IAS Mains Mathematics 2012

7/29/2019 IAS Mains Mathematics 2012

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12-S-DTN-M-NUIA

MATHEMATICSPaperI

Time Allowed : Three Hours Maximum Marks : 300

INSTRUCTIONSEach 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 statedclearly on the cover of the answer-book in the spaceprovided for the purpose. No marks will be given for theanswers written in a medium other than that specified inthe Admission Certificate.Candidates should attempt Question Nos. I and 5, whichare compulsory, and any three of the remaining questionsselecting at least one question from each Section.The number of marks carried by each question is indicatedat the end of the question.Assume suitable data if considered necessary and indicatethe same clearly.Symbols/ notations carry their usual meanings, unlessotherwise indicated.Important Note : Whenever a question is being attempted,all its parts/sub-parts must be attempted contiguously. Thismeans that before moving on to the next question to beattempted, candidates must finish attempting all parts/sub-parts of the previous question attempted. This is to bestrictly followed.Pages left blank in the answer-book are to be clearly struckout in ink. Any answers that follow pages left blank may notbe given credit.arrq : arct f61ko4i-d(rP-qq 3 RUA tptv Oil I


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SectionA

1. (a) Define a function f of two real variablesin the xy-plane by

f (x,

Check

1X 3 COS- +y3cos-yor x, y # 0X 2 +y2

0 , otherwisethe continuity and differentia-

bility of f at (0, 0). 12(b) Let p and q be positive real numbers

such that -+ -1 = 1. Show that for realPnumbers a, b

at'qab


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1. () xy-k-istic-t A z1 c r i t . c f4 (h--71x3 cos + y3cos-y

f (x, y) = x2 + y2

0 , ST R2llgRI ER 1 4 1141 tr771 (0, 0) ER f11cfK r atRmar 414*tr-A 72

(4) 147 .ati-7 p afiq kit UM04T aihiract ,I FT 1+ 1= 1 tiTte-47 fa 11*-ctra t f ieA ral lPa baft +ti2P

(A ) r a1 ( 1fH.5, tlf-A apra r 38chf taunt12

arz R 5M q B =2 , b3 , b4 , b5}'A tmfR V kch t-fcRfief .314+ILIFE f R 5 2 ,

a1 v~r- T R I T i 6-1411q1 B kda FRF11144-4t pn 6111

(A) T iR:R3 > R 3 1,a, 1171-t41-cR t,T(a, 13, y) = (a +213-3y, 2a +513-4y, a +4(3+y)

1 - T 1 E r F o r r r E m t I TT NSfedr411antiR atR lam f m 1I2

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(e ) Prove that two of the straight linesrepresented by the equation

x3+ bx2 y + cxy2 +y3 =0will be at right angles, if b + c = -2.2

2. (a) ( 1 ) Let V be the vector space of all2 x 2 matrices over the field ofreal numbers. Let W be the setconsisting of all matrices with zerodeterminant. Is W a subspace ofV ?Justify your answer.

(ii) Find the dimension and a basis forthe space W of all solutions ofthe following homogeneous systemusing matrix notation :2

x +2x2 +3x3 -2x4 +4x5 =02x +4x2 +8x3+x4 +9x5 =0

3x +6x2 +13x3+4x4 +14x5 =0Consider the linear mapping

f : I1 2 2 -)1 R 2 byf tx,3x+4y, 2x-5y)Find the matrix A relative to thebasis {(1, 0), (0, 1)} and the matrix Brelative to the basis ((1, 2), (2, 3)}.2

(ii) Ifs a characteristic root ofa non-singular matrix A , thenprove that 1 is a characteristicroot of Adj A.

F-DTN-M-NUIA/ 15


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r f f -4 c=re--A 7 I t ei-ntorx3 +bx2y+cxy2 +y3 =0

slu fRecd ate iurail A A14-ich)uil E R tril ,gCb+c=-2t122 ()hf-Ag v qR.ciPich titseiraA 9 4-q tr{

Aifi 2 x2 all&kt r2r TAU wilt t I AR#11A 7 W 7F :1 T I T T R IT 1 :1 F-0 P'41 alF & Ochitl Flt WRT-t347I aci{RT T Z cici, tr-7IPiHRiRdd HHVId fiTh-P 4ift 5(1. *arfz W31T&O 1:47qr r Faxc.qf - 1 4 4 (ia , 34M17 old 4r t r-:2

x1 +2x2 +3x3 -2x4 +4x5 =02x +4x2 +8x3+x4 +9x5 =0

3x1 +6x2 +13x3+4x4 +14x5 = 0

(13) (i) f (x, y) =(3x +4y, 2x -5y) 5,10 WINT:R2 ,R2 tft .ft-R 11-71311T T R {(1 , 0) , (0 , 1))ltsga aii

3-TTVII )(1, 2), (2, 3)}711*:tra34r & 30 B2

(ii)a ?c oeirsto-iung 3-117 A31-NuRTrui-F er t, a. fay tir-4II1;41i f i T F I euT TAdj A T I

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(c)et (1 i 2 + i`H= 2 1-

2 - 1 + i 2be a Hermitian matrix. Find a non-singular matrix P such that D = PT HPis diagonal.

3. (a) Find the points of local extrema andsaddle points of the function f of twovariables defined by

f (x, y) = x3+y3- 63(x + y) + 12xy

(b) Define a sequence s of real numbersby(log(n + i)- og n)2 E

Does lim sn exist? If so, compute then cc

value of this limit and justify youranswer.0Find all the real values of p and qso that the integral To X P (log )1 3 dxconverges.0

4. (a) Compute the volume of the solidenclosed between the surfacesx2 +y2 =9 and x2 +z2 =9.02020=1+ ii

F-DTN-M-NUIA/ 15


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T rR1+ i

H=1i2 i 1+ i 2

W f )-11W t I la'=1c-',n+iuflil 341e Pv c r+ D = PT HP f aT cr I f t I0

3. () (x, y)= x3 + y3 63(x + y) + 12xyUt rforrrtm% P-utrzr tik143341i& -w irt f-4T310

(15) snlog(n + i) log n)2i+ iSi trreurm T E r riim sn fir 31 t? ifa tHr %, n T t i l m rn1R W f T 4-kW-RirftRt 3U4 4711 VET A c14,sR7 trft7ipitt aR.Ach4 H f l T h ' rW7,WftP (log+)q dx 3 1 R Y W u T a ,20

4. () '71 X +y2 =9 344 X +Z2 =9tifcc d111 3uzr-d4 a qft-*-d706,1114 1 - 1 1 20

FDTN MN UIA / 15P.T.O.


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(b) A variable plane is parallel to the planex yz- + - + - = 0a b c

and meets the axes in A, B, Crespectively. Prove that the circle ABClies on the coneyz(b + c) + zx( c + a) +xy(a +b) 0

1c b)a c ) a)(c) Show that the locus of a point from

which the three mutually perpendiculartangent lines can be drawn to theparaboloid x2 + y2 +2z = 0 is

X2 + y2 + 4z =1

SectionB

5. (a) Solvedy _.xyeklY) 2dx y(1+ ek/Y1 2 )+2x2 e(x /Y ) 2

(b) Find the orthogonal trajectories of thefamily of curves x2 + y2 = ax.2

(c) Using Laplace transforms, solve theinitial value problem

y" +2y' + y =e , (0) = -L y'(0) =12202012F-DTN-M-NUIA/15


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0 1.,ich 97 k-1 4 4c1c4 3774 1-14(4X yza b c

9J-11-4{ t 3717 Shi-Pf: A, B, C14 3 4 0 1 ? 1 fA9-1t I -r frd ABC, *T

yz bc+ ZX c + a--) + Xy[2+-)=0,c b c) aT R Prd t

01) T1 1s -F T kchx2 +y2 +2z = 0 i A17 1: 1 1 1 : 7 M W V A1 - 1 7 7 1 T 44M1 i%T fq3794 x2 +y2 +4z=1t I

isiug-17

5. () 6ci tINR :2dy _xy , 2dx y(1 + e (x/Y ) 2 ) + 2x2 e( x /Y ) 2

ash-TO x2 + y2 = ax 4 5149)tfiR12M P R1FEITall airu rnT T H I

y" +2y' + y = e-t , y(0) = 1, 00)=1Qcfl21rida77 2020F-DTN-M-NUIA/ 15P.T.O.


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(d) A particle moves with an accelerationa4X + x3

towards the origin. If it starts from restat a distance a from the origin, find itsvelocity when its distance from theorigin is a.2

(e) If A . x 2 yd -2xz3+x21 7B =2zi +y.) - x2-)k

a2find the value ofA xS)at (1, 0, -2).axay6. (a) Show that the differential equation

+ y2 ,62 ) dy_ 0(2xylog y) cb c + (x2is not exact. Find an integrating factorand hence, the solution of the equation. 20

(b) Find the general solution of theequation y-- y" = 12x2 + 6x.0

(c) Solve the ordinary differential equation

x(x -1)y" - (2x-1)y' +2y = x2 (2x -3)011 1212

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3/4 14 194 3T1TI 4 34)1*kat t I qC 3/4gf3/4r1rITTPan 3714340 a14 -4T*A, z7r],u4

tru34r-Ri22-)A = x 2 yzi -2xz3 j + xz2 k

-B=2z +y-xk(1, 0,-) c2 (A xB)1 7 1t ic i

ax ay*r*i26. (T) -T 4 1 15 7 f@ F 313/4*@ f11 4 , 11 1 1

(2xy log y) eh( + (x2 +y2 il2 dy 07rI1u11 1 04 Iju iT ;11 c( TherNg 37I131-R a (-14 -11*tul *1 6c.-10.f4414b(Iff y"' -y"=i2x2 +6 x wr 94 1 4 4 ,r

I0(10 14 11 -173131 3/4*-F (4414ntu1

x(x -1)y" - (2x -1)y' + 2y = x2 (2x -3)Fr irtNR0

F-DTN-M-NUIA/ 151.T.O.


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7. (a) A heavy ring of mass m, slides on asmooth vertical rod and is attached to alight string which passes over a smallpulley distant a from the rod and has amass M (> m) fastened to its other end.Show that if the ring be dropped from apoint in the rod in the same horizontalplane as the pulley, it will descend a2distance Mefore coming to rest.Ai 2 _mam2 20

(b) A heavy hemispherical shell of radius ahas a particle attached to a point on therim, and rests with the curved surfacein contact with a rough sphere ofradius bat the highest point. Prove thatif -b > f -1, the equilibrium is stable,

awhatever be the weight of the particle. 20(c) The end links of a uniform chain slide

along a fixed rough horizontal rod. Provethat the ratio of the maximum span tothe length of the chain is

141+pwhere p is the coefficient of friction.0

8. (a) Derive the Frenet-Serret formulae.Define the curvature and torsion for aspace curve. Compute them for thespace curve

x = t, y = t 2 , z =2t33Show that the curvature and torsion areequal for this curve.0p. log

F-DTN-M-NUIA/152


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l ' h LicaITN 4-ccidi4 fa IR 111:4 43 4 A 4 4 2 1 - ) TP3 ITT &CM PO I UlI*- >1 t, al chur aT 4R -=QaA rnuraTar P-Trzit et M I0kq' -Q,44,414 taFT (A) 3/4 i4 3/4144 -7 3 / 4arra w-ct 4Faara-Ern A t3/4att % I fa .G4lf-13/4 *T- /4 a4 fates T 1

23111TIM glog 1+11+,tl

8. () z-tllz WltA 1:M * 1 1 A 1 3 1 7 *- 8ncir 3117 faAlaq tribute Stra7 3A44

x = t,y = t2 , z = t34 +4.4 c ftm--d4*tri zifF f3/43 / 4 3 / 4

fog , nit $t fdttlaq WkIWl t I

20

(g)

(T)

20

20

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(b) Verify Green's theorem in the plane forfc((xy + y2 ) dx + x2 dy)

where C is the closed curve of the regionbounded by y = x and y = x2 .0

(c) If F = y i + (x - 2 xz) j -xyk, evaluateJJ(V x )

where S is the surface of the spherex2 + y2 +z2 = a2 above the xy-plane.0

F-DTN-M-NUIA/ 154


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(N) fc[(xy +y2 ) dx + x2 dy]- 1 O slim W I F ictlit14 tfm7, MCk .iqd ash t, y = x31'1( y = x 2 flutra4 TaTr m- I0

-(7 )F = y i + (x -2xz)j - xyk, MxF) n ci-'

4 - 1 1 - 1 1 c hix2 +y2 +z2 = a2 I xy-(44ciei .3)4(t(tl0

* * *

F-DTN-M-NUIA/155S3-1100


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F'DTNMNUIA

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TIT : 300

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CS(M) EXAM2012S I . No. F-DTN-M-NUIB

MATHEMATICSPaper H

Time Allowed : Three Hours Maximum Marks : 300INSTRUCTIONS

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 bestated clearly on the cover of the answer-book in thespace provided for the purpose. No marks will be givenfor the answers written in a medium other than thatspecified in the Admission Certificate.Candidates should attempt Question Nos. 1 and 5 whichare compulsory and any three of the remaining ques-tions selecting at least one question from each Section.Assume suitable data if considered necessary andindicate the same clearly.Symbols and notations carry usual meaning, unlessotherwise indicated.All questions carry equal marks.A graph sheet is attached to this question paper for useby the candidate. The graph sheet is to be carefullydetached from the question paper and securely fastenedto 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 nextQuestion to be attempted, candidates must finishattempting all parts/sub-pans of the previousQuestion atteMpted. This is to be strictly followed.Pages left blank in the answer-book are to be clearlystruck out in ink. Any answers that follow pages leftblank 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 thegroup of order 16 generated by a and b suchthat the order of a is 8, the order of b is 2 and-ibab = a 1 .2

(b) Le t0 if nf,,(x).Show that f (x) converges to a continuousfunction but not uniformly.2

(c) Show that the function defined by

{x3 y 5 (x + iy)f (z) = x6 + y1 0 Z *0, z0

is not analytic at the origin though it satisfiesCauchy-Riemann equations at the origin. 12

F-DTN-M-NUIBContd.)


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1. CO a aft b WRTwftie a1r 1 6 k figA ' ctre2%ffm14)A tat, ke) ft aft4P.2*311Tba/;1 =iiierl2

1

0,ift x < n +1

1sin, ffr., LE X tS .1x n + 11

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 7x3 y 5 (x+ iy

Z #0f(z) =6 4' y100, 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 wli7T h-K -r t2

F-DTN-M-NUIBContd.)


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(d) For each hour per day that Ashok studiesmathematics, it yields him 10 marks and foreach hour that he studies physics, it yields him5 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 dayhe 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+1gent .2

2. (a) How many conjugacy classes does the per-mutation group S5 of permutations 5 numbershave ? Write down one element in each class(preferably in terms of cycles).5

+ )1)2 I,2f) # (0, 0)x + y-1 , if (x, y) = (0, 0).

af afShow that and exist at (0, 0) thoughaxyf(x, y) is not continuous at (0, 0).5

(c) Use Cauchy integral formula to evaluate,3zf where c is the circle I z I = 2.c (z +1)4

15

(b) Let f(x, y) =

F-DTN-M-NUIBContd.)


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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 ich5 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 k7VFWTE W il" 40 31- W 7 1 1 :1 c h t 1 I 3 1 1-4"4/1 7 t I1 , 1 4 1,1: Wm eF47 f* 3174 31-4*r3TiWItTTat (=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'a2

( T ) T 4 T 1-47 l*ft 1(5 31fitflTel I 12tz=1I

2. (W ) 5 *MA:T 5h4ti4 q H14 k19 S5 ifitinn. 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, v2

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3 z4z,, < 1tz+0Entsth4TFA 9C 77,-O c rIz1 =2t I 15

F - D T N - M - N U I BContd.)


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(d) Find the minimum distance of the line givenby the planes 3x +4y + 5z =7 and x z= 9from the origin, by the method of Lagrange'smultipliers.5

3. (a) Is the ideal generated by 2 and X in the poly-nom ial ring z [X] of polynomials in a singlevariable 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 that0x 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


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(w) TM*3x+4y+5z=7311Kx-z=9*TRT4WW , nivi4t*tivic a;ir yru, .dtioi

norpi1 -t1 #1 177 I53. (W) Tea Z*aciti ilultel a" 7W tr Xtittelg* '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 94a- -dT*tra449eja 4frgi5

(w ) 4r9. mt (x) 3rwg-ffizr [o,f(1)=f(0)=031t( ff 2 (x)dx=1.Rintfrg

0

fxf(x)1(x)dx =50(r) treq (z) = (z +1)

1(z + 3) V,

(i) 1


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4. (a) Describe the maximal ideals in the ring ofGaussian 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 Riemannintegrable. Justify.0

(c) By the method of Vogel, determine an initialbasic 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 todest ina t ions D I , D2 and D3. The cost of send-ing product P i to destinations Di is C, wherethe matrix

1 C 0 [1070

03

11

1569

5 -1513

The to al requirements of destinations D1 , D2a nd D3 are given by 45, 45, 95 respectivelyand the availability of the products P1 , P2 , P3a nd P4 are respectively 25, 35, 55 and 70.20

Se c tion 'IV5 . (a) Solve the partial differential equation

( D 2 E 1 ) (D D 1 2 z = e x" 2Use Newton-Raphson method to find the realroot of the equation 3x = cos x + 1 correct tofour decimal places.2

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4. (T) 4[74)4 9JTr1r Eij={a+bila, be z}a l4P4T3TIAN mg' W tire! Q1 77 I0(TO th-9. f(x)t* (7" er47,

'+&towboi471 * 31-ff 6kr4v, I0

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[10 0 15 5[Cul= 7 3 6 15

0 11 9 131i-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 ft71-di sto-RT: 25, 35, 55 3rF 70 t I0

075. (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 I2

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(c) Provide a computer algorithm to solve ano rd ina ry d if fe ren tia l equa t io n C I --) indxthe interval [a, b] for n number of discretepoints, 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 iterativem etho d :3 x + 20y z = 1820x + y 2z = 172 x 3 y + 20 z = 2 5 .0dy7. (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 0.960420F-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 + 1dx[a, hi 4, siticin n & LITe t e l l ,

OITITR etKi F,b c ; ir wFTt4hIcur 5.7 4tr-47,mtrITT 41-4Y(d) = a t I2(IT)ThAtIl 410*4WWT q,4r4t 1 + 1 1 ,mu1 in Sim of g I2() 1 ;1 4v-ti a k4;n v i. T R T4 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 rccir4r; I2

6.id a lifiTW 3r4W 7 Iikac htut1qft*:px+9Y =3z0(T4) eitwrrt 3 T 9 4a il trr itrr Fr t,1 t4 w a r1tict: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 rCAI04 1 1 ti-MaTc hrfiT 4 7 1 T 4 1 1 1 7 . 4 - A . Z itrg,Cr-473x + 20y z = -1S20x + 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 :dxx : 0.1 0.2 0.3 0.4y : 0.9975 0.9900 0.9776 0.960420

F-DTN-M-NUIB 1 1 (Contd.)


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(b) The edge r = a of a circular plate is keptat temperature f(0). The plate is insulated sothat there is no loss of heat from either surface.Find the temperature distribution in steadystate.0

(c) In a certain examination, a candidate has toappear for one major and two minor subjects.The rules for declaration of results are : marksfo 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% andabove marks in each of the three subjects, thecandidate is declared to have passed theexamination in first class with distinction. Ifthe candidate obtains 60% and above marks ineach of the three subjects, the candidate isdeclared to have passed the examination infirst class. If the candidate obtains 50% orabove in major, 40% or above in each of thetwo minors and an average of 50% or above inall the three subjects put together, thecandidate is declared to have passed theexamination in second class. All thosecandidates, who have obtained 50% and abovein major and 40% or above in minor, aredeclared to have passed the examination. If thecandidate obtains less than 50% in major orless than 40% in any one of the two minors,the candidate is declared to have failed in theexaminations. Draw a flow chart to declare theresults for the above.0

F-DTN-M-NUIB2Contd.)


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(ti)thcbit OF* R71k r = a 71917 f(0)Tur.ticti t tTratheurftft4 )-*5 - 4ditml flr r a ft 4 4 K r 4 a-Fr finHit{ 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 ft4 -1tiurrlaft2 3 1t( M 3 k gic i N1-0f*7t I LIR14 -4 1q1i* d ell)* f iat i l 475% 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 fkW AM i t 14 k ,14-1-+1qc,K f t 4-K 4 5 0 % 7 1

7 1-04 40% ?Tr 6,4k3 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 rFrthfia- -kg' 4 1 1 [ 1 1 t 1397- f t i4 -1- 1 1 q 1 1,tlkzlt 50% zIT thtt 3fr 'art' Tet k 4 0% 71 thmtifTtd 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 trfrurrz4thurrR 7 1;4 . 7 7 M "sNiV" I0

F-DTN-M-NUIB3Contd.)


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8. (a) A pendulum consists of a rod of length 2a andmass in; to one end of which a spherical bob ofr ad iu s a / 3 an d ma s s 15 m i s a t tac h e d . F in d themoment of inertia of the pendulum :(i) abo ut an ax is thro ugh the o the r end o f the

rod and at right angles to the rod.5(ii) a bou t a pa ra lle l ax is thro ugh the cent re o f

mass of the pendulum.[Given : The centre of mass of thepend ulum is a/ 12 a bove the centre o f thesphere.]5

(b) Show that 0 = x f(r) is a possible form for thevelocity potential for an incompressible fluidmotion. If the fluid velocity 7/as rfind the surfaces of constant speed.0

F-DTN-M-NUIB4


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8. (T) STW ci pi* 2a visit{ SW UT34gnAi mqm. pir t .1tA crwr( knii a/3 ai/K%04111 15 m P V Trucbk (itch's Tr Tairt IK a m a )tca 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 w41a arnz arg*0114 I

[AV : ei)eichm)** a/12 WTK1 I]5

wi-17ft 0=x fir) srwaff i f t ruK9. 7A*kirfi14*sraiTqnttzrRNr-4.,44.kr r) 3Frftwalvtrik*titIcy Alf4 7 I0

FDTN-M-NU1135


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F-DTN-M-NUIB

rfr

+il ulciSPW-trl H

F i f r i r :0 2- :300ar/hT

.7*c7 owt tsR4 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

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IAS Mains Mathematics 2012