Optional Paper

Subject: MATHEMATICS-I 11f01it-l

Total Pages : 32 Time : 3 Hours Maximum Marks 200

Answer Booklet No. ｾｾｾｏ｟ｏ｟ｬ｟＿＠ __

RSM-08 Roll No·----=-=---,------

(In Figures) Roll No. ___________ _

(In Words)

(Signature of the Invigilator) (Signature of the Candidate)

FOR EXAMINER'S USE ONLY INSTRUCTIONS FOR CANDIDATES Marks Obtained

PART-A PART-E PART-C 1. Write your Roll Number in the space Q. Marks Q. Marks Q. Marks

provided on the Top of this page. No. Obtained No. Obtained No. Obtained I 21 33 2 22 34 2 Read the instructions given inside carefully. 3 23 35 4 24 36 5 25 37 3. Two pages are attached at the end of the

6 26 38 Test Booklet for rough work. 7 27 39 8 28 9 29 4. You should return the Test Booklet to the 10 30

Invigilator at the end of the examination II 31 12 32 .

and should not carry any paper with you 13

outside the examination hall. 14 15 16

5. A candidate found creating disturbance at 17 18 the examination centre or misbehaving with 19

Invigilation Staff or cheating will render 20

Total Total Total himself liable to disqualification.

Marks Obtained : Part-A : Part-B : Part-C :

Total :

(Marks in Words) (Signature of Examiner) (Signature of Head Examiner)

I

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(2) ｾ＠ \i;it '1'1 m ＢＱｈＧｊＺｾＧｨ＠ ｾ＠ I

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(4) ｾＧｦｩｲ･Ａ＼Ｉ［Ｍ WI'[ -.\1 wnfilr 'R <l'if{-'ffi<m ＢＧｨｩｾ＠

ｾ＠ "'hi "fitcRT iMT ""' 'fire! ""'" "it "'TQ1: "lffi WI'[ ｾ＠

>fr "'h1'fi! 3£'R "ffi"l 'ltft <of "'RT iMJ I

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t <if """ m if 3Jql><lctl <);- \W( ､ｾﾫｩｬＧｬＱ＠ irJr I

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[This question paper contains 32 pages]

Time : Three Hours

ｾ＠ : liR"""'"

RSM-08

MATHEMATICS-I

•lf01<1-l

IMPORTANT NOTE

<i (\ 'i<t 'i 0 i f.$r

Maximum Marks : 200

'f"Tlq; : 200

(a) The question paper has been divided into three parts - Part A, B and C. The number of questions to be attempted and their marks are indicated in each part. --'<'!"Of', ''<r" ｡ｩｒﾷｾＢ＠ 'ffi'l 'IT'if ｩｦｾｾ＠ I mil<o 'WI if <l Wl "fH "'""'lR'if <ii\ ｾ＠aiR "3'fil;- ai<'!> ｾ＠ 'WI if ｾ＠ Wl 1Jit t I

(b) Attempt answers either in Hindi or English, not in both. ow ｾ＠ "T oilMt '100 ii <t llPm 1('1' ii ｾＮ＠ $IT ii 'lit

(c) Write the answers in the space provided below each question. Additional Booklet or Blank Paper will neither be provided not allowed.

mil<o - if; <l\it R>t S'< m ii ｾ＠ ow ｾ＠ 1 3lfctR'Ii1 ｾ＠ "T 'lilu ｾ＠ " <it ｾ＠ <t Km ｾ＠ aiR 'l ｾ＠ ｾ＠ 31'plfu ift "ITirft I

(d) The candidates should not write the answers beyond the limit of words prescribed in Parts A, B and C, failing which the marks can be deducted. ｾ＠ ｱ［ｾ＠ 'WI "or', '"' .. ｡ｩｒﾷｾ＠ .. ii .mow f.!>Tifur ｾＧｉｩｩ＠ m-.rr <t ｾ＠ ii 'lit jffig.l • I ｾ＠ dffi>H m 'R ai<'!> q;R: "'T ｾ＠ t I

(e) In case candidate makes any identification mark i.e. Roll No./Nameffelephone No./Mobile No. or any other marking either outside or inside the answer book, it would be treated as using unfair means. The candidature of the candidate for the entire examinations shall be rejected by the Commission, if he is found doing so. ｾ＠ ;;m ow ｾ＠ if; 3R;1: 3l'I'IT ｾ＠ ｾ＠ ｾ＠ <r'IT "00 'I'<RRmi'11'll{<1 '!'<Rt'ef<14il'l '!'<R <rr 3l"'' ｾ＠ f.mR ｾ＠ m& "!H 3l'I'IT ｾ＠ Wl "'H ｱ［ｾ＠ ｾ＠ m'R "" wm llHT

ｾ＠ I 3l1'l)rr ;;m ｾ＠ wl "fH 'R ｾ＠ <ii\ ｾ＠ 'fihHr if ｾ＠ "" '!i1: ift "'l'Mt I

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Marks: 40

PART-A 'IWT- 3l

'""': 40

Note : Attempt all the twenty questions. Each question carries 2 marks. Answer should not

exceed 15 words . .no <Jlffi! 20 ｬＡｬｾｦ＼ｴ＠ O<R ｾ＠ I mil'!> lfR ii;- 2 ai9; R>Tifur ｾ＠ I O<R 15 W<<ih'f ｾ＠ ｾ＠ iRJ

"'1fuit I

1. Give definition of nullity of a linear transformation.

1('1> ｾ＠ <"41'<1{01 qi\ ｾ＠ ｾ＠ '!ilf;rit I

2. IfV(F) is a vector space, then prove that

-.\l:: V(F) 1('1> Wrn ｾ＠ i nT ｦｵ＼ｲｾ＠ 1'1;

A.x ｾ＠ 0 => ａＮｾ＠ 0 or !'IT x ｾ＠ 0

3. Prove that {f, j} is a basis for IR2.

ｦｵ＼ｲｾｦＧｬ［＠ {f, J}, ｒ Ｒ ｩｩ＾ｾｾ＠ 31l'!T{i I

4. Give an example of a non-commutative group.

1('1> ＳＱｓｨＧＱｦｱｦＮｪｾＧｬ＠ WW'Iil d,l.{'l ｾ＠ I

4

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5. Give statement of Lagrange's theorem for a subgroup of a finite group. 1('1> '1Rfim ww i!>l('l> aq B>io i!>tffi( "ffili"f wl'l 'l>f 'l><R ｾ＠ 1

6. Define a prime field. 1('1> ｾ＠ <:tf'li\ ｾｾ＠ I

7. What is a zero sequence ?

1('1> W'l ｾＭ ｾ＿＠

8. Why does a real sequence cannot have more than one limit ? 1(<!i "1 «1f«ili ｾ＠ 1('1> <'t "!Rnfilll'i 'I'll 'ltit <ll!<1T ?

5 P.T.O.

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9. Why does the function f{x) =eX sin xis continuous ?

'WR f(x) ｾ＠ e' sin x'l'ifmR<i?

10. State Rolle's Theorem.

UR JP1<r 'l>f 'P'R ｾ＠ I

11. Write down envelope of

f-1"1f<1f@i1l:1sJ-¥! 'l>f 3l"'"''"l"'oil);oq ｾＺ＠

ｹ］ｲｮＮｸＫｾ｡ＲｭＲＫ｢Ｒ＠

12. Give definition of a limit point of a subsetS ofiR.

ＢＧＺＡＢｾＧｉｉｒＭｩｬ［ＭｬＨＧｬ＾､ＴＢＧＺＡＢｾＢ＠ S Ｍｩｬ［ＭｬＨＧｬ＾ｾｦｬｆＺ［Ｇｦｩｴｾｾ＠ I

6

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13. Prove that the set IR - Q of irrational numbers is not open. ｾ＼ｩＦｲｲ｡ｩｔＧｬ＾ｦ＠ B1"'1'liR- Q ｾＧｬｩｦｾ＠ I ｦｵ＼ｲｾ＠ I

14. Define absolute convergence of a series of real numbers. 111«1f""' <i&rrarr 'li\1('!> >Wm *- ｾＧｬｩｴｾｾ＠ 1

15. Write down definition of a hannonic function. 1('!> 01tilfeili Ｇｬ＾＼Ｇｒ･ｩｬｾｾ＠ 1

I 16. Write down equation of a normal at a point (r 1, 8i) to a conic;= 1 + e cos 8.

ＱＨＧＡ＾ｾｾ＠ ｾ＠ 1 + e cos 9 i!> ＱＨＧＡ＾ｾ＠ (r1, 91) 'R 3!M<'I"' 'lif B4iili<"l Wifun( 1

7 P.T.O.

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17. How many normals, in general, can be drawn to an ellipse from a given point in the

plane of the ellipse?

ｷｾ＼ｬＢＧｬ､ＧｬｬＬ＠ ｾｾＢＧﾷ＠ ｾｾｾＭｩＡｗｭｾｾＢｩｴＮ＠ ¥fol>eR ｾｭＢＧｬｷｲｮ＠ｾ＿＠

18. Write down centre and radius of the following sphere:

Wor 'l1<if 'Of iR:: >( 1lF<rf !Rfl9:i! I

ax2 + ay2 + az2 + 2ux+ 2vy+ 2wz+ d = 0, a ':f. 0.

19. Write down sufficient conditions for a complex function to be analytic.

ｾ＠ ｾＧｬｦ＠ 'I><'R ｾ＠ f•<H"' ir.'t .:I 'l'll'<f'lrif lRflrt I

20. State the fundamental theories of Algebra.

awpiOO ｾ＠ 'i""l'f w'i<r 'Of 'I'<R ｾ＠ I

8

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Marks: 60

PART-B 'IWT-"1

.;.;-: 60

Note : Attempt all the twelve questions. Each question carries 5 marks. Answer should not exceed 50 words.

-.ffi: : "WI«! 12 'W'it <);- O<R <i\f;r<! I J«$n lR'f <);- 5 3l'li f.l>rifur ｾ＠ I O<R 50 ｾ＠ <f ｾ＠ otf ｾ＠ｾｉ＠

21. Let V be an n-dimensional inner product space. Prove that any orthonormal set in V is linearly independent. Also prove that if x = (x1 , •••• ,xn) and y = (y1 , •••• ,y11) relative to an orthonormal basis ( ei) then (x, y) =xi yl + .... + x11 yn.

lJAT V l('li ｮＭｾ＠ >'R liWR: "ffir ｾ＠ I fu:;;; ｾ＠ W V -.1' offl <'iomWH"' ｻｬｾＢＧｱ＠ ｾ＠ｾ＠ tWrr I 'If<; llPm \'1'"1"{11'11"1 "lT'lT{ Ｈ･Ｉ＼Ｉ［Ｍｾ＠ ｘｾ＠ (x1 , ••.. ,x") q y ｾ＠ (y1 , .... ,y") mfu:;;; ｾＢＧＭ - 1 1 x" " ｾＢｾＧ＠ I'!>(X, y) -X y + .... + y.

22. Let V be a finite dimensional real vector space. Then prove that any two bases ofV have the same number of elements.

fu:;;; ｾ＠ lln '%" Ｇｾｒｦｬｭｾ＠ <mef<"l'ii <fur wrli?: -.1' llfRit <;l3ll'lRI-.t m ¢l<f'l'l -,Tit 1

9 P.T.O.

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23. In an inner product space V(IR) prove that

1('1> W l!WIO<'ffi V(IR) it fu<r ｾ＠

!<x, Y)l,; lxl· IYI·

10

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24. Iff : G ｾ＠ G is a group homomorphism, then show that f is one-to-one if and only if ker(f) ｾ＠ {e}.

ｾｦＺ＠ G--> G' l!'l'lj'l til'il111i1i>"'1 ｾｩＧｩｴｾｦｵｦｾｯＧｆｲｲｾ＠ ､ｨＺｴ＼ｭｾｫ･ｲＨｦＩ＠ ｾ＠ {e} I

25. Prove that a finite integral domain is always a field.

m..r ｾ＠ ffil{'li'lftflm 'oil"'" iliR"" &tr -.Frr 1

--··----------------------------------------------

11 P.T.O.

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26. Prove that

ｦｵ［Ｆｾ＠

!(sinx)=cosx

27. A variable chord PQ of a conic L = 1 + e cos 9 subtends a constant angle 2 ｾ｡ｴ＠ the focus r

S. Show that PQ always touches a conic having the same focus and directrix.

Ｇ＼ＢＧＧｉ＼ｾ＠ PQ '<"' ｾ＠ ｾ＠ ｾ＠ I + e cos 9 'lit '!lN S 'R '<"' 3l'R eitur 2 p 'f'l1<i\ t I fq€it><l fif;

ｾ＠ PQ ｜ｏｬｭＧ＼ＧｬＧｾＬ＠ ｾ＠ ＧＡｬｎｾｗｦｦｩｲ＼ｦｦｩｩＬ＠ '!if<'ffi<rt>ft I

12

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28. Let <I> be the empty class of open subsets of JR. Then prove that the intersection of <I> is IR. llFl!il!> <I>, IR ｩｦ＾ｾ＠ ＳＴｦｬｾﾷｾ＼ｩｬ＠ 'l>fff<ffi"'Wi I ｦｵﾫｾ＠ ill> n ＼ｉ＾ｾ＠ 1R I

29. If a real sequence (an) is convergent, then prove that the set {a0

} is bounded.

>W::l('l'«lwf•'" ｾＨ｡Ｌ＾ｾｭｭｦｵﾫｾｩｬＡ＾＠ ﾫｾﾷｾＢ＠ {a"J ｾｭ＠ 1

13 P.T.O.

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30. Find the sphere having a great circle

aqq'ffi ｾＧ｛ＧｦＢ＼ｉｔｒｦＧｬＧｬＢｦｦｩＭＬ［ｭｾ＠ 1

X2 + y2 + z2 - 2x + Sy + 2z-4 ｾ＠ 0, 2x-2y-z ｾ＠ 5.

31. Find out the analytic function, whose real part is eX(x cos y- y sin y).

;rn f<H<il&l Ｇｬｩ＼ＧｒＧＡｩｴＭＬ［ｭｾＬ＠ ｾ＠ <11«1f•'li 'I1'T e'(x cosy-y sin y) ｾ＠ I

14

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32. If f(z) is continuous on a contour C of length I and j f(z}j ::5: M for all z E C, then show that

J f(z) dz ,; MI.

c

J f(z) dz ,; MI.

c

IS P.T.O.

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Marks: 100

PART-C

ＧｉｗｔＭｾ＠

ai<n: 100

Note : Attempt any 5 questions. Each question carries 20 marks. Answer should not exceed

200 words.

33. Let V be a finite dimensional vector space and W 1 and W2 be subspaces of V. Then

prove that dim(W1 + W2) ｾ､ｩｭ＠ W1 +dim W2 -dim(W1 n W,)

llRT Ill> v ｾ＠ tffiflm fi:ri\<nwm ｾ＠ t mr w I 1'[ w 2 ｾ＼ｩｴ＠ 34B'lR t m fu<r ｾｉｬｬ＾＠

dim(W1 + ｗ Ｒ Ｉｾ､ｩｭ＠ W1 +dim W2 -dim(W1 nW2)

16

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17 P.T.O.

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34. Let I be an ideal of a commutative ring IR with unity. Then prove that R/1 is a field if and only if I is maximal.

llA! f<l;- I l('l'"ffit "Tffi" "'<1>4mfqal:f"""il"'• "i'l\'l'f R ＧＡｩｬｬＨＧｾＧ＠ "l!>f5q\'l ｾ＠ I ftr.o: ｾ＠ f<l;- R/I 1('1' 1\trm-.J<: <'f *'m -.J<: I 4i<l") 4\'1 i- I

18

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19 P.T.O.

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35. Let f: IR ---t JR be a real function. Prove that f is continuous if any only if x0

---t x implies

that f(xn) --> f(x).

'IRif<l> f: U<--> U< ｾ＠ 'llffifqij; Ｇｬｩｦｦｩｾ＠ I ｾｾｦ＼ｬ＾＠ f<Rl<lffl'IK ｡ｦｴ＼ｾＬ＠ 'IF< Xn--> X

..,m f(xn) --> f(x) 'fit Wm '01: I

20

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21 P.T.O.

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36. Find the asymptotes ofthe curve

f1"1Mf&ct mn<); O!+*H'llff ｭﾫＡｾ＠ I

x2+2x-l ｹｾ＠

X

22

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23 P.T.O.

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37. x'+t Ifu = tan-1 then prove x-y

x' +y' 'Ill; u ｾ＠ tan-1 <it fu<r ｾ＠x-y

ilu . ilu . X ax +yay=sm2u

24

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25 P.T.O.

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38. If PSP' and QSQ' are two mutually perpendicular focal chords of a ｣ｯｮｩ｣ｾ］＠ 1 + e cos 9

then prove that

＼ＡｒｐｓｐＧ＼ｲｑｓｑＧｾｾｾｾ＠ 1 Ｋ･｣ｯｳ･Ｇｬｩｴｴｭ＼ｒｾＧｬｬＧｩｬ＼ｲｾｭｭｦｵ＼ｲｾ＠

(SP + SP' + SQ + SQ')(SP · SP'-SQ · SQ') _ j_ (SP + SP')-(SQ + SQ') (SP · SP' + SQ · SQ')-2

26

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27 P.T.O.

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I 39. (a) Jff(z)has a poleoforderm at ｺｾ＠ a, then prove that f(z) has a zero oforderm at ｺｾ＠ a.

I 'lfll: z ｾ＠ a 'lrf(z) 'PI m l!il! 'PI ll;'fi' titB ohit ftr.;" ｾ＠ 1l!;- z ｾ＠ a 'R f(z) 'PI ll;'fi' m l!il! 'PI

ｾＭ［ｩｭｲ＠ I

(b) If f(z) has a pole at z ｾ｡Ｌ＠ then prove that I f(z)l->oo as z--> a.

'lfll: z ｾ＠ a 'R f( z) 'PI ll;'fi' titB ｾ＠ <it ftRr ｾ＠ ｾ＠ I f( z) I --> oo 'l'llf z --> a I

28

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29 P.T.O.

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30

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Space For Rough Work ｉｾ＠ <61lf <); #!1[ "1'1\f

31

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Space For Rough Work /..,..r 1ffilf ｾ＠ ｾ＠ "!'10

32

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