Departrncnt of Mathernatics

National Institute of Technology, Tiruchirappalli

l\1AIR34 Real Analysis and Partial Differential Equations

Class Test - I
Ti1ne: one hour
Maximum Marks: 20
~.\ ns,,·er all t he four questions . Each question carry 5 rnarks.

( 1 ) Shov., that the set of all rational numbers q such that q2 < 3 has no greatest member in it . Show
that the set of all rational number is countable.
OR
Sho\\i' tha t t here is a rational number between any two distinct real nun1bers . Show further t hat
the set of all irrational nun1bers is uncountable .

J-2') S how that ( -x ) ( -y) == xy for any two elements x, y in a field F . Prove all results used and
ind icate t he result/field axion1s used in each step. ,

ftJ Let X be a metric space and p E X . Define the following terms: neighbourhood of p, open set .
limit point , closed set . Show that a subset EC X is open if and only if Ee is closed.

1 / ~et X be a rnet ric space. Show that a closed subset F of a compact subset K C X is con1pact . If
!4 ) ~ is closed subset of X and I( is con1pact subset of X, what can you say about the compactness
of F n K? Justify your answer .
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An swer all the !1'!£,stions

)· \\lhu1 i., ,·om11c11satl'<I Scrnfronduclor? Dm·, its rcsi<ti,·ity al,u equal that of :111
intrinsh.· -;l'lni4..'(UldUl·tor? If nol , wh~ nol'!
t In consid ering Ill<' condu,·1h•it,· of metal, only l'lcrlron mnnmcnl is of im portance.
Why hole, un· nol cun,idrn•d'!
\Y. What an• the ,,ff,•cts of hii:h dnpini: and ,fri:cu,•r,oc)•?
,~ Find lh,· n ·,islh·it~· of ntrin,ic silicon dopc,I " it h th,• donor impurity In the rxlenl of
I in 10' a tom<. llsc >larulunl """"'' nf ~ i.
, S. Why iron impuritie, arr 11111 '"''d 10 mat-r N-typc Si'/
,{,. Ht"• lo l'limin:rh' the ri,e 1i11w dda~ (I,) :rnd ,torai:e tim•· ddnJ· (t.) m :m Nl'i\'
1ransistor'!
7. A uniform har of IJ p,· ,ilicon
II· ,.,,1, :,
cru"•wdional :11·,•a of I 111111 X Imm and a
length of I cm i, conncct,·d 10 ., !\' hall,·n ,nppl~ :rnd ,, rnrrying :r nirrrnl of 2m A
nt .\ IIOK
a) C ak ulalc the 1h..r111al eq uilibrium d cctrun arrrl huh: d,•11 ,ili,·, in the bar.
h ) Cakulatt the dopant concl•ntrntion prcsumin,t th:U only donors ,,ith an
ca ert:.)' h•,·t•I o rO. J-UlrV hdoo- the t·onJutuou tuu,d un· present.
/For si o'ic= 1.8 X 10 1' 1<-mJJ
'8. If Vo = 0.65 V :II l n= Jm,\ for :r 1e01pern1ur<' of 2~0 1'. d,•1rr111inc the diode vohugc at
lu = I mA for T= J 25°C.
9. An :'o/l'N trnnsh hrr ha,\ »r = O.X\' and rnlll'<'lu r curn·nt of I ,\. \\ h:tl \\oultl he the
,·alue of\1"1 for I, ~ Jllrn;\ a nd 5.-\.
Lp., obiain the n ,lu,•s of\' :md I in lig. I a,,u mini: lh c 1ll'OJ1 :H'ross the diode uf0.7\ '

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SUBJECT:ECPC 11 NETWORK ANALYSIS AND SYNTI JESIS
DA TE:28/08/2018 TIM E:4 PM TO 5 PM~

ASSESMENTI · t<.-rH ::. l ,.2_.JL

( 1). Obtain the Thevenin ' s equivalent circuit at the termina ls PQ in the network of Fig. (1' f3]
@. In the circuit of Fi~.2 the switch s_ is_ in position I till steady state condition~ a'.e reached
and then moved to 2. Fmd the energy d1ss1pated in the two resistors. Show that this 1s equal to
the energy stored in the capacitor before moving the switch. [3]
aii~d the values for these currents in the circuit ofFig.3 at t= Sms (a) iL , (b) ix_; (c) iy . [4]
/use the superposition theorm on-the circqit-shown in Fig.4 to find i. [4]
5. Use nodal analysis with tree-branch volt;~;~/ oJJ the circuit of Fig.5 to determine what value
ofV2 will cause v= =0 - -- __ ----7_./ -;/' [4]

6. Obtain the Delta connected equivalent circuit for the network of Fig.6 [2]
\ '

BA ,.,
s

-----<lQ
1A2 {iw' i 2000
1
lOOV::i: ·
1 30/V',
vu
I 1·
" - - - + -_ _ _J JOOµF
Fig.I Fig.2

soon 15 A

iWn I
jl
4 n.

t =0
zoon 6n
120mA f h
lo.zH
h ;- 12 n

Fig.3 Fig.4

- 3A /
" A (2+j 3)0
~
(2+j 16)0 B

/ sn (3 - ; 2)0
24 VJ_ -=- V2

0- 1 -0

Fig.5 Fig.6
 Semester: 3 Branch: ECE Section: B

Date : 28.08 .2018 Time : 09:30 a.m . to 10:30 a.m. Max. Mark s: 20

ECPClO: Signals and Systems

Answer ALL questions

/ suppose that {x, y, z} is a linearly independent subset of vector space V. Determine whether the
subset {x+2y, y+2z, z+2x} is a linearly independent or not. (2)

;('Find a basis and dimension ofV = {(x, y, z, u) E JR.4 / x - y- z = O, x + z - u = O} (1.5)

3. Find an orthonormal s~~ in R 3

c~ntainin~ J\ k_, p; ,. g:) . M-¾ 6
~ ~~
~ ~
et x = (I , I , I , I) and y = (I , I , -1, 0) be two vectors in R 4 Dete~ine two v~c~or: J! ! v' m· ·
- lR.4 such that (i) x = u + v , (ii) u is parallel toy and (iii) vis orthogonal toy (2.5)

-~ A periodic signal x(t) with a period 4 seconds is described over one fundamental period by

x(t) = 3-t for O:St :S 4

Plot the signal and find its trigonometric Fourier series representation. Using these Fourier series
coefficients, verify the Parseval ' s theorem. (4.5)

~ 2sin(700m)
(!J) Determine y(t) = x(t)*h(t), where x(t) = 2cos(500m) + 4cos(900m) and h(t) = -----'-
m
(5.5)

***END***
 J\AT10NAL l NSTrn rn : OF TFCIJNOLO(;Y, TfRl"CIIIRAPALU

ll E P ,\RTMENT OF FLECTIHlNIC'S AND C'Ol\1Mt IN I CAT IO i\ J,:;-,,.<; JNEE J{I ~(;

DEPT : ECE MAX :\1AJ<K~: 20
\'EAR / SEC : II - B DA TE : 27-08-2018
Sllll COllE : ECPC12 DURA TION : l fir.
SUB NAME : ELECTRODYNAMICS AND ELECTROMAGNKrJC WAVES
Answer all Questions :

La. Check the di\'ergence theorem using the functi on V= y2 x + (Zxy + z 2 )y + (Zyz)z and
the unit cube situated at the origin.(Ref figl) (0.5)
b. Suppose V=((2xz + 3y )y + (4yz )z . Check Stokes' theorem for the squa re su rface
2 2

show · 11 fig.2 (])

:z ,r-j__'
· - -----

1 -_:1 X \

*e ..i ?i-o · 2
c. Prove that the electric field as the negative gradient of potential. (0.5)
d. Calculate the electric field intensity of continuous distribution of charges. (1)
2. Using Gauss's law, find the electric field due to uniformly - charged
sphere. (2)
3. a. Find the expression for the capacitance of a spherical capacitor, with the outer shell
earthed and the inner shell is given +Q charge. The spherical shells are with radii 'a'
and 'b' respectively. (1)
b. Find the capacitance per unit length of along cylindrical capacitor which is made of
two coaxial cylindrical tubes of radii 'a ' and 'b' and given las the charge per unit length.
(1)
4. An inverted hemispherical bowl of radius R carries a uniform surface charge density
G. Find the potential difference between pole and the Centre. (2)
S. Calculate the energy and energy density in terms of field quantities. (2)
6. Explain the method of images to determine the electric potential. (2)
7. Define polarization and find the electric field produced due to a uniformly polarized
sphere. (2)
8. Explain: Multipole expansion of a dipole. (2)
ff.Write the relationship between susceptibility and dielectric constant. (1)
10. Derive the boundary conditions at the interface of two different dielectric mediums.(2)
 , ...-
BRANCH ElfCTRONJCS ANO COMMUN fCA HON EN G INHRtN G
nrlE: ECPC1 4 Digital C,rcuttc. an d Sy<,tem~
DATE : 24 -08 -1018 TIMf 2 10 Noon t o 1 JO P M MI\X MMH(5 .7 0

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.1 I k fo h .' .-rn.,h,~ d1,1.1x' l1.' 1m1c .111d d1 !! 11 ,1I c. 1!,! 11 ,1l c.

b I l('" m.rn , b 1h ,He 1'-·q111rcd t n 11.· p r1.' '-l' lll all the kc~<. Pll ., KC\ hn.trd ,.., •lh 10)(
(2)

,e ,\ \\ h,ll :u ,.: , omc n! the a<.h :1111.1gc, d1 µ 1tal ~~ '-!Clll'- u 1111p,1rrd tn t1naln1! .. ,c.tcm,')
t, I \ J' l.un the d1f1i..·1cn(c hct \\ ccn pn'1 tl\c IP!l lC arHi nc gat1H' ln!-! 1c (2 )

!kl \\ the (ll\.' lllt pf' a nc n adder using 4-h,t bmar::,. adder,
~- Pn.,, c step b~ step th e rul e of Boo lean algebra : (A -~B)(A !-(') J\ • RC (2)
4

-,
J· /\ 2-bit bi nan. · add er sum s nvo numbers, A I AO and B 180 to -, ·ield the unsigned
result Y2Y l YO . ,vhere the zero subscript indicates the least sign ificant bit (LS B)
~ \ 'rite down the truth table for the required outputs Y2 . Y I and YO .
,{\ _Usi ng a K map or otherwise. determine a si mplified produ ct of sum s expression for
Y~ and shov,, hov,1 the circuit can be implemented usi ng only NOR gates (of an!
number of inputs) . (3)

,S: Imple ment the following function with the best suitable decoder based
combinati onal circuit f(x1 ,x2.x3)=I( I,5,6,7) )'\ 11fi:_(( 11 1,,, S (2)

Explai n the non -degenerate and degenerate forms with suitable examples (2)
Assume that lhe inverter in the network below has a propa gati on delay of 5 ns and
the AND gate has a propagation delay of IO ns. Draw a timin g di agram for the
network sh owing X, Y. and Z. Assume that X is initially 0, Y is initially I. X
becomes l for 80 ns, and then X is O again . (2)

X
[''>,-)
~~ .. _
y

.I '
~"\

~#'
\I
I
,/
i
l z
0

/ Des ign the PLA Di agra m for the BCD to G ra y code com -c rtcr (2)
•

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