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B T P S R T 0 8 0 1 7B. T e c h D e gre e II Se m e s te r Ex am ina tion i n Po l y m e r S c ie nc e andR u b b e r T e c h n o lo g y , A p ril 2 0 0 8

PTF 1201 ENGINEERING MATHEMATICS II(N e w S c h e m e )T im e : 3 H o u r s ax im u m M ar ks : 100

PART A( An s we r ANY FIVE ques t ions)( Al l que s t ions c a r ry EOUAL m ar ks ) (5 x 5 = 2 5 )I . a) ind the volume of the t e t rahedron form ed by the po in ts

(1 ,1 ,3) , (4, 3 , 2) , (5 ,2 ,7)an d( 6,4,8) . b ) i n d t h e v a l u e s o f k for which the equat ionsx+y+z=1,2x+y+4z=k,4x+y+10z=k 2have a so lu t ion and so lve them

comp le te ly in each case .12 3 0-2 4 3 2(c) i nd the r a nk o f 3 2 1 36 8 7 5

U s e N ew t o n R ap h s o n m e t h o d t o f ind a r o o t o f x S i n x + C o s x = 0 w h i ch is r e a rx = .A pply Gauss e l iminat ion m ethod to so lve :x+4y z=5;x+y6z=12 ,3x y z=4.A pply Eu l e r s me thod to so lve y = x + y ; y (0) = 0 c h o o s in g t h e s t e p l e n g t h = 0 . 2(car ry ou t 4 s teps)Use graphical method to so lve :M a x i m i ze z = 30x + 40y

S u b je c t t o 5 0 x + 3 6y 100,00025x + 36y 5. 91,000; x y O .

PART -

a) i th usua l nota t ions, show tha t V 2 (r ) =n (n +1) r n - 2 6)(b ) vec tor f ie ld is g iven by f ( S i n y ) i + x ( 1+ C o s y ) j . Evaluate the l ine in tegral

ove r a c i rcu lar pa th x 2 + y 2 = a 2 z =O . 9 )OR(a ) i nd the va lue o f a i f a x 2 y + yz )i+(x y2 _ xz2) j +( 2 x y z 2 x y2 ) k h a s z e r o

divergence . 5)(b) er i fy G auss d ivergence theorem for the funct ion f = y i + x j + z 2 k over thecy l ind r i ca l reg i o n b o u n d ed b y x 2 + y 2 =9,z =0 a n d z = 2 . 1 0)

(a) o lve by ma t r ix me thod x + y + z = 6, x + 2y + 3 z = 4, x + 4y + 9 z = 6 . 6 ) T u r n O v e r )

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6 2 2 2 3 and the eigen vectors. What are the

2 3( b ) ind the eigen values of A =

2

eigen values of A 1

7+2=9)

R

V . State the Cayley Hamilton Theorem. Verify the same for A =21a n df i n d i t s in v e r s e . 9 )Show that the equations 3 x + 4 y + 5y = a , 4x + 5 y + 6 z = b , 5 x + 6y + 7 z = c

do not have a solution unless a + c = 2b .6 )V I. ( a ) Find a real root of the equation x 3 2 x 5 = 0 b y t h e m e t h o d o f f a ls e p o s i ti o n ,c o r r e c t to 3 d e c i m a l p l a c e s . ( 7 )( b ) F i n d t h e c u b e r o o t o f 1 0 u s in g N e w t o n -R a p h s o n m e t h o d . ( 8 )

V I I . S o lv e b y G a u s s -Jo r d a n m e t h o d :2 x -3y+z= -1;x+4y+5z=25;3x-4y+z=2. ( 8 )

Find by Newton s method, a root of x 3 3 x 5 = 0 . ( 7 )V I I I . ( a ) Using Runge Kutta method of order 4, compute y ( .2 ) a n d y (.4) from

1 0 y = x 2 y 2 ;y ( 0 ) = 1 , h = 0 .1 .d x ( 8 )( b ) Solve y = y 2 + x , y ( 0 ) = 1 using Taylor s method and compute

y ( 0 .1 ) a n d y ( 0 .2 ) ( 7 )R

IX . ( a ) G i v e n t h e t a b le o f v a l u e s o f x a n d y .x.00.05 .1 0 .1 5 .2 0 .2 5 .3 0y.0 0.0247.0488.0723 . 0 954.1180dy d2 yfmd at x =1.00 . 1.1401 ( 7 )dx2x

6x( b ) b yvaluatef 2s i n g0 1 X( i) r a p e z o i d a l r u l e i i ) addle s rule ( 8 )X . o l v e t h e L PP b y s im p l e x m e t h o d :M a x Z 4 x, + 3x2 + 4x 3 + 6x4S u b j e c t t o + 2 x 2 + 2 x 3 + 4x 4< 80

2x , + 2x 3 + x03x, + 3x 2 + x 3 + x4603x, + 3x 2 + x 3 + x480, x i x , x , x 4 > O . 1 5 )

X I. f ir m p r o d u c e s p r o d u c t A a n d B a n d s e l l s t h e m a t a p r o f i t o f R s .2 /- a n d R s .3 / -respectively. Each product is processed on machines G and H. Product A requires1 m i n u t e a n d G a n d 2 m i n u t e s o n H w h e r e as p r o d u c t B re q u i r e s 1 m i n u t e o n e a c hof the machines. Machine G is not available for more than 6 hours 40 min/day wherea s t h e t im e c o n s t r a in t f o r H i s 1 0 h o u r s . So l v e t h i s p r o b l e m u s i n g s im p l e x m e t h o d f o rm a x i m i zi n g th e p r o f i t . 1 5 )

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