PART 01 ENGINEERING MATHEMATICS

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1 PART ENGINEERING MATHEMATIS (ommon to all canddates) (Answer ALL questons) 4 4. If the rank of a matr b s, then 9 9 b value of b s ) ) ) 6 4) 4. If the rank of non-square matr A and rank of the augmented matr of system of lnear equatons are equal, then the system ) s nconsstent ) has no soluton ) s consstent 4) does not have soluton. I f the s ystem +y+za, y+zb, +y zc, where a, b, c are constants, s consstent, then t has nfnte solutons only when ) a+b+c ) a b+c ) a+b c 4) a+b+c 4. If A, then the algebrac and geometrc multplcty are respectvely ), ), ), 4),. The sgnature of quadratc form y+yz+z s ) ) ) 4) 6. If ulog y, then u +yu y s equal to ) u ) u ) 4) 7. + y If utan y, then u +yu y +y u yy equals ) ) sn u cos u ) sn u cos u 4) sn u cos u 8. If uyz, v +y +z, w+y+z, then (, y, z) ( u, v, w) s equal to ) ( y)(y z)(z ) ) ( y)(y z)(z ) y y z z 4) yz ) 9. The partcular ntegral of (D +D)y ++4 s ) +4 ) + + ) 4) ( +4). In the equaton (t)+y(t) sn t, y (t) (t)cos t, gven () and y(), f cos t sn t cos t, then y s equal to ) cos t sn t+sn t ) cos t+sn t sn t ) sn t cos t sn t 4) cos t+sn t+sn t. If mnmum value of f() +b+c s greater than mamum value of g() c+b, then for s real, ) <c< b ) no real value of a ) c > b 4) c >b. For m the par tal dfferental equaton by elmnatng the arbtrary constants a and b from b y za log as ) p yq ) p+qp+yq ) yp q 4) p+q z. The partcular ntegral of (D DD +D )ze +y s ) e+y ) e +y ) e +y 4) e +y 4. If ftan y, then dv (grad f) s equal to ) ) ) 4). If F a + byj + czk, then F. ds, where S s the surface of a unt sphere, s 4π ) ( a + b + c) ) 4π ) (a+b+c) 4) S 4 π(a+b+c) SAKTHI TANET - ENGINEERING MATHEMATIS- e/tancet-mba-ma--8/maths-tancet-8-8p/ts

2 6. The value of y sn d + cosydy, where s the plane trangle enclosed by the lnes y, π y and y, s π ) 8 ) π 4 π (π +8) ) 8 π (π +4) 4) π + 7. If f(z)u+v s analytc, then ts frst dervatve equals ) u v ) u + v 8. The value of z, s ) v u 4) u u z + 4 dz z +, where s the crcle ) π ) π ) π 4) π 9. The value of ellpse +y, s ) ) ) 4z + z + dz z 4, where s the. The pole of cosz snz s π π ) ) ) π 4) t. The value of ) e sn t dt t s 4) log ) log ) log 4) 4. The soluton of (D +9)ycos t, y() and y(π/) s gven by ) y (cos t+4 sn t+4 cos t) π 4 a. The Fourer sne transform of e s ) tan (s/a) ) tan (s/a) ) tanh - (s/a) 4) tan (s/a) 4. If Z(u ) n ( z ) z + z + 4, z >, then the value of u s equal to ) ) 9 ) 46 4) 9. As soon as a new value of a varable s found, t s used mmedately n the equatons, such method s known as ) Gauss-Jordan method ) Gauss-Jacob s method ) Gauss Elmnaton method 4) Gauss-Sedal method 6. The value of for the data (, ), (, ), (, 9), (, ) and (4, 8) s ) ) 8 ) 7 4) 6 7. If y(), y()4, y()8 and y(4), then y() s equal to ) ) 6. ) 8 4) 8. The jont probablty densty functon of a random varable (, y) s gven by f(, y)kye ( +y ), where, y>. Then the value of k s ) ) ) 4 4) 9. The two lnes of regresson are perpendcular to each other f the co-effcent of correlaton equals ) ) ) 4) ±. Let the random varable X have the probablty densty functon ) y ( cos t+sn t+cos t) ) y (cos t+4 sn t+4 cos t) 4) y (cos t 4 sn t+4 cos t) > e for f otherwse Then the moment generatng functon s ) t ME ENGINEERING MATHS : ANSWERS * * * ) t ) + t 4) t SAKTHI TANET - ENGINEERING MATHEMATIS- e/tancet-mba-ma--8/maths-tancet-8-8p/ts

3 PART ENGINEERING MATHEMATIS DETAILED SOLUTIONS b a + b c b. () 4 4 b 9 9 b 4 b b + b b 4 b + b b b 4 b + b b b b 6 b Snce the rank s any determnant of order 4 + b b b 9 + b 6 b ( b)(6+b) b 6 (or) b. () If ρ(a) ρ(a, B) then the gven system s consstent.. () [A, B] a b c 4. (). () 6. (4) b a + b a + b + c Snce the system has nfnte solutons mples rank s less than. a+b+c Algebrac multplcty Geometrc multplcty A A [ ]+[ ] + D a D D A Dfference between postve square terms and non postve square terms Sgnature u u u log y. y y b a c R R y. y SAKTHI TANET - ENGINEERING MATHEMATIS- e/tancet-mba-ma--8/maths-tancet-8-8p/ts

4 7. (4) Let f(u)z u y u. y y u +yu y tan u y y y + y y + y y learly z s a homogeneous functon of degree. Formula : 8. () g(u) nf u f ' u tanu sec u snu cos u cos u snu cosu sn u u u u + y + y g(u)[g (u) ] sn u(cos u ) sn u cos u sn u sn 4u sn u sn u cos u u yz u u u yz ; z ; y v +y +z v v v ; y ; z w +y+z w w w ; ; Now u u u u, v, w v v v, y, z w w w yz z y y z y z yz ( y)(y z)(z ) Now (, y, z) ( u, v, w) ( u, v, w) (, y, z) z y y y z z a b c bc ca ab ( ) ( a b)( b c)( c a) 9. () Aullary equaton s m +m m(m+) m, m.f Ae +Be A+Be + If P.I., then Soluton y A+Be + +4 then dy d Be + +4 d y d (D +D)y orrect opton s () Be + d y d + dy d ++4 SAKTHI 4 TANET - ENGINEERING MATHEMATIS- e/tancet-mba-ma--8/maths-tancet-8-8p/ts 4

5 . (). (). () () (t)+y(t) sn t... () y (t) (t) cos t... () cos t sn t cos t (t) sn t cos t+sn t sn t cos t+sn t+ y(t) sn t y(t) sn t+ cost sn t y(t) sn t+cos t sn t f() +b+c f () +b f () f () +b b f ( b) > b gves mnmum Mnmum value ( b) +b( b)+c b b +c b +c g() c+b g () c g () g () c Now g ( c) < c c gves mamum Mamum value ( c) c( c)+b c +c +b c +b Mnmum value of f()>mamum value of g() b +c >c +b c >b c > z p z b b y a log a b( y ) b y ( ) ( ) ( ) ( ) ( ) ( ) a b y a b y SAKTHI ( )p a... () q z a b( y ) ( ) b ( ) a b a b y y (y )q a... (). () 4. (4) From () and () ( )p (y )q p p yq q p+q p+yq PI Formula : Now D e + y DD' + D' + y + y e e 4D D' 4 e + y dv (grad f). f f f + f f tan y f f y y + + y y ( + ) + ( + y ) y ( + y ) y y. f. y +. + y + y y ( + y ) TANET - ENGINEERING MATHEMATIS- e/tancet-mba-ma--8/maths-tancet-8-8p/ts

6 . () 6. (*) 7. (4) f y dv (grad f) f ( + y ). ( y) ( + y ) y ( + y ) SAKTHI 6 f + f y y ( + ) ( y + y ).F + j + k. ( a + byj+ cz k) a+b+c By Gauss dvergence theorem F.nds ˆ.Fdv S π y V ( + + ) V a b c dv (a+b+c) dv V (a+b+c) volume of the unt sphere (a+b+c) 4π () y π 4π ( a + b + c) O y y, y π, y π wll not form a trangle The data gven n the problem are not correct. f(z) u+v 8. () f (z) u + v u u By R equatons u v z Z z Let f(z) z Resdue at z Now y z + 4 z + 4 z + z + z s a smple pole z + 4 lm z z + z + 4 lm z z + 4 dz z + f z dz π (sum of resdues of poles wthn ) [By auchy s Resdue Theorem] π π 4 TANET - ENGINEERING MATHEMATIS- e/tancet-mba-ma--8/maths-tancet-8-8p/ts 6

7 9. () +y y + z sn t S L t ds S S + 4 S If f t t has a lmt as t and L(f(t))F(s), then f t L F ( s) ds t S log s log s + 4 S O z z4 s log 4 s + 4 s s log 4 s + 4 z 4 les outsde of the ellpse f(z) 4z + z + s analytc nsde z 4 f z dz By auchy s theorem 4z + z + dz z 4 sn t L t st sn t e t Put s s log 4 s + 4 s log 4 s + 4 s log 4 s + 4. (4). () To fnal pole of cos z sn z cos z sn z tan z z π 4 Pole s z π 4 L(sn cos t t) L s put cos z sn z [L() L(cos t)] S + S S 4 SAKTHI 7 t sn t e t log 4 log 4. (*) Aullary equaton s gven by m +9 m 9 m ±.F A cos t+b sn t P.I. cos t D + 9 cos t cos t y(t) A cos t+b sn t+ cos t TANET - ENGINEERING MATHEMATIS- e/tancet-mba-ma--8/maths-tancet-8-8p/ts 7

8 y() y π 4 A + A B B + 6 B 8. () Now y() If we use any nterpolaton method we get the value near to 6. ( y ) + k ye d dy. () y(t) 4 6 sn t cos t cos t + (4 cos t 6 sn t+cos t) a Fourer sne transform of e s tan s a. (4) Requred method Gauss sedal method. 6. () 7. () (, ) (, o ) (, ) (, ) (, 9) (, ) (4, 8) (4, 4 ) (, ) (, ) 7 y() y() 4 y() 8 y(4) y() +.e., 9. () y k ye dy e d.e., k 4 k 4 If the two regresson lnes are perpendcular to each other, then the coeffcent of correlaton s equal to.. (*) Moment generatng functon s t ( t) e e d e d e ( t ) ( t) ( ) ( ) e d t t ( t) e t ( ) t ( t ) SAKTHI 8 TANET - ENGINEERING MATHEMATIS- e/tancet-mba-ma--8/maths-tancet-8-8p/ts 8