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7/27/2019 Civil Main Mathematics_2
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CS(M) EXAM2012
S I . No.F-DTN-M-NUIB
MATHEMATICS
Paper H
Time Allowed : Three Hours Maximum Marks : 300
INSTRUCTIONS
Each question is printed both in Hindi and in English.
Answers must be written in the medium specified in theAdmission Certificate issued to you, which must be
stated clearly on the cover of the answer-book in the
space provided for the purpose. No marks will be given
for the answers written in a medium other than that
specified in the Admission Certificate.
Candidates should attempt Question Nos. 1 and 5 which
are compulsory and any three of the remaining ques-
tions selecting at least one question from each Section.
Assume suitable data if considered necessary and
indicate the same clearly.
Symbols and notations carry usual meaning, unless
otherwise indicated.
All questions carry equal marks.
A graph sheet is attached to this question paper for use
by the candidate. The graph sheet is to be carefullydetached from the question paper and securely fastened
to the answer-book.
Important : Whenever a Question is being attempted,
all its parts/sub-parts must be attempted contiguously.
This means that before moving on to the next
Question to be attempted, candidates must finish
attempting all parts/sub-pans of the previous
Question atteMpted. This is to be strictly followed.Pages left blank in the answer-book are to be clearly
struck out in ink. Any answers that follow pages left
blank may not be given credit.
iszn9- :rc1 timn T i g 9-47-3v *P IS ga-gflsvl t1
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Section 'A'
1. (a) How many elements of order 2 are there in the
group of order 16 generated by a and b such
that the order of a is 8, the order of b is 2 and
-ibab = a
1.2
(b) Le t
0 if <1 n +1
1sin—, if — x 1
n +1
1
0,
f x > n
f,,(x).
Show that f„ (x) converges to a continuous
function but not uniformly.2(c) Show that the function defined by
{x
3 y 5 (x + iy)f (z) = x6 + y1 0 Z *°
0, z—0
is not analytic at the origin though it satisfies
Cauchy-Riemann equations at the origin. 12
F-DTN-M-NUIB Contd.)
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1. CO a aft b WRTwftie a1r 1 6 k figA ' ctre
2%ffm14)A tat, ke) ft aft4P.2*311Tba/;1 =iiierl21
0,ift x <
n +1
1sin, ffr., —LE X tS .1
x n + 1
1
0,a. x >—,
fn(x) =
T 4 6-47 k fn (x) tioo T ER ff1firiTiC a" rdT t,
t f t9 t;chtt i l iciI2(Ti) Tfri-4 7
x3 y 5 (x+ iyZ #0
f(z) =6 4' y10
0, z=
kr i rg r rEf f E F F 9 -fa'Ach 9*
4%14 W f .1ttet 1:R40 4 1 4 1 1 1 4 - 1 4 1 c h t u i 1 1 wli7
T h-K -r t2F-DTN-M-NUIB Contd.)
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(d) For each hour per day that Ashok studies
mathematics, it yields him 10 marks and for
each hour that he studies physics, it yields him
5 marks. He can study at most 14 hours a dayand he must get at least 40 marks in each.
Determine graphically how many hours a day
he should s tud y ma thema t ics and physics ea ch,
in order to maximize his marks ?2(e) Show that the series i —g nn6 is conver-
„=1(g+1
gent .22. (a) How many conjugacy classes does the per-
mutation group S5 of permutations 5 numbers
have ? Write down one element in each class(preferably in terms of cycles).5
+ )1)2I
,2f) # (0, 0)
x + y-
1 , if (x, y) = (0, 0).
af afShow that — and — exist at (0, 0) though
axyf(x, y) is not continuous at (0, 0).5
(c) Use Cauchy integral formula to evaluate
,3z
f where c is the circle I z I = 2.c (z +1)4
15
(b) Let f(x, y) =
F-DTN-M-NUIB Contd.)
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(r) Raft* 315N9. t t1 UUT h z 1 7 9. c l v k10 349: tt 6rtvir-dft k alum * raw t r i t -R 7 ‘It ich
5 a r e? it ftr t1 a1 t i a6 9-F-dfqff 36 7 4a trEw 1 4 E* 3TETR F T.( 9cht0 311T 3041 k7
VFWTE W il" 40 31- W 7 1 1 :1 c h t • 1 I 3 1 1-4"4/1 7 t I
1 , 1 4 1,1: Wm eF47 f* 3174 31-4 *r
3TiWItTTat (=IA *M7,gfa - R 9 ka4E r t : r .1 13 1 d afr 474 ErE4 qtra fr1 } - A E R . 1 1 < c i l
'a 2( T ) T 4 T 1-47 l*ft 1(5 31fitflTel I 12
tz=1I2. (W ) 5 *MA:T 5h4ti4 q H14 k19 S5 i
fitinn. a7161“?9** .047*MR
(4-d-c 6 1 4 o 1 5 1 . 7 1 n 4)5(w ) 4r9 011;
(x +):rq (x, Y )* (0, o)f(x, v2I
, trft (x, y ). (0, o).
'R 1 . I W 7 Wt
Tro,o)tmarritio3ax ay
ftx, tidd 71 t (0, 0) C R I53 z
4z,, < 1tz+0
Ents“th4TFA 9C 77,-O c rIz1 =2t I 15
F - D T N - M - N U I B Contd.)
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(d) Find the minimum distance of the line given
by the planes 3x +4y + 5z =7 and x — z= 9
from the origin, by the method of Lagrange's
multipliers.53. (a) Is the ideal generated by 2 and X in the poly-
nom ial ring z [X] of polynomials in a single
variable X w ith c oe fficie nts in the r ing of inte -gersz, a principal ideal ? Justify your answer.
1 5
(b) Le t f(x) be differentiable on [0. 1] such that
f(l) = f(0) = 0 and I f 2 (x) dx = 1. Prove that0
x f (x)f s(x) dx =
5
(c) Expand the function f(z)—•(z +1 )
1
(z +3)in
Laur e nt se r ie s val id for
(i) 1<lzi<3 (ii) izi>3 (iii) 0<lz+11<2
(iv) Izi<15(d) Evaluate by c ontour inte grat ion
/ = jdo
<1.0 I— 2acos0 + a'
2 ir
1 5
F-DTN-M-NUIB Contd.)
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(w) TM*3x+4y+5z=7311Kx-z=9*TRT4W
W , nivi4t*tivic a;ir yru, .dtioi
norpi
1 -t1 #1 177 I
5
3. (W) Tea Z*aciti ilultel a" 7W tr X
tittelg* 'S Ora q m q z [x] 4 2 3fR X qi ; 1 T T
f t c r i j u r w r a-l azrr ikEzr Irma *sln1 U ? 31 94
a- -dT*tra449eja 4frgi5(w ) 4r9. mt (x) 3rwg-ffizr [o,
f(1)=f(0)=031t( ff 2 (x)dx=1.Rintfrg0
fxf(x)1(x)dx =
50
(r) treq (z) =(z +1)
1
(z + 3)V,
(i) 1<lz1<3zI>3 (iii) 0<lz+11<2
(iv) Izl<1 * f A V, vilkl 4# 4 TriKI5(4) cw tiflichwf * fli irviari 4)177
2n
/- 5de
0 1-2acos0+ a a <11 5
F-DTN-M-NUIB Contd.)
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4. (a) Describe the maximal ideals in the ring of
Gaussian integers z [i] -= {a + bi I a, b E Z}.
20
(b) Give an example of a function f(x), that is
not Riemann integrable but I A.01 is Riemann
integrable. Justify.0(c) By the method of Vogel, determine an initial
basic feasible solution for the following trans-
portation problem :P r o d u c t s P l. P2, P3 and P4 have to be sent to
dest ina t ions D I , D2 and D3. The cost of send-
ing product P i to destinations Di is C, where
the matrix
1 C 0 [
10
7
0
0
3
11
15
6
9
5 -
15
13
The to al requirements of destinations D1 , D2
a nd D3 are given by 45, 45, 95 respectively
and the availability of the products P1 , P2 , P3
a nd P4 are respectively 25, 35, 55 and 70.20
Se c tion 'IV
5 . (a) Solve the partial differential equation
( D — 2 E 1 ) (D — D 1 2 z = e x" •
2
Use Newton-Raphson method to find the realroot of the equation 3x = cos x + 1 correct to
four decimal places.2F-DTN-M-NUIB Contd.)
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4. (T) 4[74)4 9JTr1
r Eij={a+bila, be z}a l4P4T3
TIAN mg' W tire! Q1 77 I0
(TO th-9. f(x)t* (7" er47,
'+&
towboi471 * 31-ff 6kr4v, I0(it) f;ImPaci tirt-479. f14-Itql*r97, a*Th fa*
4i 61t1, ar t-kr4 al-raft 4rt4iuftur
Teud P1 , P2 , P3 .4 011dan D I , D2 3 1 1 7
D3 ffT 47- 1. t I d c 4 - 1 1 q Pidc4lDi dctMa
fr e lliic t 1 -1-7
[10 0 15 5
[Cul= 7 3 6 15
0 11 9 13
1i-aft DI , D2 AK D3 Q 1 1 W 3 q o h d I s b 4 - P T :
45, 45, 95 t, an-t 3?:fTd P i , /31, P3 A t< P4 ft
71-di sto-RT: 25, 35, 55 3rF 70 t I
0
07
5. (T) slifiTT 31 7-*-ff tili c htul
(D - 1)1 )(D - D92 z = e A ± YiA 7 I2(V) titilcbtui 3x = cos x + 1 W, q r< T414-Icti PiT4
ffTaltdkchr-ria9-
kcFr9 fdfF7 W T T44[6- QN-7 I2F-DTN-M-NUIB Contd.)
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(c) Provide a computer algorithm to solve an
o rd ina ry d if fe ren tia l equa t io n C I --
) indx
the interval [a, b] for n number of discrete
points, where the initial value is y(a) = a.using E ule r 's method .2
(d) Obta in the eq ua t io ns go ve rning the m o t io n o fa spher ica l pend ulum.2
(e) A r ig id sphere o f rad ius a is p la ced in a st reamo f f lu id w ho se velo ci ty in the und isturbed stateisV. Determ ine the ve loc ity o f the f luid a t a nypo in t o f the d is turbed s trea m.2
6. (a) Solve the partial differential equationpx + qy = 3 z .0
(b) A string of length I is fixed at its ends. Thestring from the mid-point is pulled up to aheight k and then re lea sed f rom res t. Find thedef lect ion y(x, t) of the vibrating string. 20
(c) Solve the following system of simultaneous
equations, using Gauss-Seidel iterative
m etho d :
3 x + 20y — z = — 18
20x + y — 2z = 17
2 x — 3 y + 20 z = 2 5 .0dy
7. (a) Find dx at x = 0.1 from the following data :
x : 0.1 0.2 03 0.4y 0.9975 0.9900 0.9776 0.9604
20
F-DTN-M-NUIB 10 (Contd.)
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( I T ) * I T E R I I T 4 4 W-4 4i4kbelif(x•y) W ‘ T s i c k i + 1
dx
[a, hi 4, siticin n & LITe t e l l ,
OITITR etKi F,b c ; ir w
FTt4hIcur 5.7 4tr-47,mtrITT 41-4
Y(d) = a t I2(IT)ThAtIl• 410*4WW°T q,4r4
t 1 + 1 1 ,mu1 in Sim of g I2() 1 ;1 4v-ti a k4;n v i. T R T
4 K w r, 144-4r thcrea anwr 4 44 vtuff tw i R r €44pu Erru Phki tr r k i r z f i r
ccir4r; I26.id a lifiTW 3r4W 7 Iikac htut1q
ft*:px+9Y =3z0(T4) ei“twrrt 3 T 9 4a il trr itrr Fr t
,1 t4 w a r 1tict:i "Et 41,11 T9T t 1 3 4 1 3I 41 q fit19voir itirt 4,4iii art W T R *c irIT y(x. t) 4 1 1 e k e r
CAI04 1 1 ti-MaTc hrfiT 4 7 1 T 4 1 1 1 7 . 4 - A . Z i
trg, Cr-47
3x + 20y z = -1S
20x + y – 2z = 17
2x – 3y + 20z = 25.0
d7. (W ) 1 4 - 1 ti t q c r 3 riT-at — } X = 0'1 Hier f :
dx
x : 0.1 0.2 0.3 0.4y : 0.9975 0.9900 0.9776 0.9604
20
F-DTN-M-NUIB 1 1 (Contd.)
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(b) The edge r = a of a circular plate is kept
at temperature f(0). The plate is insulated so
that there is no loss of heat from either surface.
Find the temperature distribution in steadystate.0
(c) In a certain examination, a candidate has to
appear for one major and two minor subjects.
The rules for declaration of results are : marks
fo r m ajo r a r e de no te d by M 1 and for minors byand M 3 . If the candidate obtains 75% and
above marks in each of the three subjects, the
candidate is declared to have passed the
examination in first class with distinction. If
the candidate obtains 60% and above marks in
each of the three subjects, the candidate is
declared to have passed the examination in
first class. If the candidate obtains 50% or
above in major, 40% or above in each of the
two minors and an average of 50% or above in
all the three subjects put together, the
candidate is declared to have passed the
examination in second class. All thosecandidates, who have obtained 50% and above
in major and 40% or above in minor, are
declared to have passed the examination. If the
candidate obtains less than 50% in major or
less than 40% in any one of the two minors,
the candidate is declared to have failed in theexaminations. Draw a flow chart to declare the
results for the above.0F-DTN-M-NUIB2Contd.)
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(ti)thcbit OF* R71k r = a 71917 f(0)
Tur.ticti t tTratheurftft4 )-*5 - 4ditml fl
r r a ft 4 4 K r4 a-Fr fin
Hit{ ftr- 7 10(r) s T w 1 4 -1 4 riew 4, d 4 4 - 1 4 ,4k07ilt 311-c 4 )
T r t z-9T ksitiifthir01 t 14Ptuil4l ft
4 -1tiurr
laft2 3 1t( M 3 k gic i N1-0
f*7t I LIR14 -4 1q1i* d ell)* f iat i l 4
75% iff 39* allt aiT7 7 1 ch*ci l t th 3fra
'7 1 W f% 1 19.' E l )-P ra. t I zrfi .14-9 q1K1 1 1 fi44): 4 60% AT
t .14 -4 -nqc i I t3-4 -Pr 4 E 4 )4 gm' shiErf f
kW AM i t 14 k ,14-1-+1qc,K f t 4-K 4 5 0 % 7 1
7 1-04 40% ?Tr 6,4k
3 frc 4 w) P Ii t i) 47 rr 3 11F r a - 4 50% t ir no t
91Fr ill al 3ft0 'R ag 4rT4 4 r t rFr
thfia- -kg' 4 1 1 [ 1 1 t 139 7- f t ‘i4 -1- 1 1 q 1 1,tlkzlt 50% zIT thtt 3fr 'art' Tet k 4 0% 71 thmt
ifTtd t `7 1f r ' te t ra ' 14 .Ln 4 44)qcM. 7 1 C 4 50% 4 air RA A H iZ .fl *
40% 4 W IT 3 t- 91'7l 3fr i++ilG glt 33
' -9) 4 )17c 171:TT3745 trfrurrz4
thurrR 7 1;4 . 7 7 M "sNiV" I0F-DTN-M-NUIB3Contd.)
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8. (T) STW ci pi* 2a visit{ SW UT34gnAi m
qm. pir t .1tA crwr( knii a/3 ai/K
%04111 15 m P V Trucbk (itch's Tr Tairt I
K a m a )t§ca a m p f1 1 ; 1 q :
a ) u fl t ti tR r7 * *a 1 ,14 ta T O a f t QnsfLi k WO I5on9Y f f w
41aa«rnz arg
*0114 I
[AV : ei)eichm)*
* a/12 WTK1 I]5wi-17ft 0=x fir) srwaff i f t ruK9. 7A*kirfi14*sraiTqnttzrRNr-4.,44.kr r) 3Frftwalvtrik*tit
Icy Alf4 7 I0FDTN-M-NU1135
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F-DTN-M-NUIB
rfr
+il ulci
SPW-trl H
F i f r i r :0 2- :300
ar/hT
.7*c7 owt t
sR4 1 9i drl< 3 e 1 T rra 4 4 1 % 4 4 1 4 a tuir 37473r7trkr4V-47 r c h q / 7 / 7 1-t, zTian -r FlEr 34 (et,irt<1-(4-74 ER- 344fir4 ER-wr4rarr-67 9-h 4 r-tff cr-cRio grurrr arrufcff 3r4r ch ef
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air
77 *R- PR Y ,Irt( q7rit2T R- 3rmzrw ff) urrgw *W A: wr wq.etr77mud G I )
977 0 31It -7 4 1 4 4 3 r 2 fe n 3? -4,4 4 4,6i
;Fir e
?TNT PR I k 34- offir e
F r PR " c ff kmv,144/7gclijkarrzfrir07grcifft2 firlu
t Ipyrffi rqurqw& 9R if r/7 3 r 7 7 r TT um-Rrwr Fatr 317. c 1 < 5
rig • arm y - 6 t faszgs77fitam a •zir
urr spy f rwi. wrinjurr-griff re upz-frrar a:/
Frzwr ar2 f r(FgarT4 7$7 Th" T r er fq4 Srf47 3rr+Ta-4- Ti Pos4 stag S trrwt gur/3q-sTruY Sr Ter
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zifri1ffY aTd f r url<l 3rd 4 R7.44, FrrF7 emit t
Note : English version of the Instructions is printed onthe front cover of this question paper