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 (!'\

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"' ｌ｟ｌ｟ｪｾｾ｟｟ｪ｟｟｟｟ｌ｟｟｟ｌ｟｟ｌ｟ｌ｟｟｟ｬ｟ｾ｟ｌＮＮ＠_____ _J, ,.

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[Contd.

111111111111111111111111111 • ' • - 1 '

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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 ---

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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 . ' . ' - .

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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

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,---

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 '

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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 . ' . - - ' .

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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

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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＠ • -·-

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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

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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＠ - ' .

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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 •

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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 . ' . - .

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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 - ' -

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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

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I I

I I j

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•

' ' -----

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

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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

-

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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 ------ -

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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

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.. .

I I I

. . 0-I) 19 [Could ...

llllillllll IIi ｾｉｉｉｉｉ＠ ｉｾ＠ 1111 . ' . .. ' .

.. .. .. ·-

-

---· ----- --------- __ ,_ ---ＭＭｾﾷ＠ - ---- -- .. ---------···• . .-·.:' ·.- '

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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 . , ' -

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-

-

. ' ,_,-.

-

-

.

.

. 0-1)

21 [Contd: .. I lllln IIIII m ＱＱＱＱＱＱＱＱｾ＠ 1111 • 0 - ' -

. ' ' .

---- -· ｾ＠ ---- ＭｾＭＭＭＭＭﾷ＠ --

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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

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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>¥)

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----------------- ···--·-----

--. ----·

---------------------·---

-------------------- ---------

20 -I] 25 [Contd ... I IIIII ｾｾｾ＠ ＱＱＱＱＱＱＱＱＱＱＱｾ＠ 1111

• ' ' 0 - ' •

," . .>:::-"c:. __ ;_ ._ -.;,;.:,;;.,, ＬＬ［ＺＬｾ［ＬＬＮ＠ i ''• ｾＭＧＭＭＭｾＭｯ＠

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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＠. ' ' -

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I I

1-- -· r- ·-'· - 1-·--, .

.

. -: .

'

I· ' . .

-

I _,

i . ' .

0-11 27 [Contd ... 1111!111111 m lllllllllllli

• ' 0 - ' • .

'

. . -- "·--··--' ' .... ' -' ＧｾＭＭＭＭ- --- .

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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

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.

.

---------------------------------------------

-------------------------------

ＭＭＭｾＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭＭ

.

0 -I] 29

[Co11td ... ..

Ill IIIUIIIIII m lillllli ill • ' a - ' •

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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 • ' •

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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＠

[

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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＠ .

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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＠

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'

20- j 20-II 1 2 [C rrmr11111m 111111

• ' 0 -

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... ··

.. .-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 •

-------- .:<-_. - . . -.. ' . --·

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' '

'!

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 •

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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 . . . -

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!.

·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-·

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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

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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

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' 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 ' •

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____ ...

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＠< ' •

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'• .. ··--··--

' . - . .. .•

.. ·-- ＭＭｾ＠ --' ·- ' -.. , ·-· -· .

ntd ... . \, Ｍｾ＠ -;

1111111111 ' . 13 [Cpntd ••.

ＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱｾ＠ 11111111111 • > 0 - ' ' •

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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

'

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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

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'- 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＠

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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 . , . - ' ' .

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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

.

. -

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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

'

i

I I

. ' 20-llj 26 [Con

ＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＱＡＱＱｾ＠' , 0 - '

www.examrace.com

. 20-11]

,. 27'

' ,, '

' [Contd ...

I IIIII ｾｉｉｉｉｉｉｉｉｉｉｉｉｉｉｉｕｉｉｉｉｉｉｉｉ＠• ' 0 - T 1 •

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. 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＠

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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.

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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＠

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(b) A particle falls from rest under gravity in a medium whose resistance varies a:

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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- ............... .

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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

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2 2 4 v 1 v0 · 1 v0 2::::}--2+-----:t-········· v0 2V 2·3V

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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

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20-II] 35 [Contd •.• 11111111111 m IIIIIIIIUIIIIIIII

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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.

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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

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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

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20-II I 48 622

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