of r b 0 '

is 26

Quest Tutorials


57. (9) r c b 0

) ) ) r c b 1 j 2 j 3k


151/163

If

r xi yj zk

, then r.a 0 x z 0 x z

Also, x 1 , y = 2 + , z = 3 -3 = 1- = 4 ) ) ) )

r 3j 6 j 3k

' r b 3 6 9

58. The number of distinct real roots of

4 3 2

x 4x 12x x 1 0 is

58. (2) Let us assume that all four roots are real and distinct

Hence, f(x) = 0 must have 3 distinct real roots and

f " x 0 must have 2 distinct real roots, but that is not true as

2

f x 12 x 2x 2 99 with D < 0

Hence, f(x) = 0 cant have all four roots real

As, f(0) = -1, f(1) =9 and f(-1) = 15


152/163

f (x) 0 must have two distinct roots one in (-1, 0) and the other one in (0, 1) 27

Quest Tutorials



(MULTIPLE CORRECT ANSWERS TYPE)

This section contains 2 questions. Each question has four statements (A, B, C and D) given in Column I

and five statements (p, q , r, s and t ) in Column II. Any given statement in Column I can have correct

matching with ONE and MORE statement(s) given in Column II. For example, if for a given question,

Statement B matches with the statements given in q and r, then for the particular question, againststatement

B, darken the bubbles corresponding to q and r in the ORS.

59. Match the statements given in Column I with the values given in Column II

Column I Column II

(A)

If

a j 3k,b j 3k

and

c 2 3k

from a triangle, then the

internal angle of the triangle

between a


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and b

be

(p)

6

(B)

If

b

2 2

a

f x 3x dx a b

, then

the value of f

6

is

(q) 2

3

(C)


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The value of

5

2 6

7

6

sec x dx

ln 3

is

(r)

3

(D) The maximum value of

1

Arg

1 z

for z 1, z 1 $ is

given by

(r)

(t)


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2

59. A q , B p , C s , D t

For (A): Since, side lengths are 2, 2, 2 3 , hence angle between a

and b

is

4 4 12

cos 1/ 2 2 / 3

2 2 2

' '

For (B) :

b

2 2

a

1

f x dx b a f x x f / 6 / 6

2

For (C) :

5/6


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7 /6

1

ln sec x tan x ln 3 ln

ln 3 ln 3 3

, -

For (D) :

i

2 2

1 1 cos isin

z e ,

1 z 1 cos sin

1 sin

arg f

1 z 1 cos


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, which is maximum when 228

Quest Tutorials


60. Match the statements given in Column I with the intervals/ union of intervals given in Column II

Column I Column II

(A) The set

2

2iz

Re : z is complex number, z 1, z 1

1 z

1 6

4 $ + 7

5 8

(p) . : . , 1 1,

(B) The domain of the function

x 2

1

2 x 1

8 3

f x sin is

1 3


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(q) . : . ,0 0,

(C)

If

1 tan 1

f tan 1 tan

1 tan 1

, then the set

f : 0

2

1 6

4 7 2

5 8

is

(r) 2,.

(D)

If

3/2

f x x 3x 10 , x 0 - , then f(x) is

increasing in


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(s) . : . , 1 1,

(t) . : . ,0 2,

60. A s Let k =

i

2 2i 2

2iz 2ie 2i cos isin

Re Re Re

1 z 1 e 2sin 2isin cos

=

2i cos isin 1

Re

2isin cos isin sin

It is defined only when k * . : . , 1 1,

B t Since,

x 2

2x 2


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8 3

1 1

1 3

'

2 2

x

2 x

8.3

1 1

9 3

2 2

Put

x

y 3

2

8y

1 1

9 y

2 2

2


161/163

8y y 9 y 1

1 0

9 y y 3 y 3

2 -

y 9

0

y 3

-

( y + 1 and y + 3 are always (+) ve)

y 3 and y 9 -* . : . x ,1 [2, )

and

2

8y y 9 y 1

1 0

9 y y 3 y 3

2 -

y 1


162/163

0

y 3

-

2 y 1 and y 3 29

Quest Tutorials


* . : . x ( ,0] 2,

The common solution is x * . : . ( ,0] 2, .

C r

Since,

2

1 tan 1

f tan 1 tan 2sec

1 tan 1

* . f [2, )

D r Let f(x) =

3/2

x 3x 10

1/2 3/2


163/163

f ' x 3 / 2 x 3x 10 x 3

For increasing,

f ' x 0 -

- 3 x 3x / 2 5 x 0

' - - 3 x 5x 10 0 x 2

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