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    PRIST UNIVERSITY (Estd. u/s 3 of UGC Act, 1956)

    V!!", T#$%&u' 613*3_______________________________________________________________________

    _

    +UESTIN -AN

    .T0c#. CUNICATIN SYSTES.T0c#. APP2IE E2ECTRNICS

    .T0c#. E-EE SYSTES

    Cou's0 0t4!s

    Cou's0 Cod0 T4t!0 1778S11-/1778S11-P APP2IEATEATICS

    :R E2ECTRNICS ENGINEERS

    R0;u!t4o$s 7*17 R0;u!t4o$

    (:o' Stud0$ts d"4tt0d f'o"

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    SY22A-US

    A==!40d t#0"t4cs fo' E!0ct'o$4cs E$;4$00'4$;

    (Common to Mtech programs in Applied Electronics, Communication systems engineering and embedded systems-Effective from 202 on!ards"

    U$4t I C!cu!us of &'4t4o$s

    #unctional $ Euler%s e&uation-'ariational problems involving one unno!n function-

    several unno!n functions-functional dependent on higher order derivatives-several

    independent variables-isoperimetric problems)

    U$4t II I$t0;'! t'$sfo'"s $d >&0 0?ut4o$s

    #ourier transform pairs, *roperties $ #ourier +ine and Cosine transforms, Convolution

    integrals, Evaluation of integrals using #ourier ransform)iscrete #ourier ransform-properties)

    Application of #ourier transform to !ave e&uation)

    .-transform-properties-inverse transform- solution to difference e&uation)

    U$4t III 24$0' P'o;'""4$;

    +imple/ algorithm-t!o phase method-duality-transportation and assignment problems-inventory-scheduling)

    U$4t IV R$do" ='oc0ss $d ?u0u4$; t#0o'@

    Classification $ auto correlation-cross correlation-ergodicity-po!er spectral densityfunction-*oisson process)

    +ingle and multiple server Marovian &ueuing models- customer impatience- &ueuing

    applications)

    U$4t V T0st4$; of #@=ot#0s4s

    +ampling distributions-esting of hypothesis of normal, t, chi s&uare, # distributions fortesting mean and variance- large sample test) Analysis of variance $ one !ay

    classification)

    R0f0'0$c0

    1. G'0>0!.-.S. 4;#0' E$;4$00'4$; t#0"t4csB, #$$ Pu!4ct4o$s, 7**5.

    7. =oo'.

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    U$4t I

    CA2CU2US : VARIATINS

    P'tA

    ) efine #unctional)

    2) ive the commutative character of the operators anddx

    d

    1) ive the Euler agrange differential e&uation)

    3) ive the necessary condition for the integral ( )dxxx

    yyxFI =2

    4,, to be stationary)

    5) 6rite do!n the second order Euler *oisson differential e&uation)

    7) 6rite do!n the 2 order Euler- *oisson differential e&uation)

    8) efine isoperimetric problem)

    P't-

    ) (i"#ind the e/tremals of the functional dxxx xyyyxyv

    += 2

    sin2

    242

    "9(: (;"

    (ii"ed? (;"

    2) #ind the path on !hich a particle in the absence of friction !ill slide from one point to

    another in the shortest time, under the action of gravity) (;"

    1) #ind the e/tremals of the functional ++=

    2

    0"2

    24

    24

    ("9(",(:

    dxyzzyxzxyvgiven that

    "2

    (,0"0(,"2

    (,0"0( ====

    zzyy ) (;"

    3) etermine the e/ternal of the functional

    +=a

    a

    dxyyxyI "

    244

    2

    ("9(: that satisfies the

    boundary conditions) 0"(,0"(,0"(,0"( 44 ==== ayayayay (;"

    5) 6rite the ostrogradsy e&uation for the functional

    dxdyy

    z

    x

    zyxzI

    D

    22

    "9,(:

    +

    = (;"

    7) *rove that the sphere is the solid figure of revolution !hich for a given surface area has

    ma/imum volume) (;"

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    U$4t II

    INTEGRA2 TRANS:RS AN AVE E+UATINS

    PART A

    1. efine #ourier transform pair)

    7. 6rite do!n the formula for # @f(/" cos a/)

    3. #ind the #ourier sine transform ofx

    )

    . #ind the #ourier sine transform ofxe )

    5. *rove that the #ourier sine transform of ,

    ""((

    =a

    ssF

    aaxf

    sF )

    6. #ind { })"1( nZ

    . #ind )

    n

    Z

    8. #ind

    Bn

    aZ

    n

    )

    9. f { } ",("( ZFnfZ = then prove that { } )"(

    =a

    zFnfaZ

    n

    1*. #ind nanZ "( )

    PART -

    11) (i" #ind the #ourier transform of xae and hence

    deduce that

    =

    +022 2

    cos xae

    adt

    ta

    xt ) (8)

    (ii" *rove that the #ourier ransform of2

    2x

    e

    is

    2

    2s

    e

    and deduce that

    )"( 2

    2

    2

    2sx

    eisexF

    =

    (8)

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    11. (i" #ind the #ourier series for the function

    "2,0()2"("( lrangetheinxlxf = and deduce that

    =2)

    n n

    (8)

    (ii"

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    U$4tIII

    2INEAR PRGRAING PR-2E

    P'tA

    ) 6rite the follo!ing linear programming model in the standard form

    Ma/imi>e 12 312 xxxz ++=

    +ubect to) 512 >++ xxx

    82 2 =+ xx

    D125 12 e 12 325 xxxz ++=

    +ubect to) 02 12 ++ xxx

    ;12 12 =+ xxx

    0,, 12 xxx )5) efine ransportation problem and state the different methods available for determining

    the F#+)

    7) Gse Horth 6est corner method to obtain the initial basic feasible solution of a

    transportation problem !hose cost, supply and demand is given belo!)

    Origin/Destination D1 D2 D3 Supply

    O1 2 7 4 5

    O2 3 3 1 8O3 5 4 7 7

    O4 1 6 2 14

    Demand 7 9 18

    8) istinguish bet!een transportation model and assignment model)

    ;) +olve the assignment model in

    20

    23

    52

    by Iungarian method)

    D) efine +et $up cost)

    0) efine se&uencing problems)

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    P't-

    ) Gse simple/ method to solve the **

    Ma/imi>e 2 03 xxz +=

    +ubect to 502 2 +xx

    0052 2 + xx

    D012 2 + xx

    0,, 2 xx )2) +olve the follo!ing ** by using t!o-phase method)

    Minimi>e 2 75 xxz +=

    +ubect to 50052 2 + xx

    2001 2 +xx

    0,, 2 xx )

    1) +olve the follo!ing transportation problem to minimi>e the cost of transportation byusing M

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    7) (a" here are five obs, each of !hich must go through the t!o machines A and F in the

    order AF)

    *rocessing times are given belo!)

    e the total elapsed time)

    (b" escribe the method of processing n obs through t!o machines)

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    UNIT IV

    RAN PRCESS AN +UEUING TERY

    PART A

    ) efine Jandom processes

    2) efine Auto Correlation

    1) efine +tationary processes

    3) *rove that "(ZRXX is an even function of )

    5) #ind the mean s&uare value of the random processes !hose auto correlation

    is cos2

    2A

    )

    7) efine Cross correlation)

    8) efine *o!er spectral density function

    ;) #ind the Auto correlation function of the process { }"(tX

    if

    ero mean of !ss process

    @N(t" is given by