0 mathematics :paper-i - examrace optional paper (!'\ .0 mathematics :paper-i 0 0 0 0 i 0 0 0 0...
TRANSCRIPT
o;:L[ ____ セ]]]MセfセoセrlNセevセaセlセvセatセoセrセGセsセuセsセeセoセnセlセyセMMMMMMMMMセM i 0 ;: Sub. Code : \20 I fJ //.S ( fY}) "( () j <..._, 0 Optional Paper (!'\
.0 Mathematics :Paper-I 0 0 0 0 0
i 0 0 0 0 0 0 0 0 0 0 0 0 0
! 0 0 0 0 0 0 0
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
·22
23
24
25
26
27
28
29
30
31
32
34
35
36
37
38
.39
' l
I
('.! -0 1
6
"' l⦅l⦅ェセセ⦅⦅ェ⦅⦅⦅⦅l⦅⦅⦅l⦅⦅l⦅l⦅⦅⦅ャ⦅セ⦅lNN@_____ _J, ,.
www.examrace.com
i
i, ;
PART- A
'Wl- 3!
Marks 40
セ@ 40
Note : Attempt all the -twenty questions. Each question carries 2 marks. Answer should not exceed 15 words.
ollc:
1
2
WR<f '1 o j{セ[ヲヲ@ in iN\ <Mfi\ I 1[(irq;- ャヲセGャ@ in fW1" '1 ai<l; セ@ i I iN\ 9 セ@ セゥFゥ|@
セ@ セ@ 'lt'f Wn セi@
If t is a linear trallsformation from a vector space V(F) to a vector sPace V'(F), then
show that 1(1!)=-t(v)VveV. セ[@ .
'lfe: t セGit@ セ@ V(F) セ@ セセt@ WJi£ V'(F), 'R '<'ff 'l:fu!q; \'\YI'<I <'I i, <it セ|ッ@ q\]furi'r ill; t(v)=-t(v)VveV.
Show that {l,i} is a basis for the field C of complex numbers, considered as vector
space over the field R of real numbers, where / = -I .
セ|ッ@ q\]furi'r ill; @<'l! -&&IT in !if,r C, ;;fr 'llffifll'h -&&IT in !if,r R 'R '<'ff セセt@ セ@
i, in ful1; {l,i} '<'I> <m>m: i, セ@ ;2 =-I i I
20-1] 3 [Contd ...
11111111111 m 11111111111111 • ' ' - 1 •
MセM <,- NMNセM ---
www.examrace.com
3 Let t: R2 -t R2 be a linear transformation 、・ヲゥセ・、@ by :
'lf<:: t : R2 -> R2 'l'O 'lil;rq;- セTQGu@ ('I セ@ ;>i\ f¥1 ll'lil<: "\l "4ftmilrn セ@t(x,y) = (x+ y, x-y) V (x,y) E R2
Then find the matrix oft with respect to basis B={(I,O), (o, I)} of R 2
cit R2 it; an>m: B = {(I, 0), ( 0, I)} it; Mキゥエセ@ t >t\ セ@ -.n<r 'lft1W\ I
4 Show that every subgroup of an abelian group is normal subgroup. セイョ@ 'lft1Wl flo ""' セ@ Wi& Oil セ@ BGセMwゥF@ ""' セBBB@ "'1-·Wi& c<tn %,
-----.,------------------·-------·-.. -
5 Write the conditions for a ring to be an integral domain. 'l'O 1'!w:r it; GサGャヲ\エャセ@ lfRf 1?Ji\ >t\ セヲ\ャ@ セi@
20-I] 4
·' ...
(Contd.
1111111111111111111111 n II . ' . ' - .
www.examrace.com
! :j .,
I i ! I I
I i
1 i
1
• !
6 Define field and give an example of a finite field. ' · iBf q\j mm セ@ eltt qf{fllct .lffsr 1l>T '<'I' <J<IWI セi@
--- M MMMMMMMMMMMMMMMMMMMMセᄋセᄋ]M]ᄋᄋ]ᄋ]]セ]MセMMMMMMMMMMMMMMMMM
7 Find the value of /(-Yz). rr:rz) "" lJR Wff セi@
8 . (I) sm-
Evaluate lim x . クセo@ _!.
X .
lim sinG) クセo@ 1
X
20 -I] 5 [Contd ... QQQQセQQQQQQQQQQQQQQQQQQQQQQQ@
" ' ' - ' -
· .. l1 Ci
www.examrace.com
,---
l ·---
9 Find the radius of curvature at the point (s, 'V) on the curve s=.a log(sec'V+tan'f').
'Ill> s = a log (sec \jl +tan 'I') it セ@ ( s, \jl) 1f1: GAャA^ヲヲゥMセ@ W<f <6\furir I
I
10 Evaluate J lxldx. -2
I
J lxldx 'li1 'IT'! W<f <6\furir I -2
11 PrOve that the series
ヲエキZセュセ@
zP 3P 4P 1+ セ@ + [;! + l1 + ...
is convergent for all values of p.
p it '\Nr '!AT it fu'l セ@ 'i-1
20-I] 6 (Contd •••
111n111111 m 111111111 1111 • ' 0 - ' •
- MMセBG@ l '
www.examrace.com
12 Prove that the set Q of rational numbers is not an open set. fu<r セ@ flo 'lfW1l ffi§<rreif Oil ij!jiil<! Q '<'" f<!'[<l ij!Jiil<! 'ltf セi@
13 Let fbe a real valued bounded function defined on closed interval [a, br Then for any partition P of [a, b ], prove that
L(J,P):W(J,P). 'lk f セ@ aRfThil [a, b] '!<: •m<rl'l<6 liT'! セ@ 'l><iR l <iT [a, b] 'il> M セ@pエセヲオ\イセヲャッ@
L(f,P)W(f,P).
14 Define uniform convergence of sequence of functions. 'lWf'iT 'il; "':Jll'l 'il; .fWl '<1l> <fllR <Jii'!ij{UI <fi\ qf{'lll'll セセ@
20-I] 7 • [Contd ... 11111111111 11111111111111111 . ' . - - ' .
www.examrace.com
i .. ; ;
'
I
,
セ@ セ@ Write the equation of right circular cylinder whose axis is x-axis and radius ts r
'<'li ゥゥャBャGヲM、ャセ@ セ@ 'Ill etll'li\'1 セ@セ@ ana x-0\'ll t <I'll セ@ r t 1
. .
----------------------------·
_Hi if x-1=0 is the directrix of the parabola. y2-kx+8=0, then find the values of k
·i. 7 Prove that the lines
x=ay+b,z=cy+d and x=a'y+b\ z=c'y+d' are perpendicular if aa'+cc'::.:--1. '
QG|Nセ@ セ@ N 1:lmi( x=ey+b,z=cy+d altt x=a'y+b', z=c'y+d' セ@ セ@ <il.: /
aa'+ cc' = -1.
----------------------------[
------------------i
MMMMMMMMMMMMMMセMMMMMMMMMMᄋ@
20-I] 8
. MMセMᄋMセエMZセTエZBZセMM ._:_ ___ ------ -----
I [Contd .• 1
ll!lii,MII AQセ@ 1111111. HIll
I -I !
I 1
www.examrace.com
I 18 Show that the function f(z)= x2y-iy is not analytic any where. . セイョ@ セ@ f<l;- '!iWI f(z)=x2y-iy 'litf 11J ヲ\ャャセ@ 'lt1 セ@ I
---
19 Find the radius of convergence_ of the power series. セ@ 2-11 L .n ( 2)" -11:=0 I+in
セ@ 2-n 'ffiffl L z" q\1- セセMNイゥo@ セi@! n=o(l+in2
)
---
20 Find the nature of singularity of function
f(z)= smz2 (z-n)
at z==n
'!iWI f(z)= smz (z-n)2
q\1- z=n trr ヲ\ャセ^セ。ャ@ q\1- l!Wfu -.no セ@ 1
--,-
.
o ..:r 1 9 [Contd ... llmllllllllllllllllllllil
" , ' " ' .
----- - "---.,..- _"' _______ . NNLNjKᄁGセfェ@ • -·-
www.examrace.com
I
' '
Note :
PART- B m。セォウ@ : '60
セZ@ 60
Attempt ail the twelve questions. Each question carries 5 marks. Answer should
not exceed 50 words.
;iR WR<f q セ@ l! :fiT it "''T セ@ I Wlq; QAセGQ@ it セ@ oi<l; f.!!llfur セ@ I "''T セ@ o . セャBゥヲ@ -€!"
セ@ 'ftr WIT セi@
21 If B={vl> v2, ..... ,vn} be a basis of a vector space V(F), then prove-that each element
of V can be expressed uniquely as a ·linear combination of elements of' B, where F is
a field.
<till: B =h. v2, .... , Vn} M セt@ セ@ v(F) qrr <ll1"lR t <i'r m セ@ fu Vqrr mifq;
セ@ ャゥャセ」ゥBャGャ@ Bit セ@ it セGi^uャ」エ@ セ@ it WI i\ <llflra\q m -€1- "1'R! fu'IT "'T mT 'I;,
セfセ@ i'fs!tl
10 [Contd.
11110 セiャャ@ m ュQQQエセ@ 1 • ' 0 - ' '
MMMセᄋGGZ@_,·_;_ ..
....... 1
www.examrace.com
2 2 Prove that :
2
; a+b+2c.
c c
0-Ij
a b b+c+2a b =2(a+b+c)3
a c+a+2b
li
MMLMセMMMMMセセᄋ@
--- -----------,------- -----
[Contd ... 1 illl mセ@ m 11111111! n11 . ' セ@ - ' .
www.examrace.com
23 Let (Z,+) be the additive group of integers and H=3Z={3xlxez} be its subgroup,
then ·find quotient group ZIH.
'Wil (Z,+), セ@ 1fil 1l;;;-.: セ\wi@ 'it ヲオイ\セセ@ oft<: H=3Z={3x\xEZ} セ@
'Q'i$ il<1WJ! t <iT '"'"" セ@ Z\H W0 セi@
-----······-----------------------··--·
----------·---------------------
24 Prove that a field has no proper ideal.
flhs セ@ \lf; '1'!'-*' q\\ セ@ '>l\ セ@ !J'I"'I<lJ\ ;ffi' -rniY セi@
MMMMMMMMMMMMMMMセMMMMMMMMMMM
MMMMMMMMMMセMMMMMMMMMMMMMMMM
------------------------, .. ,,,,,,_, ___________________ --
MᄋMMMMMMMMセMMMMMMMMMMMMMMMM
20-I]
---------.,-------------·--
12
, MMMMセセMMMMN@ ,--,-. -----,-.;
[Contd.,
11111111111 m 1111111111111 • ' ' - 1 •
www.examrace.com
25 Change the order of integration m the following integral f.'rr <r'll'!Wf if bセャゥ「wャ@ <liT WI qf{qf<fff セ@4a z..{;;; J J f(x,y)dxdy 0 x2
4a
----··-·· -····--·------------------·-.......
----------------------------·--------------------'-------------MMMᄋMMMMMMMMMMMMMMMMセMMMM
............. . -- ................ ··---------------------· -- --------------------------·----MMMMMMMMMMMMMMMMMMセMMMMMMMMM
----------·· -·---·----- .... ___ : _____________________________ ............... -
--------------------------------------··-·· e3z
26 Evaluate J ( ') dz, where C IS a circle /z -l/ セ@ 4 c z -m
J e3z
--dz <liT 'lR OlRf セi@ ;;;of C, /z MiOセ@ 4 'l'Ji '[ff %-r c(z-rri)
セMMMMMMMMMMMMMMMセMMMMMMMᄋMMᄋMMMM
----------------------"-··· ........ . ----------------------------MMMMGMMMセMMMMMMMMMMMMMMMMMMMMMMMMM
------------------·-·----
---·-------------'--------·--MMMMMMMMMMセMMMMMMMMMMMMMMMMMM
------,---------------------------, ' ·------------·--·· .......... ___ , ___ .. -------20-I] 13 iContd ...
lim! lli!l m 1111111 !IIIII . ' . - .
www.examrace.com
27 Pfove that every finite subset of the set R of real numbers is compact.
セエイ[。@ セ@ lln "1'"'il<'" mr.m it e!"J'*'4 R"" セ@ m!lRr "'e!"J'*'4 mer -.to %1
28 ( 3 3) . セQ@ X + Y au au
If u =tan , then find the value of x-+ y-. x-y ax oy
au au x-+ y- '1>1 111'1 -.no セi@ax ay
20-I] 14 [Contd ...
I On! IIIII In QQQQQQQQセ@ 011 • > 0 - ' -
www.examrace.com
I
29 From a point P tangents are drawn to the parabola / = 4ax. Ifthe chord of contact ofthese tangents touches the rectangu1ar hyperbola x2 - y2 :::: cl-, then ーイッカセ@ that the locus of Pis · -
2 2 X y 2+-2 =I. a 4a
M セ@ P "\\ q \'IW-l / = 4ax Giエ\GAセ|@ 'twf( w.lt 'l'l\ セ@ 1 'lll( n \GAセ|@ 'nsnoi'i 1/!r \GAセA@ ;;\1<n -l<-bGゥ」エI|オセア@ セriThゥゥNGヲア@ x
2 -/ =if 'liT \GAセエ@ セ@ i, m fu<l: セ@ ll!; セ@ p <OJ セGi@
x2 y2 -+-·=] il
· a2 4a2 . •
.
'
' ' ;
.
'
.
. . 20-1]
15 [Contd ...
IIIID Mil m セャゥd@ D ill . ' . - ' .
. . .
MMLNLNセ@
MMMMMMMMMMMMセMMMMMMM ... ---. MGセエ\GyBGセMGZ@ セMMMM
www.examrace.com
I I
I I j
'
•
' ' -----
30 Find the equation of sphere whi-ch passes through the point H。NLセN@ y) and the circle
セ@ H。LセN@ y) aih: '[<f x2+i セ。 R [@ コセッ@ -\1" i'r<lr<: セ@ -.mil <iWr '1>1 セエヲイゥヲゥ\Gャ@ -.ml
セi@
20-I] 16 [Contd
I !llllllllll! lllllllli • ' ' - t
www.examrace.com
31 Find the envelope of the family of straight lines xja+ yjb =I, where ab = 4. m<'! 't1!1NIT t セN@ xja+ yjb =I ilfOi ab = 4 t <or i3f•<il\ii't<nrrer セ@ 1
-
'
'
32 Find the Taylm's -series expansion of f(z) = ( )!( ) z-1 z-3_ in the powers of (z- 2).
( z - 2) 'Ill! 'lKil it '!0\iA I
f(z)= (z-I)(z-3) <or im: lll"i't 11m\ Wff セ@ I
.
'
'
'20-1) 17 [Contd,. I 11m1 ョセ@ m 111111111 1111 . ' ' - ' '
' MMセセ@ ----- --- --"-T -·--- --- . - --------- -- セMMMMM ------ -
www.examrace.com
I
I I I
' '
Note :
PART- C
'11'1 - "«
Marks : 100
oi'l; : 100
Attempt any five questions. -Each· question carries 20 marks. Answer should not
exceed 200 wotds.
'Iii{ '1\'t セ@ |{セBGA@ セセ@ ][(i\q; |{セBGヲゥョ@ fu!'l '(o ;;iq;" f.rft t1 \UIT 'tOo セ@ B
セ@ '1t\' -.1'11 セi@
33 Determine the eigenvalues and the corresponding eigenvectors of the matrix A, where :
セ@ A in fu!'l セセ@ セ@ <I'll "\M<! セセ@ \ャゥGZセt@ -,mr セ@ • .m :
[ 2 2 0]
A= 2 1 1
-7 2 -3
20-I] 18 [Coni<
IIUU セi@ Hllllllllll • ' 0 - I
www.examrace.com
.
.. .
I I I
. . 0-I) 19 [Could ...
llllillllll IIi セiiiii@ iセ@ 1111 . ' . .. ' .
.. .. .. ·-
-
---· ----- --------- __ ,_ ---MMセᄋ@ - ---- -- .. ---------···• . .-·.:' ·.- '
www.examrace.com
!
! j i
I I I !
I
I
I
34 Define prime ideal and prove that an ideal I of a coffimutative ring R with unity is prim
if and only if R/1 is an integral domain.
セ@ セoiBGiGャゥゥ|I@ q\1 セ@ セ@ <I'll fu<r セ@ f<n ilm1t ゥャ^セヲ\ャヲGゥゥGjᆱ@ 」エ|bセア|ャ@ """"" 1
q\1 'l'O セGャBGャGャゥGゥャ@ I セ@ セGャBGャGャゥGゥャ@ セ@ ..W oin: t<r.r ..W RJI 'l'O GサBョGAゥャセ@ >rRr セ@ 1
20-I] 20 [Con
I IIIII セiiiiiA@ 111111 . , ' -
www.examrace.com
-
-
. ' ,_,-.
-
-
.
.
. 0-1)
21 [Contd: .. I lllln IIIII m QQQQQQQQセ@ 1111 • 0 - ' -
. ' ' .
---- -· セ@ ---- MセMMMMMᄋ@ --
www.examrace.com
i
35 Find a cylinder of greatest volume. Which can be inscribed in a cone.
'1'!'. セセ@ ;j セ@ Nセ@ il!ll«R qf.r i\\iR q!r >ml セi@
20 -I] 22
-----
... -, ·,,
[Co
llllilllli m 1111 • ' 0 -
, ""., "' ""'*1 1 www.examrace.com
.
. .•
-
.
.
.
. ..
..
-• .
.
.
I セ@
- . • . 20-IJ
23 [Contd •.. i
I IIIII I H! セヲエャゥ@ U Ill . , . - ' .
Mセ@ - -·--MMセMBBGᄋMセMBBBBG@ .• <<'- ZNセMM[⦅[NLNNLNNNNセMM
www.examrace.com
I
I :
36 (a) Show that)he eight points of intersection of the curve xy(x2- i )+x2 + i -a2 セ@ 0
and its asymptotes lie on a circle x2 + y2 = a2 .
セtッ@ セ@ llo w xy(x2- Y,2 )+ x2 + y 2 -a2 セ@ 0 <I'll セ@ MセGェェ@ it ano
セ@ Rl<il• セ@ 1('!i 'l'f x2 + i セ@ a2 '!\ ft<m i'till
(b) Prove that every real valued function which is differentiable at a point a E R i'
Continuous at that point. Give an example to show-that the converse of this resuli
is not true.
#Rl: セ@ llo セ@ <maf<!ifi lll'l'r! 'I5WI -.!r llo セ@ a E R TI <>J'I<ii\i\4\'1 'I; .q;; "'
セ@ '!\ Wri! >1\ Wrrl1('!i セGiwi@ セ@ llo"' qセGiib@ <61 セ`ZA@ '1t'f fl
'
- -_'""'."_.N·d,,:--:-·· "·-•:' '·":
24
....
[C01
lllnl セQQQAQQQQ@. ' ' -
OM>¥)
' www.examrace.com
----------------- ···--·-----
--. ----·
---------------------·---
-------------------- ---------
20 -I] 25 [Contd ... I IIIII セセセ@ QQQQQQQQQQQセ@ 1111
• ' ' 0 - ' •
," . .>:::-"c:. __ ;_ ._ -.;,;.:,;;.,, LL[ZLセ[LLN@ i ''• セMGMMMセMッ@
. www.examrace.com
i i !
I . ' i
I
i I !
37 (a) If a function f is continuous in. closed interval [a, b] and ce(a,b) such tlu
f(c)<O, thenprovethatthereexistsa 15>0 such that f(x)<O\Ixe(c-o,c+o:
セ@ 'li\iJ'f j *Ju <lRTTh'! [a, b] i'i mra 'I; a'll ce (a, b) -qq> セ@ セ@ 'I; ilo I (c) < c
<it fu;;;: セ@ ilf; -qq> o>O Gャゥtセセ@ ilf; j(x)<OWe(c-o,c+o).
(b) lf/(x)=(x-4)logx, then prove that tloe equation x1ogx=4-x is satisfied I
at least one value of x e ( l, 4) .
20-I]
セ@ f(x)=(x-4)Iogx, <it ftwr セ@ ilo セT|ゥ「HGャ@ xiogx=4-x "l>'' 'i\ "l>'' セ@
'l1'! X E (1, 4) if; \W!; '6W .til '/;J
26 [C01
11m1 セiiiiiA@ セセセ@. ' ' -
www.examrace.com
I I
1-- -· r- ·-'· - 1-·--, .
.
. -: .
'
I· ' . .
-
I _,
i . ' .
0-11 27 [Contd ... 1111!111111 m lllllllllllli
• ' 0 - ' • .
'
. . -- "·--··--' ' .... ' -' GセMMMM- --- .
www.examrace.com
38 Prove that the equation -./(;; + .JbY + ..J(; = 0 represents a. cone which touches
coordinate planes. Find the equation to the reciprocal cone.
fu;;;: セ@ ffi ../aX+ .jbY + .kz = 0 'l'O セ@ <if セ@ -.miT 'I; "'] キ[セャゥア[@ B'ict\i\l
ᆱ\セ|@ -.miT 'I; I ;Wi; セ@ セゥ_ャ^@ <til B<l'i<wl W0 セi@
.
20-I] 28 [C IIIIUIIIU
• ' 0
www.examrace.com
.
.
---------------------------------------------
-------------------------------
MMMセMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM
.
0 -I] 29
[Co11td ... ..
Ill IIIUIIIIII m lillllli ill • ' a - ' •
www.examrace.com
39 By the method of contour integration prove that : q[{):(!ll |ゥセiGi^wゥ@ l<ifu "i\ ftRl 'li'rtWl lli; . :
=
f cosmx dx = セ@ -ma ( > ) 2 2
e , m_o. o-x +a 2a
'20-1] 30
I
I -I
I -I -I
I
[Contd... 20
111111 n11 m 1111111111111 • > 0 • ' •
www.examrace.com
31 [Contd ... 11!111 セセ@ !D セiw@ U l!fl
\ セ@I
1 i ' I ' I ' ., ' I
'I ' I -. \ i I
.J, r.,: l セ@
[
• • www.examrace.com
SPACE FOR ROUGH WORK iGサセA[@ "i6f1[ if; f.TII セ@
20- I I 32
IIIII! llllllllll!llllllll!ll ' , ' - _, .
MMMMMMMMセLNNNNNNLMMNMMセ@ .
www.examrace.com
Optional Paper . Mathematics : Paper-DJ Time · 3 Hours I Maximum Marks · 200 I Total Pages · 48
. ,, ,·.... . . ' .. ·. . . .. · ... .. e|H。ャャゥセエゥHAョGG{ヲA「ゥ・@ > ·. ᄋエセッG@ E"'"''"''s Uso Only)
PART-A ..
.j·.PAi<fcil ... ·. . .. ,,y---·pART,.c .. . Gt"!rld Total . '•' "' ' '••· ' ,.
QN E-1 E-2 __ AC ON E-1· .E-2 AC QN I E,1 'E-2 ·Ac セaセtMa@ I •
1 21 33 .·. PARJ-B ..
.. 2 22 34 "tt'r'b ·. 3 23 35 Toil> I .
. . · .. .. 4 24 36 'H'Marks' . . . 5 25 37 Final Mセqエ。ャ@
6 26 38 Marks in Words
7 27 39 • . 8 28
9 . 29 . 10 .
30 . r・セエォs@ ッセ@ Meセセセセ。⦅セセイOcエ|ゥセヲ@ eセ。⦅ャオ。エッイ@
11 31 .
12 32
13
14
15
16
17
18 .Remai'ks ッョウ」イオGセゥョゥウ・イ@
19 .
;w
To tar
EvaiO ator's Sign
·- ZBBBGセZセM •.- ..
I
' 0 N -"' セ@
www.examrace.com
... ··
.. .-IIIII
; :;· ; ·, セ@
I
i '
I I j I ' 'j ' ,;
PART - A I '1FT - ill Marks/ of<O : 40
Note : Attempt all the twenty questions. Each question carries 2 marks. Answer should not exceed 15 words.
;i'tc : Wl«r ' o ャヲセ^ゥ|@ il; i3U1: セ@ I セ@ ャヲセMゥイ@ il; fui\ ' oi'l> セ@ % I ;;m 9 セ@ セt\g}@セ@ illfllq; ."'li) -.RJ セ@ I
I Find the unit vector nonnal to the surface x2 y + zxz ;:: 4 at the point ( 2, - 2, 3) .
• '\ 2 Prove that the vector rn r is an irrotational vector for all values of n.
20-11] 3 [Contd ... 11m111111 m QQQQQQQQセQQQQQQQQQ@• > , 0 - I I •
-------- .:<-_. - . . -.. ' . --·
www.examrace.com
' '
'!
I, ,
i' '
' I 20-1
3 lf a+ E + c = 6 and \a\= 3, lEI= !i, \<'\ = 7, then find angle between a and ,b .
4 Define angle of friction.
U't!uJ <WJ セ@ セ@ I
20-II] 4 [(
llllillllll m 1111 • ' 0 -
MMセ@
www.examrace.com
\ \
:',
/
.. セ@llil
5 Write the principle of virtual work for a syster'n of coplanar forces acting at different points of a rigid body.
M "" m i\; NlR flr-:;® '1\ ャャャゥュセャヲヲゥ@ bセッ|ゥ|QセN@ <[(if セ@ i\; ゥwャᄋセ@ <011! 'l>l ' ' .. fuiSI'tl セ@ I
6 What are the conditions of equilibrium of a freely floating body in a liquid ?
.ll:'l· ii セ@ W! '\1 <ffir fQu6 '1\ bゥエセQ@ i\; セ@ 'f'IT % ?
20·;,·IT] MZセZ@ セ[セZᄋM :.';;,• _,_.
5 [Contd ...
111111111111 m 111111111111111111 • ' 0 - ! 1 •
www.examrace.com
j
I
20 j I
'i
7
. 8
' If the angular velocity of a particle· moving in a plane .curve be constant about a !
origin, show that its transverse acceleration is proportional to its radial velocity.
<!fu M B'lCfiif 'llli i'i -qq; • rfil'll;, Gセ^Bセ@ <li1 'I'! セ@ セ@ セャャャ@ セ@ セ@ セ@ セ@
fuis <Pi\ f<n i3"\1<I>T iliJ= i'IT"T iJct'l<:r セ@ セ@ セGャャェTQヲヲゥ@ iMT I )
A particle is moving with simple harmonic motion of atllplitude a. At what 、ゥセ@
from the centre will its velocity be half of the maximum ?
-qq> '1>"1 ""' 01 I 'I J• 11li . i'i セ@ 0IT'Wl a t 'rfll '11'! % I 'io.9: it 1lmR\ セ@ 'I':
セ@ セ@ セ@ "i1i1 011'<! -;Wrr I
20,.-II 1· 6 [( iiiuiiiiセ@ m 1111 . . . -
www.examrace.com
!.
·d 9 Prove エィセエ@ the areal velocity in a central- orbit is independent of time ..
ftR:: セ@ fll; セ@ セ@ . 'l>>al it セ@ ii<r "Wf'[ セ@ ffif;! -.tiT セ@
'ce
d •••
QQセQQ@
10 Define the basic feasible solution of a linear programming problem.
'tfurq; ャゥャAャャセGャ@ WR'll <o w>rrtt セ@ -..r <fiT セ@ セ@ 1
11 What do you meau by slack aud surplus variables in L.P.P. ?
L.P .P. it "['ld 1'{(0> 'l'i セ@ "<lit io -.it it i3ll'l 'I'll "fFri\ i' ?
20-,IIJ 7 [Contd .•.
1111111111111111111111111111111111 • ' 0 - ' ' •
LMMMセM ------- ·---:=...:"""' __ ;p-·
www.examrace.com
I·
!,
II
' I .
i I , I'
I! il II I! i:
I I'
12 Explain dual problem in a ャゥョ・セイ@ programming problem.
13 Solve the follOwing difference equation
f.r-1 <>RIT セ\|ャG「サ P Q@ Q\if セ@ I
20-Il] 8 iiiiuョャャセ@ . ' '
, •• ' 1
www.examrace.com
1 . . 14 Find the truncation error for ex at x = 5 if first three terms are retained in expansion.
. i 1 I x=-
5
15 Find the order and degree of the difference equation Yn+2 -7yn = 5
.J
mtd ..•
11111111111
20-II} 9 [Contd ...
11111111111111111111111111111111 • l 0 - ' ' •
'
セᄋMᄋMMMMMMMM
www.examrace.com
' i
i
20 セ@
i_
' I i' j: ' i! 'I : !, II
il ,1: I'
" ' I
!
1\' I ,I
I セ@ ' '
i' ' '
16 Explain Regula-falsi method ..
セ@ - セ@ AA <fit ゥェセJGGGセ@
17 Solve : 1R'f セ@
(x+y)(dx-dr);, dx+dy
20-ll] 10 I ·IIIIUIIIIIml
• , 0
'""'I www.examrace.com
td .• :
11111111 ' .
18 Find integrating factor (I.F.) of differential equation (x3+x/)dx+2y3dy=O.
セ@ セ\|ェ\ャャHGャ@ (x3+.xi)dx+2y3dy=O 'ffT セセQGQ^QGQGQM Tli'll ヲヲゥ\ヲᄋセ@ I
19 Find complementary function of the following differential equation f.rq セ@ (1<\j'!)(UI 'ffT 'l:W 10\iR -.ira セM I
20-II] ll [Contd ...
· lllln IIIII m セQQQQQQQQQQQQQQQQ@• ' ' - 1 ' •
www.examrace.com
____ ...
20 Solve : 1<-.r セ@
-1 ーセエ。ョHークセケI@
I I
,, ,,
RPセii}@
·-·---
12
• イイュュセQQQ@< ' •
www.examrace.com
'• .. ··--··--
' . - . .. .•
.. ·-- MMセ@ --' ·- ' -.. , ·-· -· .
ntd ... . \, Mセ@ -;
1111111111 ' . 13 [Cpntd ••.
QQQQQQQQQQQQQQQQQQQQセ@ 11111111111 • > 0 - ' ' •
www.examrace.com
I •
1
Note :
. PART - B I 'll'1 -'I
Marks/ aFi; : 60
Attempt all the twelve questionS. Each· question· carries 5 marks. Answer should
not exceed 50 words.
Ww\ q '< jAセ\ゥ|@ 'o ;J'iR セ@ I 1f<'1q; QAセGQ@ i\; セ@ ajq;- セ@ セ@ I 'l'iR セ@ o セPP@ «
0lfuq; セ@ セ@ 'l!fM I
21 Prove that : fu;;; セ@ :
20-Ilj 14
[Contd.
IIIII セhゥセセセセ@ iゥャゥゥセA@• 2 0 -, ! I
'
www.examrace.com
22 By' Stoke's t'heorem prove that div curl F = 0.
-..i\q; 'lit l!""m "i't ftRi: セ@ fto div curl F = 0.
\
15 [Contd ... 1111111111111111111111111111111111 • > 0 - I I • •
-1 -'
'I I
www.examrace.com
'- 1
'
23 A uniform chain, of length c, is to be suspended from two ·points A and B, in the same
horizontal line so that either terminal tension is n times that at the lowest point, show
that the span AB must be
セ@ ャッァサョKセHョ R@ -1)}
C セ@ <t\ 1N WIR $1\1: <!i\ 1N if セ@ 1;m 16 <iT flqoi'i A <!'IT B 16 >M \il2'"lil'il
'1; I セ@ M fut "' <A1'f 1'1'1<1'1 flq "' <Rf'[ "'liT II ;rn '1;' cit ftr.& セ@ f<l; セ@
セabセi@
. セャッァサョKセHLOMQIス@
-· . -·
20-II] 16 (Contd
QQQQセQQQQセ@ II! QQQQQQQQセQQQQQ Q@
. ' ' - ' '
BGNtセLMセL@
www.examrace.com
24 _ A quadrilateral ABCD is immersed in the liquid with CD in the free smface aod sides AD and BC vertical of lengths a and p respectively. Firid the depth of centre ·of pressUre (c.p.) in terms of a ·and セN@'%" BGァセBGャゥBAゥャサ@ 1ft\ir ABCD .<!i\ セ@ CD セ@ 'r'O if % .ofR セ@ AD <!'TT BC o;"'\m om lf>>m: a .ofR p セ@ <!i\ t 1 <[!'[ ilR: <!i\ セM a .ofR 0 ili 'lG! if ^ikセ@ セ@ 1
セMᄋᄋᄋMM
-- ........ ---co--------
MMセMMLMMMMMMMMセMMMMMM.. ·-·- ··-
セMMMセMMGMMMMMセMMMMMMM -·----'----------------···-··-'-'-----
. セM -
20-ll)
---------··-· ·---·--------
17 [Contd ... 111m 11n QQセ@ m11111111111111
• 2 0 - I I •
. セMセ@
·--.-=-'-
.,
www.examrace.com
,_ --1 '
25 A heavy elastic ball drops from the ceiling of a room and after reboundfng twice t -floor reaches a height equal to one half that of the ceiling. Prove that the 」ッM・ヲヲゥ」ゥセ@
(1)1/4
of イ・ウエゥエオセゥッョ@ is 2 ""' lffiT .J«im'' i\'<:: '5lJ1: 'li\ l9<f it. flm\1 i セ@ <i\ OR Gャゥセ|@ it セ@ セ@ 00 'li\ -"11
セ@ <f1li セ@ i, <i\ flwl: セ@ f<n ャャエ\ャゥセ\h@ :J"li'!> ct4
i I
. --- ..
ᄋMMMMセMMMMMMMMMMMMMMMMセMMMMMMMMMMMMMMセ@
20-Il] 18 [Contd ••
I huiiiiiiiiセ@ QQQQQQQQセQQQQQQQQ@• ' 0 - ' ' •
www.examrace.com
26 Two particles connected by. a fine string are constrained to move in a cycloidal tube. in a vertical plane, the .axis of the cycloid being vertical and vertex upwards. Prove .that the tension in the string is constant through out the motion. <;1- Gセ^BA@ '%· 1ffi\ift 1i\t -(\ <Pl セ@ a'l! '% セ@ '!o1T i'i セ@ 'lfff -(\ QrゥセQB@ セ@ 1 セ@"ii>T illlil ilt'llm: '<'[ セwヲ@ il>'IT 'Iii i3!h: セ@ I セ@ <i>(t flo q;uif ,_. •1RI•11'1 -rn\ S'( 1i\t if <fiTq i3lm: Wn I
-IT] 19 [Contd ... 1111111111111111111111111111111111 . , . - ' ' .
www.examrace.com
27 By using Newton-Raphson's method, find the root of x4 -x-10==0 ·: which is ,near
' -· . ' .
x ::= 2, correct to three places of decimaL
"l"'l (l'be'l セ@ "' m it x4 -x-10=0, '1>1 'Ff Wff セ[[ヲイゥャャ^@ x=2 in
m BGセセ@ BセM "' \セr@ Bl1'il ""' "«tt m I
. ··--
.
.
. .•
.
.
.
, .
20-II] 20 -'
. II!IU)all !U Aュャエwセ@
' ·-- --. --· ----·------··
...
. . www.examrace.com
28 Solve by Simplex method. fu"lct<ffl M セ@ &N セ@
Minimize (f.F'1<111) ---: .
• Subject to HセI@
and (<lin:)
2,0.-II] . ' .
. , ..
Z::::x 1+x2+3x3
Jx1 +2x2 +x3 :::;3
2x1 +x2 +2x3 ::;; 2
x1,x2 ,x3 ;:::o
21 [Contd ...
llmlllllllillllllllllli 1111111 • > 0 - '· 1 •
I i
' ' セ@
i ' ' i ' '
•
J セ@
www.examrace.com
1 29 Given· following data, find the value of the following integral using Simpson's
3 ru
W'f <r<r· セ@ "i\ f.r<f |QTQQAゥセGQ@ "1!iT fiJ""l\1'1 it '<"' セ@ f.r<m IDU l!R W<! セ@
•• J exdx 0
e=2·72, e2 =7-39, e3=20-09, e4 =54-60
-· i
I
.
. .
20-ll] 22 [Con
111111111111n m111111 . , 0 . ' ..
. セN@ 1 ---
M[BBスZセセM .«-----·- MNNNNNMLセMᄋN@ ., ..
. www.examrace.com
30 Fi11d the dual of the following L.P.P. • f.£"1 セ@ セGQュセQ@ WW11 'liJ セ@ WWIT W<! セM
. Minimize HヲGャBャ\ャセI@
Such that HャセwrZイI@
and (ll'TT)
Z=x1+x2+x3
x1- 3x2 + 4x3 = 5
2x1 - 2x2 53
2x2-x3S5
x1,_x2 ;?:0,
x3 is unrestricted in sign ( x3 セ@ "B" セ@ セI@
•
' . , , 23 [Contd ...
11m11111i 111 m111111111111111 . ' ' - ' . ' .
セセM ' . MMᄋMBMMᄋGMMMMᄋセᄋM
www.examrace.com
' ' ' '
31 Solve : -.<'f セ@
,d3y 2d2y dy . x ---x --+2x--2y=x3+3x
dx3 dx2 dx
.
20-II I
.
24
. ..
'
.
I NAセ_N_N@. I
.
. -
www.examrace.com
32 Solve the following equations : f.rq \セAェェ\キョᄋ@ <&r "" セ@ :
dx = -wy; dy = wx dt dt
· Show that the point ( x, y) lies on a circle.
セヲ\ャ@ セ@ flo セ@ ( x, y) t(<li <'[<! 1f1: fur<! % I
20-II] 25 [Contd ...
111111111111111111111111111111111 • ' " - 1 1 •
-·-·· .>'
I
I I i
www.examrace.com
. 20-11]
,. 27'
' ,, '
' [Contd ...
I IIIII セiiiiiiiiiiiiiiiuiiiiiiii@• ' 0 - T 1 •
, . www.examrace.com
. I
!
·,
.! '
'
l ' ' Mセ@
PART - C I 11Ft - '6 Marks/ ai<l; : 10
. . Note : Attempt any 5 questions. Each question carries 20 marks. Answer should not ex(:tf€
.... 200 words. · ·.
;j]e : アLGヲセ@ セ@ ャャセGヲ@ セi@ llirl; セ@ セGゥ@ il; fu<l; ' セ@ oi<f; ?.!'llfur セ@ I OW 'o o Hoi'f '\'! att\ •nl't' tiT'fl 'llfi!'( 1
33 Verify Gauss' theorem for
. ヲB]Hク R MケコIゥKHケ R MクコIjKHコ R セクケIォ@
taken· over the rectangular parallelepiped.
OSxSa, OSySb, OSzSc .
f.= (x2- yz) i +(/ -xz)J +{z 2 -xy)k
io fu'( 7Jfu セ@ <liT at<l/41 セ@ ;>i\ AQセQQ|ャゥッ^@ !1<1/'i\{ BセBBGBG@ 0 :S x :Sa, 0 :S y:S
O:Sz:Sc. '(\ セ@ セ@ I
MMMMMMMMMMMMMMMMMMMMMMMᄋセ@
---'---------- ------------ ---·=
MMMMᄋMセᄋMᄋᄋMᄋᄋセ@
MMMMセMMMMMMMMMMMMMセMM..
--.. ·----------
---- ----------- -·--------'------
セPM II] 28
--- -. MLLMNLNLセZZ^]ᄋMM··,_.,
--··--·-.-.,..., ·-·· ,,..,,., セ@
www.examrace.com
., ' 'j:
-zoo' II'] 29 [Contd ...
I Ill II IIIli Aャャャャャャャャャャセャャャゥゥ@ n11 Mセ@ J-.h-: ' .. ';- '
• ' 0 - 1 ' •
·r---- Lセ@www.examrace.com
34 (a) A cylinderical tumbler, half filled with a liquid of density p is completely fi.lled wi't liquid- of density p' 'which does not mix with the former one. If h is the height an r be the radius of the base of cylinder, then find the ratio of the pressure on the bas of the tumber and the whole pressure -on its curved surface. '<'!> セセQQ\Q^QH@ V<i1'IT p 'A(<[ it '<'!> lr'f ·'if an'IT 'Rf :san t p' 'l'!i'l it '<'!> セ@lr'f 'if "'T セ@ "'ffi lr'f 'if セ@ fliffirr t, 'iU 'Rf "ffi!T % I '1f<: セ@ <ti セ@h <f'lT illl<!R q\1 セJ\it@ r m <i'r V<i1'IT it illl'IT<: 1f{ <:1'1 <r'll ;mit 'llll T'O 1f{ """ <:1'1 '61 セ@ Olffi セ@ I
(b) Sill equal heavy rods freely hinged at their ends form a regular hexagon ABCDF:J which when hung up.by the point A is kept from altering its shape by two light rod BF and CE. Find the thurst of these rods.
20-ll]
19: '<'!> m 'lffi '!% 1!'1d>!dl'{li<l> fuff '!\ '¥f % ofr1: '<'!> セZーゥ@ ABCDEF 'l'ffil % I -,i\ A 'if セ\\ャ^ャアャ@ ;;mrr J. ofr1: <;l セ@ 00 BF <r'lT CE 1mT 'W .• セ@セ@ -niD "ffi!T % I "" 00 iJ >fUi\o: Olffi 'lfu; I
.
.
30
ᄋQQAQセQ@ )1111 !li イセゥャャA@ IU! llll!f
..
www.examrace.com
;
I I I
'
I
!
i '
35 (a) The angular elevation of an enemy's position on a hill h meter high is B. ShoJ
that in order of shell it, the initial velocity of the projectile must not be less エィ。セ@
セサ@ gh ( l+ 」ッウ・」セIス@
h 1\'r?:{ セ@ floW セ@ .'TI ".'1i セ@ 'lit ft<lfu '61 セ@ ゥfヲヲゥゥセt@ セ@ '/; · I ftRl セ@
ヲャッᄋセ@ 'li itffi セ@ <m: セ@ t ft;r:; ャヲセ@ "') セ@ セ@ セサァィHャK」ッウ・」セIス@ セ@
'!>'! '!# irrr I
(b) A particle falls from rest under gravity in a medium whose resistance varies a:
20-II]
square of the veloci_ty, It v be the velocity actually acQuired by it, v0 , the カ・ャッ」ゥエセ@
.it would have 。」アオゥイセ、@ had these been no resistance and V, the terminal velocit)
ャセ R@ 1 v5 1 v{i Prove that "J'::::l-2 v2 +
2.3 v4- ............... .
0
1),'li '1i"T セ|ゥᆱBャャ\ィッヲGャ@ t &<!R f<l<IYI<H<rr セ@ 1),'li i);i! 'll'Zl'! i'i 17mn t furo'6T mmtl
セ@ t <'f'l '61 セGャャェTQ\ヲゥ@ W I '1f<:: iJWI; 1ffiT 'IT@f i\ m flo'IT 'l'IT セ@ v %, v1
""' セ@ t ;it ""' m <I>T<!1 r セ@ <fi'r{ mmr<l\ l!TU!'l '!# i'tm t '*' v O!f.n
2 2 4 v 1 v0 · 1 v0 2::::}--2+-----:t-········· v0 2V 2·3V
32 [Conti
11111111111 m 111111111111 • , 0 - 1 I
www.examrace.com
I -
.
. .
-
,, --{ ' .
'
.
.
.
. . ..
. .
20-11] 33 [Corttd •..
lllln 111111111111111111111111111 . ' ' - ' ' . .-
. ---, -- -- -- . . -- -- -- -- .
www.examrace.com
I '
I
I '
' ' '
36 Solve the following transportation problem :
セ@ qft'l&'l ww:rr 'lit 1'\'[ セ@ :
Dt D, D. 3 n.
0 l 1 2 1 4 30
o, • • 2 1 50 J J
o, 4 -2 . 5 9 20
20 40 30 10 100
\
20- II J 34 [C
1111!111111 Ill 111111 • ' 0 -
-- MMMMMMMMセセセMMBGBGMセセNLNN@ LNGャGAスセᄋLᄋャャAャLjヲッAA[⦅IGBセGMZMNL@ セᄋ@ -- ---------..,...,""'!'1!'1!!1'1!1!!!1!11 .-.. セ@ ..
www.examrace.com
-... ·-
T"-
··' "•:.',_
20-II] 35 [Contd •.• 11111111111 m IIIIIIIIUIIIIIIII
• Z 0 - I I •
www.examrace.com
37 (a) Use Gauss1s forward formula to find ·:v-ror· クᄋセ@ 3'0-; given that--:-·.
X . 21 25 29 .. .. .33 .. . 37 . --·· -·- ... - --- -- --- --- ·-- --,
16·3432 15·5154 y 18·4708 17·8144 17·1070
'lfu om wr 'I> セ@ -\\- x = 3 o TI Y <61.-iWr -.n<f - i セ@ セ・ョ@ tc::_-- -
X 21 25 29 33 . . 37
)' 18·4708 17·8144 17·1070 16·3432 15·5154
(b) Solve the difference equation
ux+l -?ux+l + 12u;: = cosx
-.3Rf\ e:ffttfil.'!LtLx+l -7ux+l +12u;c:;::;cosx .coT セ@ CflftQ;- L.
. . .. ....
·. . . . . . . .
···•·· ........• . : \"
. . . ' .....
. ·. .
. .
'. •.
.
. "---··--------·· ---
...•...
-...
. .
.
. ,.. . '· 20
ZO-II I .. iiAgャェゥゥゥセiセiIセmャZゥゥゥ@ i :•;.
• ..
www.examrace.com
! !
' セ@ :
38 · (a) Find general and singular solution of the following equation
f.r"f - セ^ゥ|BBBG@ <61 "lN<!i crm セ@ ·-..r '"" セ@ :
クー R MRケーKTクセッ@
(b) Solve :
1R'f セZ@
dx dy セ@
dz
z(x+ y) z(x-y) x2 + y2
20-11] 38 [C 111111 セセセ@ m mn . , ' -
www.examrace.com
20-'- II] .. [C:ontd ... ·
11mi QQQQセ@ llllllllilllllllll 39
, > 0 - 1 • I "
' . -- BッMM^Mイセセ@ <-
www.examrace.com
I
I 39 (a) Solve : \<\if セ@ :
( D5 - D )y = 12 ・セ@ +8sinx-2x
.(b) Solve by the method of variation of paraineters . . 1IT'I\i! セ@ セ@ lffi1 \<\if セ@ I
20-II] 40 [Cor
IIIIU IIIII m 111111111 • ' 0' - • '
www.examrace.com
.
. -..
-
-
--
-
--. -----
. ᄋセ@
---. -
20-II] . 42 [Contd ... 20
11m111111 m 11111111!111111111 ' ' . • • .
.. MMMMMセMM _s,...., "*'*;.o;Z. -· -"·-----···--·---· ·---- サNLイセ」Nᄋ@ .• セMM
__ ,,.,_
.. , - ·"'·
' www.examrace.com
'
.
'
i-,
.,
' .
, '
43
- セMM --MセL@ ...-'--·-------·---: _____ ------ ..
!--,_.:O:AQ( uc Mセᄋ@
'
,
. -
[Contd ...
llllllllllllllllllllllllllllllllll • > 0 セ@ I 1 '
- -· ---
. i ,
I 'f 'i
f ,I
J J i
I
,,
I
,! ' ,f
ᄋMMMMMMMセi@
www.examrace.com
MMMMMMMMMMMセMM
_-=.__c:.·. ---·-·=-·· ·::..:.· MセMM ... ·_:_.::--·::r· セ@
. .. . - . .. .. -. . .. - - .
. ·
... - -·
.
.
.
. .. . . . ...... .
. .... ----·· ·- ·····- M・ZMセ@
.... ... -. -- -1----------=
-- ... - - ---- ᄋZZZQZZZZセ@
i" __ _...... ...
- --- ..... ..,- --- . --· -- .....
. ·-
------·-
MMMMMMMMMMMMMMMMMMMMセセ]MᄋMM]MᄋMMᄋ]MセQM
.
2G-IJ] 44
'' ., .....
------
- ·-· --=
--- MMMMMMMMMMMMMMBMセM ' ....... l ' www.examrace.com
.
-- . . .• .
... . .. .
: ';-:''' ..
. ..
- . .
. .
I
45
-
[Coutd .•.
11!111 jill! Aiセ@ QセQQセAQQQQQQQQAQQ@
I
. '
I
'
' 0 N -<0 \=-
:) ! i
i 'I
www.examrace.com
SPACE; FOR ROUGH WORK/<'!i セᄋ@ '11; iWJ セ@
20-11) 46 [Contd... 20-
llll!lllllllll mlllUIIIIIIII • ' 0 - 1 1 •
ᄋᄋ[MセMMMM--·----------.
www.examrace.com
-:r-
' 47 [Contd .•.
I IIIII セuiiiiiiiiiャャャャャャャャャャャャャ@• ' " - 1 r •
'T セ@0 N - I.
"' セ@ I
NLNNNNNLNNセᄋMMᄋMᄋᄋᄋ@ ------·-·--.
www.examrace.com