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



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.)



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.)



(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) =




(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.)



(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




(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




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.)



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.)



(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.)



( 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.)



(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.)



(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.)



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



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

7 7 . 2 7 7 . 4 R A 77 ‘5r1 t T T C W I 4c 34" R e 1910/PR " err 1 3r 5 ~ 0 I catch-) paY r7 v* we 4

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

F M T IT W te rl

er WrnT

ariwr

*cr I"(YrFffr ur*r rft7 4 " twe F ri wra

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

civil main mathematics_2

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