maths qb - new.doc
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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