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    UP/217/1 2088Question Booklet No ......................To be filled up by the candidate by bluel black ball-point pen

    Roll Xo.Rollfl rire the digits in wordsSerial No of OMR Answer Sheet .......................................... .Day and Date Signature of Invigilator)

    INSTRUCTIONS TO CANDIDATESese only blue/black ball-point pen in the space above and on both sides of the Answer Sheet)

    1. Within 10 minutes of the issue of the Question Booklet, check the Question Booklet to ensure thatit contains all the pages in correct sequence and that no page/question is missing. In case of faultyQuestion Booklet bring it to the notice of the Superintendent/lnvigilators immediately to obtain afresh Question Booklet.

    2. Do not bring any loose paper, written or blank, inside the Examination Hall except the Admit Cardwithout its envelope.

    3. A separate Answer Sheet is given. It should not be folded or mutilated. A second Answer Sheet shallnot be provided. Only the Answer Sheet will be evaluated.

    4. Write your Roll Number and Serial Number afthe Answer Sheet by pen in the space provided above.5. On the front page of the Answer Sheet, write by pen your Roll Number in the space providedat the top, and by darkening the circles at the bottom. Also, wherever applicable, write the

    Question Booklet Number and the Set Number in appropriate places.6. No overwriting is allowed in the entries of Roll No., Question Booklet No. and Set No. if any) onOMR sheet and also Roll No. and OMR Sheet No. on the Question Booklet.7. Any change in the aforesaid entries is to be verified by the invigilator, otherwise it will be taken as

    unfair means.8. Each question in this Booklet is followed by four alternative answers. For each question you are torecord the co1Tect option on the Answer Sheet by darkening the appropriate circle in the corresponding

    row of the Answer Sheet, by ball-point pen as mentioned in the guidelines given on the first pageof the Answer Sheet.

    9. For each question, darken only one circle on the Answer Sheet. f you darken more than one circleor darken a circle partially, the answer will be treated as incorrect.

    10. ,r-.. ote that the answer once filled in ink cannot be changed. If you do not wish to attempt a question,leave all the circles in the corresponding row blank such question will be awarded zero mark).

    11. For rough work, use the inner back page of the title cover and the blank page at the end of thisBooklet.

    12. Deposit only the OMR Answer Sheet at the end of the Test.13. You are not permitted to leave the Examination Hall until the end of the Test.14. Ifa candidate attempts to use any form of unfair means, he/she shall be liable to such punishment

    as the University may determine and impose on him/her.[No of Printed Pages: 40+2

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    l lP/217/1No. o f QueotloDll{5lT-if : 150

    Time/11Iltl: 2 Hours/l:fUZ Full Markst rlJ1f : 450

    Koter=n z: (I) Attempt as many questions as you can. Each question carries 3 marks. nemark will be deducted for each incorrect answer. Zero mark will be awardedfor each unattempted question.mt .it -.;B ifiT ' 'Ii( I _ 3 .at

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    l lP /217/1

    1lFI1 ill; W

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    l lP 2 l7 l

    (1) G (2) {e) , ..m e, G ' l it

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    l lP/217/1

    llT' T "'l G i\; """ '" a ]Ii1 n t I

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    l lP/217/1

    9 Let be the additive group of integers and H be the subgroup of Z obtained onmultiplying each element of Z by 3 Then the number of distinct cosets of in Z is'IR1 Z 41'11"''' ' l i ' t am H Z . . . t t Z iii ' -.it 3 i\ T-.i-\ ' >Ill t l Z il Hili 1 ffi-1 ffi ~ ~ , , < i 1 lit If&rr oT ft(1) 2 (2) 3 (3) 1 (4) 00

    10 Let H and be fInite subgroups of a group G. Then o{HK is equal to

    1) o(H)+o(K)3) o(H)o(K)

    o(HnK)

    2) o(H) o(K)4) o(H)o(K)-o(HnK)

    11 How many elements of the cyclic group of order 8 can be used as generators of thegroup?

    (1) 2 (2) 3 (3) 4 (4) 112 Which statement n not correct?

    (254)

    1) The polynomials over a ring form a ring2) The polynomials over an integral domain form an integral domain3) The polynomials over a field form a field4) A field has no zero divisors

    -m ..,.., _ 'I1ff t?(1 ) 1% ' ' oR {I. 1% ' iI'Iffi ~(2) 1% :IIR ' oR ~ { I 1% :IIR iI'Iffi ~(3) 1% $ ' oR ~ q { l 1% $ iI'Iffi ~(4) 1% $ iii 'lFI ~ I ~ I1ff ~

    5 P. T. D.}

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    13. Let f x) and g x) be two non-zero polynomials over a ring R. Thenl l'IT f.\; f (x) >itt 9 (x) 1

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    17. If the characteristic values of a square matrix of third order are 2 3 4 then the valueof its determinant is

    1) 6 2) 9 3) 24

    18. Let Tl and T be linear operators on R defmed as follows:1 FIl iln R2 R T a tt T ,

    TJia, b) = b, a), T2 a, b) = a, 0)rhen T,T2 defined by T T2 a, b) = Td T2 a, b)) maps ~ 2 ) into

    4) 54

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    2 -3 4 -1The characteristic values of the matrix a 1 5 420. area a 3 7a a a 42 3 4

    -1]1 5: ~ 3 l f r . n ~ 1lFa a 3a a

    (1) 1 2 3 4 (2) 1 -2 3, 4 (3) -1 2 -3 4 (4) -1 -2 -3 -4The rank of the matrixl -1 2]2l 1 . 4 is1 -1~ l ~ -1 _:] ~ t1(1) 2 (2) 3 (3) 4 (4) 1

    22. If is a linear transformation from an n dimensional vector space U to anm dimensional vector space V. then the sum of the rank of and the nullity of isequal to'If . T ll,'f; -flf I m u it ll,'f; m flf I m V i ll,'f; \

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    24. For the matrix- = r A-I is equal to0 0

    ~ A l ~ ~ A-I t(I) I 2) 3) A

    25. Which function is continuous at x = 0 ?

    (I) sin ( l /x) (2) sin l x 2 ) 3) tan-I ( l /x )

    l lP/217/1

    4) 2A

    4) tanx

    26. If r= { a 2 +b 2 and $ = an-I (bla), then the nth derivative of e cos{bx+c) is

    (3) e ~ s i n { b x + c + n ) 4) e=cos{bx+c+ n$)

    27. The coefficient of X in the Maclaurin s expansion of log cos x islog cos x ~ ~ < i \ t = it X4 fiT t

    I(I) 24I2) 12

    I3) 45 4)

    28. f we expand cos (. : +8 in powers of 9, the coefficient of ~ is4 31

    (I) /2I 2)

    254)I.J2

    93) 2 4)

    I6

    I2

    P. T. D.}

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    29 The infmite series expansion of log 1+ 1is valid for(I) x > l on l y (2) x< I onlylog 1+x) 3FR ~ : s T a R ~ tI) i\or.r x> I

    (3) i\or.r Ixld

    3) 1 1

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    33 For any curve ds is equal todB

    2) rp

    4F dB aJr any curve IS equ tods

    I) o s ~ 2) sin

    3) r p

    3) cos IjI

    35 For the curve p = ar the radius of curvature is

    36 Which curve has no asymptotes?

    254)

    A B3) y =mx c -x x

    I) 2 2) 3

    3) 2p a

    3) 411

    lIP 217 1

    4) sin IjI

    4) 1P. T. D.)

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    38. f x is a root of the equation f 9) =0, then an asymptote of the curve f 9) isrlIfu a eiftOM 1 (8) =0 1(ifi '@ iT q,n (8) 1(ifi . ,,,,,ffr(I) r . i n ( 8 - a ) ~ f ( a )3) r c o . ( 8 - a ) ~ f ( a )

    2) r s i n ( 8 - a ) ~ I / f ( a )(4) rcos 8 -a I / f ( a )

    39. The curve r sin 56 has(I) I loop (2) 3 loops

    (I) I (2) 3 ~40. An asymptote of the curve y tan x is

    I) x4

    2) ~3

    (3) 5 loops

    (3) 5 ~

    3) ~2

    41. x 1 g y/x) is a homogeneous function of and y of degreex 3 10g y/x), x am y lIlI mI ' ' t f.m

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    43.

    44.

    l lP /2 l7 / l

    a2u a2uIf u is a homogeneous function of x and y of degree n, then the value of x - +x x yIS

    ou(I) n-l ox 2) ouox 3) (n_l)Ouoy ou4) oyf x = T sin 8 cos , y == r sin 9 sin , t r o{x,y,z .z = r cos 8, en the value 0 8r,a, )

    x=rsin8coscp

    (I) sina

    y = r sin e sin 4> Z = r cos 8 c T a x, y z fiT lfRo{r,a, )2) r sin a 3) r sin a 4) sin a sin

    45. The envelope of the family of curves +Ba. +C =0 where A, B, C are functions of xand y variables, is

    46.

    254)

    I) B -AC =0 2) B - 4 A C = 0

    x yThe evolute of the ellipse - + - = 1 isa2 b 2

    I) {ax) /3 +{by) /3 =(a _b ) /3

    13

    3) C -4AB =0 4) A - 4 B C = 0

    2) x /3 +y /3 = {a _b ) /34) (ax) /3 +{by) /3 =a /3 b /3

    P. T.O.

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    47 The function x y3 -3axy has a maximum or minimum at the point1) o;a) 2) 0, 0) 3) 0; 0)

    (1) 0; a) (2) 0, 0 3) 0; 0)R 3N

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    51. If m and n are integers and n m is odd then the value of the integralI cos x sin nx x is3) 0

    52. If n 0::: I otn X dx then n 1n_2 is equal to-1I) _cot x

    n Icotn- x2) n I

    otn 2 x3) - I

    53. The value of the integral S cos 4 x dx isfl 4ICf t: cos4 X dx CfiT lJR

    I) 31t 2) 3)- -16 8 4

    54. The value of the integral r xsinx dx ISo 1 cos2 Xr x sinx dx CfiT l IR to 1 cos2 x

    I) 2) 3)16 8 4

    254) 15

    4)

    4)

    4)

    nm

    otn 2 xn I

    31t32

    2

    P. T. 0.

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    l lP 217 1

    55. lim ~ . . f n +r) is equal ton-+oo r ,1 n n T~1) - 12 (2) -2 (3) 12

    ~4) - 2256 The area included between the cycloid x 8 sin e , y::: a (1 - cos e) and its base is

    ..,..., x=a(8-s in8) , y=a(1-cos8) alR ""* 3l1m< i\ 1 ro

    57 The whole area of all the loops of the curve T a cos 48 is

    ~ a(1) 4~ a(2) 2 3) ~ a

    58 The perimeter of the curve r 2a cos 9 is

    1) ~ 2) na 3) ~ a 4) 8 ~ a

    59 The length of the arc of the catenary y =ccosh{xjc from the vertex 0, c) to the point(x yd is$I (0, c) i\ flr . (x yd ' y = ccosh (x/c) ~ ;m -. it ~1) yf -c (2) ~ Y f -c 3) yf+c 4) ~ y ~ c

    (254) 16

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    60. For the parabola 2a=1+cos6. the value of ds isr d ~2 . ds--'" ~ 1+cos 8 iii fuo: -

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    l lP 217 1

    r ry /265. The value of the integral Jl Jo dy dx isr ry / 2I 1l

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    I) y ~ 4ax ai'I< 'ail y ~ O. x ~ 4a(2) y ~ 4ay ai'I< 'ail x ~ O. ~ 4a(3) q { q ~ < : ( l y = 4ax x 2 = 4ay TU4) 'ail x ~ O . x ~ 4 a , y ~ O . y ~ 4 a

    69 The value of r ~ ) isr ~ ) q;J l H tI) 3 [ ;4

    (2) 15,[,;83n(3) 4

    70 The value of integral x m-1 l ~ x t - l dx (m > 0 n > 0) istI lICfliil xm 1 - x t - dx m>O,n>Oj fiT 1fR t(I) r(m)+r(n) (2) r m)r n) (3) r (m+n)

    5n(4) 8

    l lP/217/1

    (4) r (m)r(n)r (m+n)

    71. If V is the region given by x : 0 Y 0 Z 2: 0 X +Y +z

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    72. The sum of B m+L n) and B m, n+I) isB m+L n) aft .: B m, n+I) IiI irT ~I) B m, n) 2) B m+Ln+I)3) B 2m+L2n+I) 4) 2B m, n)

    73. he order of the differential equation

    {2}3/2I + ~ ) =p ~

    1S

    { 2}3/2 2I + ~ ) =p ~I) 1 2) 2 3) 4 4) 3

    74. he number of arbitrary constants n the general solution of the differential equation[ ~ ; r + S i o X ( ~ ) + I o g X ~ + 9 Y = C O S X

    will be

    d3 2 d 3 ddx; sin x log x 9y cos xI) 2 2) 3 3) 5 4) 6

    254) 20

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    75 The general solution of the differential equation sec 2 x tany dx sec2 y tan x dy = 0 isl; lIilCfl(U1 sec2 xtany dx+sec ytan x dy =0 fiT

    76.

    (I) tan x tany =c 2) tan x +tany =c 3) tan(xy) =c 4) tan x/y) = cThe general solution of the differential equation x y dy =2y isdx

    x(I) lo g y - x ) = c + - y - xx3) l og y -x )=c+-Y

    2) log y -x)=c+--- . lLy - x4) log y-x)=c+ Yx

    77 An integrating factor of the differential equation (1 +X2 dy 2xy;= cos x isdx

    (I) log( l+x2) (2) l+x2 3) x 2 4) x

    78 Equations of the form dy +Py = yn where P and Q are functions of x alone can bedx

    254)

    reduced to the linear form by dividing by yn and puttingdy +Py = Qy ' ; .Ijj,, ,n -.it, .m P om Q i\;

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    79 The differential equation M dx + N dy =: 0, where M and N are functions of x and y, isexact if

    4) aM=aNax ay

    80 If ~ [ a 8M is a function of y lone, say f{y), then an integrating factor of theM ax ayequation M dx. + N dy = 0 is-.fl\ ~ [ a _aM) y ' ' 1lRT f.I; f ly) m. cit .1i10; 01 Mdx+Ndy=OM ax ay

    ~ l i l t l i ? l i i f i l0I"1&og

    1) f ly) 2) Jly) dy 3) e l f l Y d Y 4) e-I f l y dy81 The singular solution of the equation y = px + a/p p == dy isdx

    82.

    (254)

    .1i10; U1 Y = px +alp. p dy OR idx1) y2 =ax 2) y2 =2ax 3) y2 = 4ax

    The general solution of the equation p = log px -y ,dyiE'l iiCfl(ll1 p=log{px-y) , p CfiT IDll RJ tdx

    1) c=log cx-y) (2) y=cx 3) y=x+c

    22

    4) x 2 = 4ay

    p == dy isdx

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    83 The general solution of the equation d y m4y:=0 isdx 4d 4* l llCfl{U1 -..-.1 ..+m4y =

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    86.

    87.

    The solution of the simultaneous equations dx - Y ;0:= t,dt841,,;{O, dx _y ~ dy +x = 1,, . . , tdt dtIl x=c1cost c2sint 2, y=-c l s in t c2cost - t2) X =C IC O St C 2Sint , Y=-CISint C2coSt-t3) x = cl cost c2 sint 2, Y = -cl sint C2 cost4) x=c1cost 2, y=-c ls int - t

    dy x= l isdt

    f 2_Px Qx2 =0, then a particular integral of d y P dy Qy =0 isdx' dxd y dy' fit 2-Px+Qx =0 m-- P - + Q y =0, , " " ' " 8'

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    89

    l lP /2I7/1

    o solve the linear differential equation d 2 y +P dy +Qy =R by the method of variationdx dxof parameters we need two independent solutions of the equation

    d d y +P dy ~I) ~ + Q Y ~ O 2)dx dx xd d d y +Q dy +Py ~3) ~ P .... t+QY ~ 4)dx dx x x

    90. he solution of the partial differential equation x y -z p +y z-x) q = x -y) z is' ~ < i 1 < O I x y - z ) p + y z - x ) q ~ x - y ) z ' ' OR ~

    I) x+y+z, x y z ) ~ O3) ~ x + y + z , y z / x ) ~ O

    2) x+y+z, x y / z ) ~ O4) ~ x + y + z , z x / y ) ~ O

    91. he complete solution of the partial differential equation p2 +q2 =n 2 is

    I) z ~ a x + n y + c3) z = x +ay c

    2) z ~ a x + v n 2 _a2) y+c4) z ~ V n 2 _a2) .x+a2y+c

    92. he complete solution of the partial differential equation z = px +qy + c J 1 +p2 +q2 is

    2) z ~ a x + b y + c4) z ~ a x + b y + c / a b

    254) 25 (P.T.O.)

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    93 If and > are arbitrary functions, the solution of the partial differential equari::::r-4s+t;;=O is

    1) z = ~ d y + 2 x ) + ~ 2 ( y + 2 x )3) z = ~ d Y + X ) + ~ 2 (y+x)

    2) z = d y + 2 x ) + X ~ 2 ( y + 2 x )4) z = ~ d Y + X ) + X ~ 2 ( Y + x )

    94 The solution of the partial differential equation s;;= eX + Y is

    1) z = ~ d X ) + ~ 2 ( Y ) + e

    95 The particular integral of{D 3 -2D 2D'-DD,2 +2D,3)z=ex+ y is

    2) xex + Y

    2) Z = ~ , (X)+ 2 (y)+e Y4) z = d X ) + ~ 2 ( Y ) + X Y

    the partial

    3) _2 yeX+ Y

    differential equation

    96 Putting x = eU y = elJ, and denoting ~ and ~ by D and D respectively, the equationu u

    x 2r_4xys+4y 2 t+6yq=x 3y 4 is transformed into the equation

    1) (D-2D') (D-2D'-1)z=e u+ 4 2) (D_2D')(D_2D'+1)z=e u+ 4 3) (D_D') (D_2D'_1)z=e3U 4 , 4) (D+2D')(D_2D'_1)z=e u+ 4

    254) 6

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    l lP 217 1

    97 If R, 8, T U and V are functions of x y Z, and q variables, the Monge s subsidiaryequations for the partial differential equation Rr+Ss+Tt+U(rt-S2)=V areR dp dy+T dqdx+U dp dq-V dx dy =0 and

    R, S, T, U 1('i V u x y Z, p 1('i qRr+Ss+Tt+U(r t -s )=V , .R dp dy +T dq dx+U dp dq- V dx dy =0 1('i1) Rdy -Sdydx+Tdx +Udpdx+Vdqdy= O

    2) Rdy +Sdydx+Tdx +U dpdx+Vdqdy=O3) R dx -S dx dy+Tdy +U dp dx+ V dqdy =04) R dx +S dx dy+Tdy +U dp dx+ V dqdy =0

    :aift

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    100. The extremal of the functional r x 2y 2 dx subject to the conditions y lf2) 1 y{1 0:= 2Jl/21S

    mT y (1/2) =1, y I)=2 t >1$rI(I) y x

    I2) y = - - 3x 3) y=-x 3 4) Y =_x 2 +3

    101. The solid of revolution which for a given surface area has maximum volume is(I) a cylinder 2) a cone 3) n ellipsoid 4) a sphere

    ( I) 1(iIi tffoI 2) 1(iIi 4) 1 iIi >i'r.rr

    102. The periodic time of a cycloidal pendulum is

    (I) ~ (2) 21t ~ 3) 3 ~ 4) 411 ~103. For a particle falling under gravity n a resisting medium if the law of resistance be

    mkun , the terminal velocity will be

    (254)

    1(iIi l flm>fi ' ' if t >1$r ilRa 1 iIi q;1lJ t ~ -.ft\ l flm mkvn it,

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    104. A body consisting of a cone and a hemisphere of radius r on the same base rests on arough horizontal table the hemisphere being n contact wit the table. The greatestheight of the cone so that the equilibrium may be stable is' t\ :;mm ' >: ' ~ ' r ili 'lT ~ ' it f.ifif

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    108 A uniform tetrahedron of mass is kinetically equivalent to four particles each ofmass M at the vertices of the tetrahedron and a futh particle placed at its centre of20inertia of massM _ 11.' ~ ' N g i i l " WI' -wi 'R W 'h i't, f.RiI "

    _ M 3'it< 11.' 'h"T i\;, .i\ i\;;j: 'R WI s,31T 3'it< t.m..T _ i20(1) 4M5 (2) 3M5 3) M5 (4) M5

    109 In-the motion of a body about a fixed axis the moment of momentum of the body aboutthe fixed axis is

    (1) Mk2 e2 dt (2) Mk2 (de)2 dt 2 3) Mk2 edt110 The periodic time of a compound pendulum is the same as that of a simple pendulum

    of length

    (1) kh

    k (2) h 3) hk

    111 The kinetic energy of a body moving in two dimensions is

    (1) u + M k 82 2 2) Mu + Mk 82

    3) u2(254) 30

    (4)k

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    l lP/2 l7 / l

    112. Forces P Q R act along the sides of the triangle formed by the linesx + y = L y - x = 1, y = 2. The magnitude of their resultant isoooil x y ~ L y x ~ L y ~ 2 ';P

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    l lP/217/1

    116 The transverse component of acceleration of a particle moving in a plane is

    (I) 5 (2) fli Hii 523) p 4) r rl i 2

    117 The normal component of acceleration of a particle moving in a plane is

    (I) 5 (2) f l i Hii 523) p 4) r r i 2

    118 The formula for angular velocity of a particle about a point 0 is

    . uPI) -r. uP2) e r u4) e P

    119 A point executes simple hannonic motion such that in two of its positions the velocitiesare U u and the corresponding accelerations are cr., 13 Then the distance between thepositions is11.'> >l'OR llW 3l1'ffi '1ffi it MRl

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    121. A particle is projected from the lowest point with velocity u and moves along the insideof a smooth vertical circle of radius r The particle will make complete revolutions if thepressure at the lowest point is greater thanT fIrR ~ x i - Y j +z k ' r ~ [ r [, iIt div rn r 'lIT l 1' t(I) 0 (2) nr n - (3) nrn

    (254) 33

    (4)[a be l

    (4) n +3) rn

    P,T.D.)

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    125 If is a scalar invariant, then 0 , are components ofox'(1) a contravariant vector (2) a covariant vector(3) a contravariant tensor of order 2 (4) a covariant tensor of order 2

    (1) ' q ; ' ~ I ~ I l : ~ ' l (2) ' , , ~ j { l ( 1 l l (3) JP 2 ' q ; ' ~ I ~ l l : l ( 1 l l (4) JP 2 ' ' l ~ I l : ~ ' l

    126 The equation of the cone reciprocal to ax by2 cz =0 is

    1) a y z b z x c x y ~ O

    127 How many points are there on the paraboloid ax by2 =2z the normals drawn atwhich pass through a given point (n, y ?q , ., ax + by2 ~ 2z 'R f4;

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    l lP 217 1

    129. A plane passes through a fixed point a h, c and cuts the axes n A, B, C. The locus ofthe centre of the sphere OABC is1( 0 " I'I1Rl 1( 0 fu a, b, c) j \ t o ,Pit -.it ~ 3 T A B, C i -.mIT t I l taOABC * M: "" t

    bc(I) =1x y z3) a x ~ b y c z = 1

    bc(2) =2x y z(4) ax by cz=2

    130. The director sphere of the central conicoid ax by cz2 1 is

    131, The generators of the hyperboloidtacosa,bsina,O are

    ( I] x a co s a y bs ina z=acosn b sino. c

    (3) x ac osn y bs ina z=-as ina b cos a c

    (254) 35

    222(2) x y z = - bc2 2 2 1 1 14) x y z = - b c

    which pass through the point

    (2) x a c o s a y bs ina z=asina b e o sn c

    (4) x-acosex y bs ina. z= =asina c bcosa

    P.T.O.)

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    l lP/217/l

    132. The number of paraboloids confocal with a given paraboloid and passing through agiven point is

    I) 2 (2) 3 (3) 4 4) I133. I fT is a linear transformation from a vector space into a vector space V, then [R Tl]o

    is equal to

    I) N T) (2) T)134. n an inner product space V F)

    ' "

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    137 If S is a subset of an inner product space V, then S.LU- is equal to

    1) S 2) S (3) SH 4) V

    138 The infinite series

    is convergent if

    (I p< 1

    1 - - - 3 4

    2) p l 3) P 1

    139 A function f is defmed n [0 11 as follows:

    (254)

    f xl = plq, when x is any non-zero rational number plq n its lowest terms andf{x) = 0 when x is irrational or O

    Then the Riemann integral of I in [ 0, I] is11 > '&'R 1,[0, I] iI t :

    J( x I = p q "'Ii x oN-i ' iI -.itf mJ: ' M'

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    l lP 217 1

    140. Iff { 1 when x is rational

    xl = -1 when is irrational" .hen I fIx) I x 1S equal to

    1) - b-a)

    f Ix ) = { 1 , "I'-1, 'ili

    (2) b-a) 3) 0 b - a4) 2

    141. The test for convergence of an alternating series was given by1) Cauchy 2) D'A1embert

    (1) 2) t\-3 R'

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    143. The value of curl tux u) iscur1 uxu) fiT 11R t

    --- - - ~ -')II uxcurl u-uxcurl u---+ ---+ ---) ---) -- ---) ---+ - )(2) vV)u- u V) v+(div v)u - d ivu ) v---? - - ---)--) ---)---) ---)03) v-V) u+ uV) v+ uxcurl u +uxcurl v---) ---lo --)0 ---)4) vV)u- u V) v

    144. The function f defmed by f x, y) =) x) +) y) is1) not continuous at 0,0)2) differentiable at (0, 0)3) continuous but not differentiable at 0,0)4) continuous as well as differentiable at 0,0)

    f x ,y)=)x)+)y) om 'IItmfur,.... f(I) (0,0) 'R 'It t(3) (0,0) 'R 3

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    147 The correct inequality for the modulus of the difference of two complex numbers z andZ2 is

    I) I z , - z 2 1 ~ l z d - l z 2 13) Iz,-z21';lzd-lz21

    148 The f u n c t i o n f z ) ~ l z I 2 is1) differentiable everywhere3) differentiabie at the origin only

    (2) Iz, -z21 >1 z,l +1 z214) I z , - z 2 1 ~ l z d IZ21

    2) differentiable nowhere4) differentiable at z and z i

    149 If LIF(t)) ~ f p ) . then Llt F(t)) is equal toL I F t ) ) ~ f p ) , iii Llt Flt)) iIUOf ~d(I) - f p)dp

    n (2) dp - ' f(p)

    150 { sin t} .he value of L 1L { t} l 1'I

    40

    3) (_I) f lpdp 4) (_I) d -', f lpdp

    4) coc [;)D/1(254)-210

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    1. m flrffi i\; 10 fi'r-rI: i\; 3 'i;< tt ~

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