iitians pace aits 1 2012 paper 1 with solutions

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PART I : Single Correct Answer Type

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

1. A conducting spherical shell of radius R is given a charge Q. The force exerted by one half on the other halfis

(A) 2

2016

QR (B)

2

208Q

R (C) 2

20

QR (D)

2

2032

QR

2.

Two infinite line charges of linear charge density are parallel to each other and moving in same directionwith velocity v each. The value of v such that the force of interaction between them is zero is

(A) 0 0

1 (B) 0 0

12 (C) 0 0

13 (D) 0 0

14

3. The switch S is kept in contact with 1 for long time and then thrown to 2. The current through the resistor atthis instant will be

(A) 1A (B) 2.5 A (C) 0.5 A (D) zero

SECTION I : PHYSICS


4. In the figure shown the area of each plate is A and distance between two adjacent plates is d. The chargeon plate 2 is

(A) 03AVd

(B) 03

AVd

(C) 0

23

AVd

(D) 0

23

AVd

5. An ideal ammeter (of zero resistance ) is connected as shown. The reading of the ammeter is:

(A) 0 (B) 3ER (C) 5

ER (D) 7

ER

6. A particle of mass m is fixed to one end of a light spring of force constant k and unstretched length l. The systemis rotated about the other end of the spring with an angular velocity , in gravity free space. The kinetic energyof the particle is (the length of the spring reaches an equilibrium value)

(A) 21 m2

2l (B)2 2

2

1 km2 k m

l

(C) 2

2 22

1 km2 k m

l (D) none of these.

7 The velocities of a particle executing S.H.M. are 30 cm/s and 16 cm/s when its displacements are 8 cm and 15cm from the equilibrium position. Then its amplitude of oscillation in cm is :(A) 25 (B) 21 (C) 17 (D) 13

8 In a photoelectric experiment, the collector plate is at 2.0V with respect to the emitter plate made of copper = 4.5eV). The emitter is illuminated by a source of monochromatic light of wavelength 200nm.(A) the minimum kinetic energy of the photoelectrons reaching the collector is 0.(B) the maximum kinetic energy of the photoelectrons reaching the collector is 3.7eV.(C) if the polarity of the battery is reversed then answer to part A will be 0.(D) if the polarity of the battery is reversed then answer to part B will be 1.7eV.

9 Let K1 be the maximum kinetic energy of photoelectrons emitted by a light of wavelength 1 and K2corresponding to 2. If 1 = 22 , then :

(A) 2K1 = K2 (B) K1 = 2K2 (C) K1 < 2K2 (D) K1 > 2K2


10 The relation between 1 : wavelength of series limit of lyman 2 : the wavelength of the series limit of Balmerseries and 3 : the wavelength of first line of lyman series is(A) 1 = 2 + 3 (B) 3 = 1 + 2 (C) 2 = 3 1 (D) none

PART II : Multiple Correct Answer(s) Type

This section contains 5 multiple choice questions. Each question has four choices (A),(B), (C) and (D) out of which ONE or MORE are correct.

11. A magnetic storm from the Sun can disrupt a satellite as well as move it, either toward or away from the Earthradially. Ground-based engineers start it back in a new circular orbit at the new position. Due to the storm(A) The period of a satellite displaced further away is more than the previous period..(B) The mechanical energy of a satellite displaced towards earth is more than the previous energy..(C) The speed of a satellite displaced further away is more than the previous speed..(D) The angular momentum of a satellite displaced towards earth is more than the previous angular momentum.

12. A series RLC circuit is driven by a generator at frequency 1000 Hz. The inductance is 90.0 mH; capacitance is 0.500 F; and the phase constant has a magnitude of 60.0 (Take 2 = 10)(A) Here current leads the voltage in phase(B) Here voltage leads the current in phase

(C) Resistance of circuit is 3

80

(D) At resonance = 41032 rad/sec.

13. Each of the following system begins moving upwards with a constant acceleration. Select the cases in whichquantity will change due to this upward acceleration :(A) time period of simple pendulum.(B) fraction of floating body submerged in a liquid.(C) time period of a spring block system.(D) pressure on the base of a container containing liquid.

14. A source is moving across a circle given by the equation x2 + y2 = R2, with constant speed 36

330m/s, in anti-

clockwise sense. A detector is at rest at point (2R, 0) w.r.t. the centre of the circle. If the frequency emitted bythe source is f and the speed of sound, C = 330 m/s. Then

(A) the position of the source when the detector records the minimum frequency is R R 3,2 2

and the

position of the source when the detector records the maximum frequency R 3, R2 2

(B) the co-ordinate of the source when the detector records minimum frequency is (0, R)

(C) the minimum frequency recorded by the detector is f3636

(D) the maximum frequency recorded by the detector is f36

36


15. Consider the situation in which ball of mass M is hanging in equilibrium with a string and a spring as shown infigure. If another small ball of mass m collides it then which of the following are correct

FS

k

FT

FGM

m

(A) Tension force due to inextensible string FT is impulsive force(B) Spring force FS is non impulsive force(C) Gravitational force FG is conservative force(D) Normal force due to collision between m and M, is electro magnatic force

PART 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).

16. A rod of length R and mass 2m is free to rotate about a horizontal axis passing through hinge P as shown in thefigure. First it is taken aside such that it becomes horizontal and then released. At the lowest point the rod hitsthe block B of mass 1m and stops. If mass of rod is 60 kg, find mass of the block in Kg if it just completesthe circle.

17. How much water would be filled in a container of height 14 cm, so that it appears half filled to the observerwhen viewed from the top of the container ? (w = 4/3)

18. A box and a solid sphere of equal mass are moving with the same velocity across a horizontal floor. The sphererolls without slipping and the box slides without friction. They encounter an upward slope in the floor and eachmove up the slope some distance before momentarily stopping and then moving down again. During the motionon the upward slope, sphere rolls without slipping and box slides without friction. The maximum vertical

heights reached by the box and the sphere are HB and HS respectively. What is the ratio B

S

HH5

?


19. Consider the circuit shown in figure. With switch S1 closed and the other two switches open, the circuit has atime constant 0.05 sec. With switch S2 closed and the other two switches open, the circuit has a time constant2 sec. With switch S3 closed and the other two switches open, the circuit oscillates with a period T. Find T(in sec). (Take 2 = 10)

L C R

S1 S2 S3

20. Two identical plates of metal are welded end to end as shown in figure (A), 20 cal of heat flows through it in4 minutes. If the plates are welded as shown in figure (B) the same amount of heat will flow through the platesin how many minutes ?

A B

SPACE FOR ROUGH WORK

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SECTIONII : CHEMISTRY

PARTI : Single Correct Answer Type This section contains 10 multiple choice questions. Each question has four choices (A), (B), (C) and (D) out of which ONLY ONE is correct. 21. Two students A & B are given (+)-2-butanol (X) for experiment

Student A 2 5C H INaX Y Student B 2 5C H OHTsClX Z Between Y and Z, optically active product will be:

(A) Y (B) Z (C) both (D) neither

22. In PBr3F2, the BrPF angle is: (A) 90o (B) 120o (C) 180o (D) 109.5o

23. O

CH3

CH2

CH3

H / H O2 X

The product (X) formed in the above reaction is:

(A)

CH3

CH2

CH3

OHOH

(B)

OH

OH

CH3 CH3

(C)

CH2OH

OH

CH3 CH3

(D)

CH2OH

CH3 CH3

24.

NH2 NH C

O

NH2 , which nitrogen attacks on the carbonyl carbon during

nucleophillic addition reaction? (A) (B) (C) (D) any


25. For a first order reaction, 3R P , conc. vs time graph is given below

20

Concentration

t (in days)

P

R

Choose correct option regarding this (A) 1 2 20t days (B) 1 2 10t days

(C) 1 2 5t days (D) 1 2 15t days

26. An acid-base indicator has Ka = 3.0 105. The acid form of the indicator is red and the basic form is

blue. The [H+] required to change the indicator from 75% red to 75% blue is (A) 8 105 M (B)9 105 M

(C) 1 105 M (D) 3 104 M

27. Which of the following a compounds, on pinacol-pinacalone rearrangement produces a compound

which gives a precrpitate with KOI ?

CH3

OH

OH

CH3C2H5

OH

C2H5OH

CH3

OH

CH3OH

Ph

OH

CH3OH

(A) (B) (C) (D) 28. For a perfectly crystalline solid Cp, m =aT 3, where a is constant. If Cp, m at 10 K is 0.42 J/K mol,

molar entropy at 10 K is: (A) 0.42 J/K mol (B) 0.14 J/K mol (C) 4.2 J/K mol (D) Zero

29. How many maximum number of electrons of an atom will have the following set and combination of

quantum numbers? n + l = 5 m = 0, 1

1s2

(A) 8 (B) 12 (C) 10 (D) 6


30.

CH2

CH3

OHCH2 H / 3

BHA B

Product (B) in the above reaction is:

(A)

CH3

BCH3

CH3

CH3

CH3

CH3

(B)

CH3

BCH3

CH3CH3

CH3

CH3

(C) CH3

H3C

BCH3

CH3

CH3

CH3

(D) H3C

BCH3

CH3

CH3

CH3

CH3

PARTII: Multiple Correct Answer(s) Type This section contains 5 multiple choice questions. Each question has four choices (A), (B), (C) or (D) out of whichONE or MORE are correct. 31. 2 2

White residueof amixture ofcompounds

H OONa A B C

2 3 2

DNa CO H O

2HClH O D E

Choose correct statement(s) regarding the properties of the products of the above reaction sequence. (A) Gas (D) is a linear molecule. (B) Gas (C) is paramagnetic in nature (C) Aqueous solution of (B) cannot be stored in aluminium vessels. (D) Compound(E) is hygroscopic in nature. 32. Which of the following equations is/are correctly formulated : (A) 3Cu + 8HNO3(dil.) 3Cu(NO3)2 + 2NO + 4H2O (B) 3Zn + 8HNO3 (very dil.) 3Zn(NO3)2 + 2NO + 4H2O (C) 4Sn + 10HNO3 (very dil.) 4Sn(NO3)2 + NH4NO3 + 3H2O (D) As + 3HNO3 (dil.) H3AsO3 + 3NO2


33. Which of the following form(s) more than one monosubstituted product with NBS?

(A)

(B)

(C)

CH3

(D)

CH3

34. Find out the correct properties for 33[ ( ) ]Co ox

(A) 3 2sp d hybridization (B) 2 3d sp hybridization (C) Chelating reagent (D) Diamagnetic 35. Choose correct statement(s) out of the following. (A) CO2 shows only negative deviation from ideal behavior at any P and T. (B) The r.m.s speed of CO2 is higher than that of N2O at constant temperature. (C) KE of 4g of N2 at NTP is lower than that of 6g of CO under same conditions. (D) The rate of diffusion of PH3 is same as that of H2S under identical conditions.

PART 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) 36. In fuel cell H2 gas is consumed at anode while O2 at cathode. If volume of H2 consumed is same as

the volume of He produced at STP from the 0.5 gm atom of U235 which is disintegrated into Pb207 after 3108 yrs. ( 1 2t of U

235 =108 yrs. ). Approximately How many faradays have left at the anode of fuel cell?

37. Calculate the molality of NaCl solution whose elevation in boiling point is numerically equal to the

depression in freezing point of 0.12m Al2(SO4)3 solution in water. Assume complete dissociation of salts. (Kf and Kb of water are 1.86 and 0.558 Kkg/mole)

38. The number of acetal (or ketal) groups in the following saccharide is:

O

O

O

OHOH

OH

OHOH

OH

OH

OH 39. How many alcohols with formula C5H12O will give an anesthetic substance on treatment with

Cl2/Ca(OH)2? 40. What is the number of unpaired electrons in the complex which is formed when aqueous solution of

NaNO2 is treated with saturated solution of FeSO4 followed by addition of small amount of conc. H2SO4

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PART I : Single Correct Answer Type

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

41. Let L be the line of intersection of the planes r i 2 j 3k 0 and 0kj2i3r . Theacute angle which line L makes with j , is

(A) cos143

(B) cos1 32

(C) cos1 21

(D) cos131

42. Let f (x) be a function defined for 0 < x < 2

such that

6f = 0 and f '(x) tan x =

x

6

dttsintcos2

,

then f (x) is equal to(A) ln2(2 sin x) (B) ln2(sin 3x) (C) ln2 xtan3 (D) ln2(2 cos 2x)

43. In triangle ABC, let b = 6, c = 10 and r1 = r2 + r3 + r then the area of triangle ABC is equal to[Note: All symbols used have usual meaning in a triangle.](A) 15 (B) 18 (C) 24 (D) 30

44. Number of positive integers which have no two digits having the same value with sum of their digits being 45,is(A) 10 ! (B) 9 ! (C) 9 9 ! (D) 17 8 !

45. The equation of the line passing through M(1, 1, 1) and intersects at right angle to the line of intersection of the

planes x + 2y 4z = 0 and 2x y + 2z = 0 is a

1x = b

1y = c

1z , then a : b : c equals

(A) 5 : 1 : 2 (B) 5 : 1 : 2 (C) 5 : 1 : 2(D) 5 : 1 : 2

46. Let xn be the sequence of numbers denoted by xn = nP4

195

1n

33n

PP

(n N) where Pn denotes the number

of ways in which n distinct things can be arranged on n different places. The sum of all possible values of n N for which xn > 0, is(A) 10 (B) 9 (C) 8 (D) 6

47. The range of values of a for which the function f(x) =

3x1,x1x0,acosx 13

has the smallest value at x = 1, is(A) [cos 2, 1] (B) [1, cos 2] (C) [0, 1] (D) [1, 1]

SECTION III : MATHEMATICS


48. Suppose f(x) = x2 2x. The value of )1i2(ff is equal to (where i = 1 )(A) 15 (B) 5i + 5 (C) 35 (D) 15i 10

49. The locus of the midpoints of the chords drawn from the point M (1, 8) to the circlex2 + y2 6x 4y 11 = 0, is equal to(A) x2 + y2 4x + 10y 19 = 0 (B) x2 + y2 + 4x + 10y 19 = 0(C) x2 + y2 + 4x 10y 19 = 0 (D) x2 + y2 4x 10y + 19 = 0

50. Given a triangle ABC with AB fixed in length and position. As the vertex C moves on a fixed straight line, theintersection point of the three medians of the ABC moves on(A) a straight line (B) a circle (C) a parabola (D) an ellipse

PART II : Multiple Correct Answer(s) Type

This section contains 5 multiple choice questions. Each question has four choices (A),(B), (C) and (D) out of which ONE or MORE are correct.

51. If f (x) =

3x,17x8x

3x2],1x[

2x,2

xx3

2

2

then which of the following hold(s) good ?[Note : [x] denotes largest integer less than or equal to x. ](A) 1)x(fLim

2x

. (B) f(x) is differentiable at x = 2.

(C) f (x) is continuous at x = 2. (D) f(x) is discontinuous at x = 3.

52. If the position vector of a point P is kzjyixr , where x, y, z N and projection of r on kjia

is 3

10 then number of possible position of P is also equal to

(A) number of natural solution of the equation x + y + z = 7.(B) number of ways of selecting two objects out of 10 distinct objects arranged in a row so that no two of them

are next to each other.(C) the total number of outcomes when a pair of dice are rolled once.(D) number of positive divisors of 1800.

53. For x, y, z

2,0 , let x, y, z be first three consecutive terms of an arithmetic progression such that cos

x + cos y + cos z = 1 and sin x + sin y + sin z = 2

1, then which of the following is/are correct?

(A) cot y = 2 (B) cos (x y) = 2213

(C) 3

22y2tan (D) sin (x y) + sin (z y) = 0


54. Which of the following statement(s) is(are) correct?

(A) If P, Q, R are 3 different points in space and they do not lie on same line, then RQPQis a vector orthogonal to the plane containing P, Q and R.

(B) If x, y, z are positive real numbers satisfying x + y + z = 10, then the minimum value of yzx

xyz

zxy

is equal to 10.

(C) The value of

n

1in nnni2nLim ll is equal to ln

2e

(D) Number of ordered pairs (x, y) of real numbers satisfying 4x 2x + 2 + 5 = | sin y | are infinite.

55. In which of the following cases unique equation of the plane can be established?(A) Given a normal vector to the plane and a perpendicular distance from origin to the plane is known.(B) Given a point on the plane and a line which intersects the plane at right angles.(C) A plane which consists of all points that are equidistant from two given points.(D) A plane which is parallel to a given line L1 and as well as line L2.

PART 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).

56. Let two circles of radii r1 and r2 (r1 > r2) in the first quadrant are tangent to co-ordinate axes.

If the length of common chord of circles is maximum, then find the value of 2

1rr

.

57. The set of natural numbers is divided into arrays of rows and columns in the form of matrices as

A1 = (1), A2 =

5432

, AA3 =

14131211109876

.................... so on.

Find the value of r 10T A671

.

58. Find the number of points of intersection of the curves y = cos x, y = sin 3x, if 2x

2

.

59. If the value of the definite integral

2n

0

)e1(n2nx dxeee2nx2x

llll =

banl where b

a is rational in the

lowest form, then find (b a). (a,b > 0)

60. Let the variable line ax + by + c = 0, where a, b, c are in arithmetic progression be normal to afamily of circles. If r be the radius of the circle of the family which intersects the circlex2 + y2 4x 4y 1 = 0 orthogonally, then find the value of r2.

Paper - I (Solution)

1. (d)

2P

2

2f P. R

22

2Q 1. . R

24 R

2

20

Q32 R

2. (a)

0E

2 d

E0

F x2 d

l repulsion

2E

0

F2 d

l ... (1)

0B2 d

20BF 2B2 d

l... (2)

force of interaction is 0

2 22

0

02 d 2 d

0 0

1

3. (a)

1 1 1 2 2L i L L i

5v2n 52

2i 1A

4. (b)

There is charge on both sides on one side it is 02

3AV

d

and other is 03AVd

5.

1 1 2 3 467eff

E Ei i i i iR R

2R & R in parallel

11

2

2 2 61 3 7

i Eii R

R & R in parallel

33

4

1 1 61 2 7

i Eii R

A 1 3Ei i i

7R

6. (c)

2xK mw l x

2

2mw lx

K mw

22 21 1K mw m l x w2 2

22

21 kxmw2 mw

2 2

21 k x2 mw

2 2

2 21 k mw lk2 mw k mw

2 22

1 kmw .l2 k mw

7. (c)

2 2A x

2 230 A 8

2 216 A 15

2 2

2 2

30 A 816 A 15

A = 17

8. (b)

photon1242eV nmE 6.2eV

200nm

maxK 6.2 W 6.2 4.5

= 1.7 eV

() at the collector = (1.7 + 2) eV

= 3.7 eV

9. (c)

c1

1

h w k

e2

2

h w k

2 1

1 2

w k 1w k 2

1 22w 2k w k

2 1k 2k w

or

2 1k 2k

21

kk2

10 (d)

1 2 3h h h

c c c

1 2 3

h h h

1 2 3

1 1 1

11. (a)

(A) T2 r3 ; (B) E = r2GMm

(C) v = r

GM; (D) L = mvr = m GMr ]

12. (b, c, d)L = 90 103 2 1000 = 180

C1

= 10002105.01

6 = 1000

circuit is inductive VL > VR

voltage leads the current

tan = R

C1L

= R

80

R = 380

at resonance, = LC1

= 63 105.01090

1

= 91945

1 = 215

105

= 410

32

13. (a, d)

(A) T = 2effg

(B) fraction = L

bPP

same

(C) T = 2km

(D) P = P0 + hPgeff

14. (a, b, c, d)

dia.

max330f f330330

6 3

6 3.f6 3

min6 3.ff

6 3

here 1cos2

and 3sin

2

max. frequency position = R cos , R sin

R R 3,2 2

min. frequency position R cos , R sin

R R 3,2 2

15. (a, b, c, d)

16. (2 kg)

Let the angular velocity of rod at the time of collision be According to the law of conservation of energyFor the rod at the horizontal and vertical positions, we get

m2gR = 2

21 RI m g2 2

2gRm2 = 2

22 3Rm

21

= = Rg3

Applying the law of conservation of angular momentum about PLet the angular speed of block about P after the collision be 0.I = m1R

20

3Rm 22 = m1R

20 0 = Rg3

m3m

1

2

Linear velocity of ball is

v0 = 0R v0 = gR3m3m

1

2

For ball to complete the circle

v0 = gR5 = gR3m3m

1

2

1

2mm

= 15

17. (8 cm)

18. (7)

mgHB = 21

mv2

mgHS = 21

mv2 + 21

I2 = 21

mv2 + 21

52

mr22 = 21

57

mv2

B

SHH

= 57

= 1.4

19. (0002)

1 = RC

2 = RL

LC = 1 2 = 0.1 sec

T = 2 LC = 1012 = 2 sec.

20. (1)

420

= dtdQ

= L2kA

(T)

t20

= TLA2k

reqd. time = 1min

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PAPER-1 21. (A) 22. (A) More electronegative element always occupy axial position in trigonal

bipyramidal geometry

P

Br

BrBr

F

F90o

23. (C)

O

CH3

CH2

CH3

H

O+

CH3

CH2

CH3

H

CH3 CH3

OH+

+2H O

CH3 CH3

CH2OH

H

CH2OH

OH

CH3 CH3 24. (A) As corresponding lone pair does not participate in resonance and thus better

nucleophillic site. 25. (B)

3

0, 020 , / 3

R Pt at d a x x

For intersection point, [R] = [P], 3, 3 4, 4a x x x a a x a 1 22 20t days

26. (C) HIn InH a

10075a

10025

]HIn[

]In][H[K a

or 25

75]H[103 5

[H+] = 1 105. 27. (A)


CH3

OH

OHCH3

CH3

OH2

OHCH3H

+

anti-group migration

CH3

C CH3

O

+COH

CH3

CH3

H

28. (B) According to 3rd law of thermodynamics,

,0

Tp mCS dTT

29. (C) 30. (C)

31. (A, B, C)

2 22 2 2 2H OONa Na O Na O NaOH O

2 2 3 2CO Na CO H O

2 2HClH O CO NaCl

32. (A, C, D) OH3NONH)NO(Zn4HNO10Zn4 23423

.)dil.V(3

33. (B, D) 34. (B, C, D) 35. (C, D) 36. 6 235 207 4 227U Pb He

3108 yrs = 3 1 2t

Thus, total moles of He produced = 7 0.5 2 0.5 4 0.5 8 3.5 8 3.06 Total moles of H2 produced for same volume =3.06

Total Faraday = 2 3.08 6


37. 1

( ) 2 4 3( )

2 0.558 5 1.86 0.12, 1

b aq f aq

b f

T NaCl T Al SO

ik m ik mm m

38. 2

39. 2 40. 3

PAPER - 1 (SOLUTION)

41. BThe line of intersection of the two planes will be perpendicular to the normal's to the planes. Hence it is parallel

to the vector = k3j2i kj2i3 = 123321kji

= k4j8i4

Also, j

166416k4j8i4

= 32

968

If is the required angle, then cos = 32

= cos1 32

.

42. A

We have f '(x) tan x =

x

6

dttsintcos2 f '(x) tan x = 2ln (sin x) + 2 ln 2

f '(x) = 2 cot x ln (sin x) + 2 cot x ln 2 f '(x) = dxd (ln2(sin x) + 2 ln2 ln(sin x))

f (x) = ln2(sin x) + 2 ln 2 ln (sin x) + C

For x = 6

; 0 = ln2 2 2 ln2 2 + C C = ln2 2

So, f(x) = ln2(sin x) + 2 ln 2 ln(sin x) + ln2 2 f(x) = [ln 2 + ln(sin x)]2 = ln2 (2sin x)

43. D

MB We have r1 = r2 + r3 + r (Given) (r1 r) = (r2 + r3) )as(s)as(s

= )cs()bs()cb(s2

1)as(s

)cs)(bs(

tan2 1

2A

452A

Hence, A = 90

Now, area of ABC = 21

bc sin A = 21

(6) (10) sin 90 = 30 (square units).

44. A1 + 2 + 3 + ............. + 9 = 45 = 0 + 1 + 2 + 3 + .................. + 9All 9 digit such numbers = 9 !All 10 digit such numbers when '0' included = 10 ! 9 !So, total = 9 ! + (10 ! 9 !) = (10) !

45. ASolving the equation of planes, we get equation of line containing planes

5z

10y

0x

...........(1)

Any point P on (1) is (0, 10, 5).

Now, direction ratios of the line joining P and M is 51,101,1

As line MP is perpendicular to line (1), so

0 (1) 10 (1 + 10) 5 (1 + 5) = 0 = 253

P

53,

56,0

So, d.r's of MP are 52,

51,1

M(1,1,1)

P(0,0,0)

x + 2y 4z = 0

2x y + 2z = 0

(0, 10, 5 )

So, equation of required line is 5

1x = 1

1y

= 21z

.

46. AWe must have,

(2n + 19) (2n 9) < 0 219

< n < 29

n {1, 2, 3, 4}

Hence, sum = 1 + 2 + 3 + 4 = 10

47. BClearly f(x) is decreasing in [0, 1) and increasing in [1, 3]So, for f(x) to be minimum at x = 1, we must have

1xLim f(x) f(1) 1acosxLim 13

1x

1 + cos1 a 1 cos1a 2

a [1, cos 2].

48. CWe have f (2i + 1) = (2i + 1)2 2(2i + 1) = 4 + 4i + 1 4i 2 = 5,and f( 5) = ( 5)2 2( 5) = 25 + 10 = 35.

49. D82MB

Clearly, mCP mAB = 1

1h8k

3h2k

= 1

Locus of (h, k) is (x 1) (x 3) + (y 2) (y 8) = 0 A BP

(h, k)

C (3, 2)

M (1, 8)

i.e., x2 + y2 4x 10y + 19 = 0

50. Aline lx + my = 1 where l, m are constants

Let C

mb1,b l

Hence 3h = b ....(1)

O

Cfixed line

lx + my = 1

A(a, 0) B(a, 0)

G(h,k)

y

xand 3k = m

b1 l....(2)

3km = 1 l (3h)

Locus of (h, k), is (3l)x + (3m)y = 1 lx + my = 31

51. A, C, D

y

O(0,0)

xx=2 x=3 x=4

(4,1)

Interpret from graph.

y=1

52. B, C, D

Projection of r on a is |a|ar

x + y + z = 10 (x, y, z N)

number of solution 101C31 = 9C2

= 36(A) x + y + z = 7, x, y , z 1 x + y + z = 4 6C2 = 15

number of non negative integral solution = 7 + 3 1C3 1 = 9C2

= 36.(B) 10 objects

2 to be selected

|||||||||OO

6C2 = 36

(C) 9C2 = 36(D) 1800 = 23 32 52

Number of divisors = (3 + 1) (2 + 1) (2 + 1) = 36.

53. A, DWe have

zx

zy,y,dy

in A.P..

Now, 1xcos cos (y d) + cos y + cos(y + d) = 1 cos y (2 cos d + 1) = 1 .....(1)

Also, 21xsin sin(y d) + sin y + sin (y + d) = 2

1 sin y (2 cos d + 1) = 2

1 .....(2)

)2(Equation)1(Equation

cot y = 2

Now, putting cos y = 32

in (1), we get

2 cos d + 1 = 32

cos d = 22

23 = cos (y x) = cos (x y)

Also, tan 2y = ytan1

ytan22

=

211

212

= 22

212

Clearly, sin (x y) + sin (z y) = sin ( d) + sin d = 0.

54. A, B D

(A) Obviously true.

(B) Using A.M. G.M. in zxy,

yzx;

yzx,

xyz;

xyz,

zxy

and add, we get

yzx

xyz

zxy

(x + y + z)

yzx

xyz

zxy

10

(C) Sn =

n

1i ni2n

n1 l =

1

0

dxx2nl

Put 2x = t

S = 2

0

dttn21 l = 2

1 20ttnt l = 2

1 022n2 l

e2nl

(D) (2x 2)2 + 1 = | sin y |L.H.S. 1 and R.H.S. 1

x = 1 and y = 2n 2

, n I ]

55. B, C

(A) There can be two parallel planes equidistant from the origin.(B) Obviously the equation of the plane is A(x x1) + B(y y1) + c(z z1) = 0 where A, B, C are

proportional to the dr's of the line.

(C) P (x1, y1, z1) and Q(x2, y2, z2)Apply condition of locus on R(x, y,. z) to get a unique plane, 2nd degree term cancels.

(D) Plane can be parallel.

56. 3Let equation of circles beS (x r1)

2 + (y r1)2 = r1

2 ... (1)and S' (x r2)

2 + (y r2)2 = r2

2 ... (2)where r1 > r2 Equation of common chord is S S' = 0 given by A

BO (0,0) x

y

2(x + y) = r1 + r2 ... (3) For maximum length of common chord,above equation (3) must pass through the centre (r2, r2) of the smaller circle.

4r2 = r1 + r2 3r2 = r1 2

1rr

= 3

57. 5First term of An is the n

th term of series S = 1 + 2 + 6 + 15 + .......... + Tn(Using method of difference) S = 0 + 1 + 2 + 6 + .............+ Tn 1 + Tn

(Subtracting) 0 = 1 + 12 + 22 + 33 + .......... + Tn Tn 1 Tn

Tn = 1 +

1n

1n

2n

Tn = 1 + 6)1n2(n)1n(

= 6

6nn3n2 23

So, T10 = 66103002000

= 286

Now common difference of the diagonal elements of A10 is 11and number of diagonal elements is 10.

Hence Tr(A10) = Sum of its diagonal elements = 1192862210

= 3355

58. 3For point of intersection, we havecos x = sin 3x

cos x =

x32

cos x = 2n

x32

(+ ve) sign, x = 82n

(n I) Possible values of n are 1 and 0.

x = 83

and x = 8

( ve) sign, x = 4n (n I) Possible values of n is 0.

x = 4

Hence, points of intersection are

8cos,

8,

4cos,

4,

83cos,

83 .

59. 1

I =

2n

0

)e1(n2nx dxeee2nx2x

llll

I = 2n

02x

x dx)e1(

12e2nxl

l

I = 2n

0II

2x

x

I

dx)e1(

e2nx2l

l

=

2n

0x

2n

0x 1e

dx1e

12nx2ll

l

=

2n

0xe1n

22n02

lll =

2n23n

22n2 lll =

3n2n

2122 ll = 4

9n2n ll

=

942nl =

98nl =

banl (b a) = 9 8 = 1.

60. 8As a, b, c are in A.P., so ax + by + c = 0 represents a family of lines passing through the fixedpoint (1 ,2). Since each member of the family is normal to a circle hence its centre must be (1, 2).So, the family of circles with centre (1, 2) will be given by(x 1)2 + (y + 2)2 = r2 x2 + y2 2x + 4y + (5 r2) = 0And using condition of orthogonality, we get2 [(1) ( 2) + (2) ( 2)] = 1 + 5 r2 r2 = 8

iitians pace aits 1 2012 paper 1 with solutions

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