Iitjee 2012 Paper i Sol

7/30/2019 Iitjee 2012 Paper i Sol

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Part I : PhysicsSection - I Single Correct Answer Type

This section contains 10 multiple choice questions. Each question has four choice (A), (B), (C) and

(D) out of which ONLY ONE is correct.

Q.1 A bi-convex lens is formed with two thin plano-convex lenses as shown

in the figure. Refractive index n of the first lens is 1.5 and that of the

second lens is 1.2 Both the curved surfaces are of the same radius ofcurvature R = 14 cm. For this bi-convex lens, for an object distance of

40 cm, the image distance will be

n = 1.2n = 1.5

R = 14 cm(A) 280.0 cm (B) 40.0 cm

(C) 21.5 cm (D) 13.3 cm

Ans. [B]

[Sol.1

f

1= (1.2 1)

14

11=

14

2.0

2f

1= (1.5 1)

1

14

1=

14

5.0

eqf

1=

20

1

14

7.0

20

1

40

1

v

1

v = 40 cm ]

Q.2 A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed , asshown in the figure. At time t = 0, a small insect starts from O and moves with constant speed v with

respect to the rod towards the other end. If reaches the end of the rod at t = T and stops. The angular

speed of the system remains throughout. The magnitude of the torque |)(|

on the system about O, as

a function of time is best represented by which plot ?

v

O

z

(A)0

Tt

| |

(B)0

Tt

| |

(C)

0T

t

| |

(D)

0T

t

| |

Ans. [B]


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[Sol. =dt

dL=

dt

d(I + mx2) = m 2x

dt

dx = 2 mv2 t [as x = vt] ]

Q.3 Three very large plates of same area are kept parallel and close to each other. They are considered as

ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at

temperatures 2T and 3T respectively. The temperature of the middle (i.e. second) plate under steady

state condition is

(A) T2

654/1

(B) T4

974/1

(C) T2

974/1

(D) T97

4/1

Ans. [C]

[Sol.dt

dQ= A 4

1

4 T)T3(

= A 441

)T2(T

2T 3T

T1

T1

= T2

974/1

]

Q.4 Consider a thin spherical shell of radius R with its centre at the origin, carrying uniform positive surface

charge density. The variation of the magnitude of the electric field |)r(E|

and the electric potential V(r)

with the distance r from the centre, is best represented by which graph ?

(A)

R

| E(r) |

V(r)

r0

(B)

R

| E(r) |

V(r)

r0

(C)

R

| E(r) |

V(r)

r0

(D)

R

| E(r) |

V(r)

r0

Ans. [D]

Q.5 In the determination of Youngs modulus

2d

MLg4Y

lby using Searles method, a wire of length L =

2m and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension l = 0.25 mm in the length ofthe wire is observed. Quantities d and l are measured using a screw gauge and a micrometer, respectively.

They have the same pitch of 0.5 mm. The number of divisions on their circular scale is 100. The contributions

to the maximum probable error of the Y measurement

(A) due to the errors in the measurements of d and l are the same

(B) due to the error in the measurement of d is twice that due to the error in the measurement ofl.

(C) due to the error in the measurement ofl is twice that due to the error in the measurement of d.

(D) due to the error in the measurement of d is four times that due to the error in the measurement ofl.

Ans. [A]


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[Sol.y

y=

L

L+

l

l+

d

d2

l = 0.25, d = 0.5 ]

Q.6 A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end

of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is

stretched by 0.2 m and released from rest at t = 0. It then executes simple harmonic motion with angular

frequency = 3rad/s. Simultaneously at t = 0, a small pebble is projected with speed v from point P

at an angle of 45 as shown in the figure. Point P is at a horizontal distance of 10 m from O. If the pebble

hits the block at t = 1s, the value of v is (take g = 10 m/s2)

v

45x

10 mO

z

P

(A)50

m/s (B)51

m/s (C) 52 m/s (D)53

m/s

Ans. [A]

[Sol. T =g

45sinv2 ]

Q.7 Youngs double slit experiment is carried out by using green, red and blue light, one color at a time. The

fringe widths recorded are G,

Rand

B, respectively. Then,

(A) G

> B

> R

(B) B

> G

> R

(C) R

> B

> G

(D) R

> G

> B

Ans. [D]

[Sol. V I B G Y O R ( increases)

=dD ]

Q.8 A small mass m is attached to a massless string whose other end is fixed at P as

shown in the figure. The mass is undergoing circular motion in the x-y plane with

centre at O and constant angular speed . If the angular momentum of the

system, calculated about O and P are denoted by0

L

andP

L

respectively,,

then Om

z

P

(A)0

L

andP

L

do not vary with time.

(B)0

L

varies with time whileP

L

remains constant

(C)0

L

remains constant whileP

L

varies with time.

(D)0

L

andP

L

both vary with time

Ans. [C]

[Sol. About point P, angular momentum changes in direction, but about O angular momentum does not change

either in direction or in magnitude. ]


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Q.9 A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass = 40

amu) is kept at 300 K in a container. The ratio of the r.m.s. speeds

)on(argv

)helium(v

rms

rmsis

(A) 0.32 (B) 0.45 (C) 2.24 (D) 3.16

Ans. [D]

[Sol. vrms

=

M

RT3

Ratio =He

ar

M

M= 10 ]

Q.10 Two large vertical and parallel metal plates having a separation of 1 cm are connected to a DC voltage

source of potential difference X. A proton is released at rest midway between the two plates. It is found

to move at 45 to the vertical JUST after release. Then X is nearly

(A) 1 105 V (B) 1 107 V (C) 1 109 V (D) 1 1010 V

Ans. [C]

[Sol.d

qV = mg

mg

qE

V =q

mgd= 19

227

106.1

1010106.1

]

Section - II : Multiple Correct Answer(s) Type

This section contains 5 multiple choice questions. Each question has four choice (A), (B), (C) and (D)

out of which ONE or MORE may be correct.

Q.11 A cubical region of side a has its centre at the origin. It encloses three fixed point charges, q at

0,4

a,0 , + 3q at (0, 0, 0) and q at

0,4

a,0 . Choose the correct option(s).

y

x

z

a

q

3q

q

(A) The net electric flux crossing the plane2

ax is equal to the net electric flux crossing the plane

2

ax


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(B) The net electric flux crossing the plane2

ay is more than the net electric flux crossing the plane

2

ay

(C) The net electric flux crossing the entire region is0

q

(D) The net electric flux crossing the plane2az is equal to the net electric flux crossing the plane

2

ax

Ans. [A,C,D]

[Sol. = charge enclosed /0

A and D because of symmetry. ]

Q.12 For the resistance network shown in the figure, choose the correct option(s)

I1444

22

2S

TQ

P I2

12V

1 1(A) the current through PQ is zero

(B) I1

= 3A

(C) The potential at S is less than that at Q

(D) I2

= 2A

Ans. [A,B,C,D]

[Sol. By input output symmetry, the current in each of the 2 resistance is the same. Similarly, the current ineach of the 4 resistance is the same. So the current in each of the 1 resistance is zero.

So, I2

=6

12= 2A

I4

=12

12= 1 AA

I1 = I2 + I4 = 3ATaking potential at negative terminal of the battery to be zero, Potential at S = 4 Volt,

Potential at Q = 8 Volt ]

Q.13 A small block of mass of 0.1 kg lies on a fixed inclined plane PQ which makes an angle with thehorizontal. A horizontal force of 1N acts on the block through its centre of mass as shown in the figure.

The block remains stationary if (take g = 10 m/s2)

Q

1N

O P

(A) = 45(B) > 45 and a frictional force acts on the block towards P(C) > 45 and a frictional force acts on the block towards Q(D) < 45 and a frictional force acts on the block towards Q

Ans. [A,C]

[Sol. Fnet

= 1 cos 1 sin ]


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Q.14 Consider the motion of a positive point charge in a region where are simultaneous uniform electric and

magnetic field jEE0

and jBB0

. At time t = 0, this charge has velocity v

in the x-y plane, making

an angle with the x-axis. Which of the following option(s) is(are) correct for time t > 0?(A) If = 0, the charge moves in a circular path in the x-z plane(B) If = 0, the charge undergoes helical motion with constant pitch along the y-axis(C) If = 10, the charge undergoes helical motion with its pitch increasing with time, along the y-axis(D) If = 90, the charge undergoes linear but accelerated motion along the y-axis

Ans. [C,D]

[Sol.

E0

B0

V

]

Q.15 A person blows into open-end of a long pipe. As a result, a high-pressure pulse of air travels down the

pipe. When this pulse reaches the other end of the pipe.

(A) a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is open

(B) a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is open

(C) a low-pressure pulse starts travelling up the pipe, if the other end of the pipe is closed

(D) a high-pressure pulse starts travelling up the pipe, if the other end of the pipe is closedAns. [B,D]

[Sol. Phase change of from denser medium and no phase charge from a rarer medium. For sound, the closeend of the pipe is a rarer medium and open end is a denser medium. ]

Section - III : Integer Answer Type

This section contains 5 questions. The answer to each question is a single digit integer, ranging from

0 to 9 (both inclusive).

Q.16 An infinitely long solid cylinder of radius R has a uniform volume charge density. It has a spherical

cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of theelectric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the

expression0

k16

R23

. The value of k is

P

y

R/2

z

x

R

Ans. 6


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[Sol. E1 2 (2R) l =

0

2

R l

01 4

RE

E2 4 (2R)2 =

0

3

2

R

3

4

02 424

RE

0021 244

R23

24

11

4

REE

k = 6 ]

Q.17 A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a shown in the figure. Both the

cylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the

magnitude of the magnetic field at the point P is given byaJ

12

N

0

, then the value of N is

aP

2aAns. 5

[Sol.

2

a32

2

a

J

2

JaB

2

00

P

12

Ja5

6

11

2

Ja00

N = 5 ]

Q.18 A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and

radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and

P is IO

and IP, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio

O

P

I

I

to the nearest integer is

O2R

2R

P

Ans. 3


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[Sol. I0

=2

1(2R)2(2R)2

2222 RRRR2

1=

44R

2

13

2

38R

IP

=2

3(2R)2 (2R)2

2222 R5RRR2

1

2

R37

2

1124R

44

3~I

I

O

P]

Q.19 A circular wire loop of radius R is placed in the x-y plane centered at the origin O. A square loop of side

a (a


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PART II: CHEMISTRY

SECTION -I : Single Correct Answer TypeThis section contains 10 multiple choice questions. Each question has four choices(A), (B), (C) and (D) out of which ONLY ONE is correct.

Q.21 As per IUPAC nomenclature, the name of the complex [Co(H2O)

4(NH

3)2]Cl

3is

(A) Tetraaquadiaminecobalt (III) chloride

(B) Tetraaquadiamminecobalt (III) chloride

(C) Diaminetetraaquacoblat (III) chloride(D) Diamminetetraaquacobalt (III) chloride

Ans. (D)

Sol. H2Oaqua

NH3ammine

Q.22 In allene (C3H

4), the type(s) of hybridisation of the carbon atoms is (are)

(A) sp and sp3 (B) sp and sp2 (C) only sp2 (D) sp2 and sp3

Ans. (B)

Sol.C = C = C

H

H

H

Hsp

2sp sp

2

Q.23 For one mole of a Vander waals gas when b = 0 and T = 300 K, the PV vs 1/V plot is shown below.

The value of the Vander waal's constant a (atm. litre2mol2) is

(Graph not to scale)

24.6

23.1

21.6

20.1

02.0 3.0

1V

(mol liter )1

PV

(liter-atmm

ol

1)

(A) 1.0 (B) 4.5 (C) 1.5 (D) 3.0

Ans. (C)

Sol. RTVV

aP

2

V

aRTPV

Slope =a = 5.12

3

02

6.246.21

a = 1.5


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Q.24 The number of optically active products obtained from the complete ozonolysis of the given

compound is

CH CH=CHCCH=CHCCH=CHCH3 3

CH3

CH3H

H

(A) 0 (B) 1 (C) 2 (D) 4

Ans. (A)

Sol. CH CH=CHCCH=CHCCH=CHCH3 3

CH3

CH3H

H

Zn

O3 2CH3CH = O +

CH3

CH3H

H

OHC CCHO + OHC C CHO

All are optically inactive

Q.25 A compound MpX

qhas cubic close packing (ccp) arrangement of X. Its unit cell structure is shown

below. The empirical formula of the compound is

M=

X =

(A) MX (B) MX2

(C) M2X (D) M

5X

14

Ans. (B)

Sol. Face centre = = 4For M,

edge centre = 144

1

centre = 1

M2X

4

i.e. MX2

Q.26 The number of aldol reaction (s) that occurs in the given transformation is

CH2CHO + 4HCHO

NaOH.aq.conc

OHOH

OHHO

(A) 1 (B) 2 (C) 3 (D) 4

Ans. (C)


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Sol. CH3CHO + 4HCHO conc. OH

OHOH

OHHO

OO||||

HCHCHCH3 )1(aldol

HO

OH|

HCCHCH||

O

22 )2(Aldol

OH/HCHO CHCH

HOCH2

HOH2

C

O

HCHO/OH

Aldol(3)

OHOH

OHHO

OHcannizaro

C CH

O

OH

HO

OH

Number of aldol reaction = 3

Q.27 The colour of light absorbed by an aqueous solution of CuSO4

is

(A) Orange-red (B) Blue-green (C) Yellow (D) VioletAns. (A)

Sol. aqueos solution of CuSO4

is blue so absorbed colour is orange-red

VI

BGY

R

O

OR

Q.28 The carboxyl functional group(COOH) is present in

(A) picric acid (B) barbituric acid (C) ascorbic acid (D) aspirinAns. (D)

Sol.

OCCH3

COOH

O

Aspirin

Q.29 The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a0

is Bohr radius]

(A) 20

2

2

ma4

h

(B) 202

2

ma16

h

(C) 202

2

ma32

h

(D) 202

2

ma64

h

Ans. (C)

Sol. mvr = nh/2p

v =a4m2

h2

z

nr

2

4

1

x

a


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Q.33 Which of the following molecules, in pure from, is (are) unstable at room temperature ?

(A) (B) (C) (D)

Ans. (BC)

Sol. Anti aromatic are unstable at room temperature

and are anti-aromatic with 4 electron system.

While is non-aromatic and is aromatic with 6.

Q.34 Which of the following hydrogen halides react(s) with AgNO3(aq) to give a precipitate that dissolves in

Na2S

2O

3(aq) ?

(A) HCl (B) HF (C) HBr (D) HIAns. (ACD)

Sol. AgF is soluble in water

AgNO3

+ HX (X = Cl, Br, I) AgX 322OSNa

[Ag(S2O

3)2]3 (soluble)

Q.35 For an ideal gas, consider only P-V work in going from an initial state X to the final state Z. The final state

Z can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are)

correct ? [take S as change in entropy and w as work done]

(A) Sx z

= Sx y

+ Sy z

(B) Wx z

= Wx y

+ Wy z

(C) Wx y z

= Wx y

(D) Sx y

z

= Sx y

Ans. (AC)


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SECTION -III: Integer Answer TypeThis section contains 5 questions. The answer to each question is a single-digit integer,

ranging from 0 to 9 (both inclusive)

Q.36 The substituents R1

and R2

for nine peptides are listed in the table given below. How many of these

peptides are positively charged at pH = 7.0 ?

HRRH

OCOHCNHCOHCNHCOHCNHCOHCNH

21

||||3

3242

2422

222

242242

2222

24222

2

3

CHNH)CH(IX

NH)CH(OHCHVIII

CONHCHCOOHCHVII

NH)CH(NH)CH(VI

CONHCHCONHCHV

NH)CH(CONHCHIV

HCOOHCHIII

CHHII

HHI21

RRPeptide

Ans. 4

Sol. Any amino acid will exist in a cationic form in solution of pH < pI (isoelectric point)

pH = 7 is less than pI given which implies pI > 7 basic amino acid.

Given peptide will have basic nature if number of basic sites are greater than acidic sites.

Peptide IV R1

= CH2CONH

2R

2= (CH

2)4NH

2

Peptide VI R1

= (CH2)4NH

2R

2= (CH

2)4NH

2

Peptide VIII R1

= CH2OH R

2= (CH

2)4NH

2

peptide IX R1

= (CH2)4NH

2R

2= CH

3

Ans. 4

Q.37 The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons, undergoes

a nuclear reaction yielding element X as shown below. To which group, element X belongs in the periodic

table ?

XH2n6HCu 11

1

0

1

1

63

29

Ans. 8

Sol. X52261

1

4

2

1

0

1

1

63

29 H2n6HCu

Ans. 8th group.

1 2 3 4 5 6 7 8

Kr Cr Sc Ti V Cr mn Fe


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[15]

Q.38 When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its

pyranose form is

OHCH|CHOH|

CHOH|CHOH

|

CH|CHO

2

2

Ans. 8

Sol.

OHCH|CHOH|CHOH|

CHOH|

CH|CHO

2

2

1

2

3

4

5

(=)

CHO

CH2

CHOH

CHOH

CH OH2

H OH

Given is D configuration for pyranose. Cyclization will happen by OH of fifth carbon.

CH

CH2

CH2OH

CHOH

CHOH

OH

H O

1*

3*

4*

5

2

Configuration at C-5 is D(glucose) this will not change. New chiral generates at C-1 and total numberof chiral center which may have D or L continue are 3. Hence total number of stereoisomers= 23 = 8

Ans. 8

Q.39 29.2% (w/w) HCl stock solution has a density of 1.25 g mL1. The molecular weight of HCl is 36.5 g

mol1. The volume (mL) of stock solution required to prepare a 200 mL solution of 0.4 M HCl is

Ans. 8

Sol. V 1.25 5.364.01000

200

100

2.29

V = ml


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Q.40 An organic compound undergoes first-order decomposition. The time taken for its decomposition to

1/8 and 1/10 of its initial concentration are t1/8

and t1/10

respectively. What is the value of

10]t[

]t[

10/1

8/1 ? [Take log10

2 = 0.3]

Ans. 9

Sol.xa

aln

K

1t

33K

1

8/1

1ln

K

1

8

1

1K

1

10/1

1ln

K

1

10

1


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[17]

PART III: MATHEMATICSSECTION -I : Single Correct Answer Type

This section contains 10 multiple choice questions. Each question hasfour choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Q.41 The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line

4x5y = 20 to the circle x2 + y2 = 9 is

(A) 20(x2 + y2)36x + 45y = 0 (B) 20(x2 + y2) + 36x45y = 0

(C) 36(x2 + y2)

20x + 45y = 0 (D) 36(x2 + y2) + 20x

45y = 0Ans. [A]

[Sol. Equation of chord of contact with mid-point of (h, k)

yk =k

h(xh) hx + ky = h2 + k2

Equation of chord of contact for point

5

204,

x + y

5

204= 9

h

=

5

204

k

= 9kh

22

4x5y = 20

(h, k) (0, 0)

x + y = 92 2

5

204,

= 22kh

h9

;

5

204 = 22

kh

k9

22 kh

h6

20 =

22 kh

k45

20 (h2 + k2)36h + 45k = 0 Locus is 20(x2 + y2)36x + 45y = 0. Ans.]

Q.42 The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that

each person gets atleast one ball is

(A) 75 (B) 150 (C) 210 (D) 243

Ans. [B]

[Sol. Number of ways =!2!3!1!1

!3!5

!2!2!2!1

!3!5

= 90 + 60 = 150. Ans.]

Q.43 Let f(x) = ,Rx,

0x,0

0x,x

cosx2

then f is

(A) differentiable both at x = 0 and at x = 2

(B) differentiable at x = 0 but not differentiable at x = 2

(C) not differentiable at x = 0 but differentiable at x = 2

(D) differentiable neither at x = 0 nor at x = 2

Ans. [B]


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[18]

[Sol. f(x) = ,Rx,

0x,0

0x,x

cosx2

At x = 0

LHD =h

)0(f)h0(fLim

0h

=

h

hcosh

Lim

2

0h

= 0

RHD =h

)0(f)h0(fLim

0h

=h

hcosh

Lim

2

0h

= 0

f(x) =

h2x2,x

cosx

2xh2,x

cosx

2

2

f '(x) =

h2x2,x

sinx

cosx2

2xh2,x

sinx

cosx2

LHD = ; RHD = f(x) is differentiable at x = 0 but not derivable at x = 2. Ans.]

Q.44 The function f : [0, 3] [1, 29], defined by f(x) = 2x315x2 + 36x + 1, is(A) one-one and onto. (B) onto but not one-one.

(C) one-one but not onto. (D) neither one-one nor onto.

Ans. [B][Sol. f(x) = 2x315x2 + 36x + 1

f '(x) = 6x230x + 36 = 6 (x 2) (x3)

Sign of f ' 0 2 3

+ +

sign of f ' changes in [0 3]

function is not one-one

O

(0, 1)

(3, 28)

(2, 29)

f(0) = 1 ; f(2) = 29 and f(3) = 28

Range = [1, 29] Function is onto but not one-one. Ans.]

Q.45 If

bax

1x

1xxLim

2

x= 4, then

(A) a = 1, b = 4 (B) a = 1, b =4 (C) a = 2, b = 3 (D) a = 2, b = 3

Ans. [B]


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[20]

[Sol. Let P =

333231

232221

131211

aaa

aaa

aaa

det.P =

333231

232221

131211

aaa

aaa

aaa

then det Q =

336

325

314

235

224

213

134

123

112

a2a2a2

a2a2a2

a2a2a2

= 222324

332

3231

232

2221

132

1211

a2a2a

a2a2a

a2a2a

= 29 2 22

333231

232221

131211

aaa

aaa

aaa

= 212 det P = 213. Ans.]

Q.49 The integral dx

xtanxsec

xsec29

2

equals (for some arbitrary constant K)

(A) Kxtanxsec71

11

1

xtanxsec

1 2211

(B)

Kxtanxsec7

1

11

1

xtanxsec

1 2211

(C)

Kxtanxsec7

1

11

1

xtanxsec

1 2211

(D)

Kxtanxsec7

1

11

1

xtanxsec

1 2211

Ans. [C][Sol. y = sec x + tan x

sec2xtan2x = 1

sec xtan x =y

1

2sec x = y +y

1 sec x =

y2

1y2

dy = (sec x tan x + sec2x) dx dy = y sec x dx dx =xsecy

dy

I =

29

2

y

xsecy

dyxsec

=

dyy

y2

1y

211

2

=

dyy

1y

2

1213

2

= dyyy21 21329

I =

211

y

27

y

2

121127

+ K


21/27

[21]

I = 21127 )xtanx(sec11

1

xtanxsec7

1

+ K

I =

7

xtanxsec

11

1

xtanxsec

12

211+ K. Ans.]

Q.50 The point P is the intersection of the straight line joining the points Q (2, 3, 5) and R (1,1, 4) with the

plane 5x

4y

z = 1. If S is the foot of the perpendicular drawn from the point T (2, 1, 4) to QR, thenthe length of the line segment PS is

(A)2

1(B) 2 (C) 2 (D) 22

Ans. [A]

[Sol. Line QR1

4z

4

1y

1

1x

= r

Let coordinates of point P are (r + 1, 4r1, r + 4) which lies on the plane

5x4yz = 1

5r + 54 (4r1)(r + 4) = 1

12r + 5 = 1 12r = 4 r =3

1

P

R (1,1, 4)

T (2, 1, 4)

Q (2, 3, 5)

S

P

3

13,

3

1,

3

4

Let coordinate of S are ( + 1, 41, + 4) ; ST QR ( + 12) 1 + (411) 4 + ( + 44) 1 = 0

1 + 16 + = 0 =2

Coordinate of S are

2

9,1,

2

3

Now, length PS =222

2

9

3

131

3

1

2

3

3

4

=36

1

9

4

36

1 =

2

1. Ans.]


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[22]

SECTION -III: Integer Answer TypeThis section contains 5 questions. The answer to each question is a single-digit integer,ranging from 0 to 9 (both inclusive)

Q.51 Let , [0, 2] be such that

2 cos (1sin ) = sin2

2cot

2tan cos 1,

tan(2) > 0 and 1 < sin e

1

e

1

e

11As

Again S = 1

21

x21

0

x dxedxe22

< 1

21

2121

0

dxedx1

S <

2

11

e

1

2

1

Also,

e

11

4

1

e

11 =

e4

1

e

1

4

3 =

e4

e4e3 > 0

S >

e

11

4

1. Ans.]

Q.53 A ship is fitted with three engines E1, E

2and E

3. The engines function independently of each other with

respective probabilities2

1,

4

1and

4

1. For the ship to be operational at least two of its engines must

function. Let X denote the event that the ship is operational and let X1, X

2and X

3denotes respectively

the events that the engines E1, E

2and E

3are functioning. Which of the following is(are) true?

(A) 16

3X|XP c

1 (B) P [Exactly two engines of the ship are functioning | X] =

8

7

(C) P [X | X2] =

16

5(D) P [X | X

1] =

16

7

Ans. [B, D]

[Sol. P(X1) =

2

; P(X

2) =

4

1and P(X

3) =

4

1

P(X) = P (atleast 2 of X1, X

2, X

3happens)

= P(X1 X

2) + P(X

2 X

3) + P(X

3 X

1)2P(X

1 X

2 X

3)


24/27

[24]

=4

1

4

1

2

12

2

1

4

1

4

1

4

1

4

1

2

1 =

4

1

(A)

X

XP

c1 =

)X(P

XXPc1

=

)X(P

XXXP23

c1

=

4

14

1

4

1

2

1

=8

1

(B)X

)happensX,X,Xof2Exactly(P321

=)X(P

X)XXXof2Exactly(P

1

321 =

4

14

1

4

1

2

1

4

1

=

8

7

(C)

2X

XP =

)X(P

)XX(P

2

2

=)X(P

)XXX(P)XXX(P)XXX(P

2

c3123

c12321

=

4

14

3

2

1

4

1

4

1

2

1

4

1

4

1

42

1

=8

5

(D)

1X

XP =

)X(P

)XX(P

1

1

=)X(P

)XXX(P)XXX(P)XXX(P

1

c3213

c21321

=

2

14

3

4

1

2

1

4

1

4

3

2

1

4

1

4

1

2

=16

7. Ans.]

Q.54 Tangents are drawn to the hyperbola 14y

9x

22

, parallel to the straight line 2xy = 1. The points

of contact of the tangent on the hyperbola are

(A)

2

1,

22

9(B)

2

1,

22

9(C) 22,33 (D) 22,33

Ans. [A, B]

[Sol. If line y = mx + c is tangent at (x1, y

1) then contact points is

c

b,

c

ma22

Equation of tangent of slope m is y = mx 222 bma

point of contact

2

1,

22

9;

2

1,

22

9. Ans.]


25/27

[25]

Q.55 If y (x) satisfies the differential equation y' y tan x = 2x sec x and y(0) = 0, then

(A)284

y2

(B)

184'y

2

(C)93

'y2

(D)

33

2

3

4

3'y

2

Ans. [A, D]

[Sol. )x(secx2xtanydxdy

I.F. = dxxtane =

xcosnel = cos x

Solution is ycos x = dxxcosxsecx2ycos x = x2 + C ; y(0) = 0 C = 0 y = x2 sec x and y' = 2x sec x + x2 sec x tan x

4

y =28

2;

4

'y =282

2

3

y =9

22

;

3

'y =33

2

3

4 2

. Ans.]

SECTION -III: Integer Answer TypeThis section contains 5 questions. The answer to each question is a single-digit integer,ranging from 0 to 9 (both inclusive)

Q.56 Let f : R R be defined as f (x) = | x | + | x21 |. The total number of points at which f attains eithera local maximum or a local minimum is

[Ans. 5][Sol. f : R R

f (x) = | x | + | (x1) (x + 1) | =

1x,1xx

1x0,1xx

0x1,1xx

1x,1xx

2

2

2

2

f ' (x) =

1x,x211x0,x21

0x1,1x2

1x,1x2

1 1/2 1/2O

1

Graph of f(x) Total number of points = 5. Ans.]


26/27

[26]

Q.57 The value of

......

23

14

23

14

23

14

23

1log6

2

3 is

[Ans. 4]

[Sol. Let x = ......23

14

23

14

23

14

23

1

x = x423

1 18x2 = 4x

18x2 + x4 = 0

x =36

441811 =

36

171 =

36

16=

9

4=

2

3

2

Hence, 6 +

2

2

33

2log

= 62 = 4 Ans.]

Q.58 Let p (x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum

at x = 3. If p(1) = 6 and p(3) = 2, then p'(0) is

[Ans. 9]

[Sol. p ' (x) = k(x1)(x3) = k(x24x + 3)

p'(x) =

x3

2

x4

3

xk

23

+

p(1) = 6

323

1k + = 6 6

3

k4

p(3) = 2

k(918 + 9) + = 2 = 2

3

k4= 4 k = 3

p'(x) = 3(x24x + 3) p'(0) = 9 Ans.]

Q.59 If candb,a

are unit vectors satisfying 9accbba222

, then c5b5a2

is

[Ans. 3]

[Sol.222

accbba

= 62 ba

= 9 ba

=2

3

2cba

= 0ba2a 2

0cba

acb

c5b5a2

= a5a2

= 3. Ans.]


27/27

Q.60 Let S be the focus of the parabola y2 = 8x and let PQ be the common chord of the circle

x2 + y22x 4y = 0 and the given parabola. The area of the triangle PQS, is

[Ans. 3]

[Sol. Parabola y2 = 8x focus = (2, 0)circle x2 + y22x4y = x(x2) + y(y4) = 0

S(2,0)

Q(2,4)

P(0,0)

circle with diametric ends (0, 0) and (2, 4)which are vertex and one end of Latus rectum.

Area of PQS

= 4221 = 4 Ans.

Aliter: Since parabola and circle both passes through origin.

one common point P (0, 0).

Solving above two equations simultaneously, x2 + 6x8 x2 = 0

By hit and trail, x = 2 satisfy above equation.

So another common point Q(2, 4)

Area of PQS = 422

1 = 4 Ans.]

Iitjee 2012 Paper i Sol

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