answer booklet no. セセセo⦅o⦅ャ⦅_@ rsm-08 · pdf file13. prove that the set ir q...
TRANSCRIPT
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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( 1) セ@ '!"' <);- """' Frlo <"'R 'R 01'RT "00 'l"''< 1ffi&it I
(2) セ@ \i;it '1'1 m BQhGjZセGィ@ セ@ I
(3) <l'if{-'jffim <l> 3RI ii - 'fill! (Rough Work) 'liB <l> Wit>: GT ¢'f (Pages) \i;it £'( i' 1
(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
(5) <if<; セ@ 3l"!'!i\ 'fire! *"' 'R - 60R セ@ t "' セ@ "0'1> <);- "ffi"l 'jal'<W セ@ t 3l'!"'l ¥HI'J:a\ BGィゥG|セ@
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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