F.Y. Diploma : Sem. II [CE/CR/CS] Applied Mathematics

Transcription

1 F.Y. Diplom : Sem. II [CE/CR/CS] Applied Mhemics Prelim Quesio Pper Soluio Q. Aemp y FIVE of he followig : [0] Q. () Defie Eve d odd fucios. [] As.: A fucio f() is sid o e eve fucio if f() f() A fucio f() is sid o e odd fucio if f() f() Q. () If f() + 5, fid f(o) + f() [] As.: Give f() + 5 f(0) (0) (0) f() () () f(0) + f() f(0) + f() 0 Q. (c) Fid if y e [] As.: y e Diff wr e d ( ) + d () [use produc rule] e + () e + Q. (d) Evlue si [] As.: si si si d () (cos ) (cos ). cos + cos cos + si + c Q. (e) Evlue [] +cos As.: cos cos sec + c Q. (f) Fid he re ouded y he curve y, is d he ordies, [] As.: A y - -

2 Vilkr : F.Y. Diplom Applied Mhemics sq. uis Q. (g) Se rpezoidl rule of umericl iegrio. [] As.: Trpezoidl rule [ mrks] The rpezoidl rule is umericl mehod h pproimes he vlue of defiie iegrl. We cosider he defiie iegrl. f We ssume h f() is coiuous o (, ) d we divide [, ] io suiervls of equl legh. usig he + pois 0, +, +,..., + We c compue he vlue of f() hese pois. y 0 f( 0 ), y f( ), y f( ),. y f( ) rpezoidl rule formul f y y y... y y 0 Q. Aemp y THREE of he followig : [] Q.() Fid if + y y + 5y 6 0 (, ) [] As.: Fid if + y y + 5y 6 0 (, ) Give y y + 5y 6 0 Diff w.r.. + y y() y y (y + 5) y + y y5 A (, ) () () () () 5 (,) 8 (,) Q. () If cos, y si, fid π As.: cos y si diff. w.r.. diff. w.r.. d d d ( cos ) d d d ( si ) [] - -

3 Prelim Quesio Pper Soluio d d (cos ) d d (si ) [ + mrk] (cos ) (si ) (si ) (cos ) (Chi Rule) (Chi Rule) (cos ) si (si ) cos By prmeric fucio / d / d (si ) cos (cos ) si si cos si cos ( / ) ( / ) Q. (c) If I d I e he curres d R d R e wo resisces i prllel o he ol curre I I + I which is cos. The he he developed i circui is give y H IR + IR. Show h he developed i circui is J miimum if I R I R where R, R,, J re coss. [] As.: Give H (I R I R ) J H I R (I I ) R J ( I I + I Diff. w.r.. I dh di J (I R + (I I ) () R ) I I I ) J ( I R IR + I R ) Diff. w.r.. I dh di J (R + R ) Pu dh di 0 J (I R IR + I R ) 0 I R IR + I R 0 I R + I R IR I R IR I R (I I ) R I R I R dh Pu I R I R i : di J (R + R ) > 0 He is miimum if I R I R Q. (d) A em is e i he form of he curve y si si. fid he rdius of [] curvure of π As.: y si si cos cos si si - -

4 Vilkr : F.Y. Diplom Applied Mhemics Pu cos cos 0 () si si Rdius of curvure / () / uis Q. Aemp y THREE of he followig : [] Q.() Fid he equio of ge & orml o he curve + y + y 5 (, ) [] As.: + y + y 5 + y, + + y 0 + y + y 0 + y y ( + y) y y y A d y m slope () () () () Equio of ge is y y m( ) y ( ) y + + y 0 Equio of orml is y y m ( ) 5 5 y ( ) y y y 0 Q. () If e y y prove h log y log y -. [] As.: Give : e y y Tkig logrihm y log y () Diff. w.r.. + log y () y - -

5 Prelim Quesio Pper Soluio y y y y y + log y log y log y log y log y y y From () y log y log y logy logy logy log y (log y) (logy) logy Q. (c) Fid if y si + ( ) [] As.: Le u si, v ( ) y u + v () Diff. w.r.. du + dv () u si Tkig logrihm log u si log Diff. w.r.. du u si + (log ) (cos ) du u si log cos du si si log cos () v ( ) kig logrihm log v log ( ) diff. w.r.. dv V sec + log ( ) () dv v sec log( ) dv sec ( ) log ( ) () From (), (), () si si (log ) (cos ) + ( ) sec log ( ) - 5 -

6 Vilkr : F.Y. Diplom Applied Mhemics - ( ) Q. (d) Evlue : I + [] ( ) As.: I Pu I + c ( ) + c Q. Aemp y THREE of he followig : [] e ( + ) Q.() Evlue : I cos ( e ) [] e ( ) As.: I cos (e ) Pu e e + e (e + e ) e ( + ) I cos sec + c (e ) + c Q. () Evlue : 5+cos [] As.: Pu cos I c + c - 6 -

7 Prelim Quesio Pper Soluio - Q. (c) Evlue :. [] As.: Le I Iegrig y Prs I (u) I d ( A (v) ) + c ( + ) + c I ( + ) + c cos Q. (d) ( + si ) ( + si ) [] As.: Pu si cos cos ( ) ( ) ( ) ( ) A + B A ( + ) + B ( + ) Pu A() A Pu B() B log ( + ) log ( + ) + c log ( + si ) log ( + si ) + c Q. (e) Evlue: As.: Le I π/ π/6 + co / /6 co / 6 cos si [] - 7 -

8 Vilkr : F.Y. Diplom Applied Mhemics I Usig propery I I / si /6 si cos () f() / /6 f( ) si ( / ) si ( / ) cos( / ) / cos /6 cos si () Addig () d () I / si cos cos si /6 I / /6 / /6 I I 6 6 Q.5 Aemp y TWO of he followig : [] Q.5() Fid he re ewee he prol y d y [] As.: y y Pu y i y 6 ( 6) 0 0 or y Y y y A (y y ) 0 / / sq. uis

9 Prelim Quesio Pper Soluio Q.5 () (i) Form Differeil Equio for y A si m + B cos m [] Y A si m + B cos m diff wr As.: y A si m + B cos m diff. w.r.. A cos m m + B ( si m) m Diff w.r.. A si m m + B(cos m) m m (A si m + B com ) m y + m y 0 Q.5 () (ii) Solve + y si As.: Comprig wih py Q p, Q si Boh re fucios oly. Give D.E. is Lier D.E. I.F. p e e p e log Soluio is Y (I.F.) Q (IF) + c y (). si + c y ( cos ) () (cos ) + c y cos + si + c Q.5 (c) The quiy of chrge of couloms psses hrough coducig wire durig smll iervl of me sec is give y dq i, where i is curre i mpere. If i 0 si 00 d h q 0, 0. Fid he chrge ime As.: dq i dq 0 si 00 dg 0 si 00 Iegrig dg 0 si 00 q 0 ( cos00) 00 q cos00 + c 0 whe q 0, 0 [] [6] + c - 9 -

10 Vilkr : F.Y. Diplom Applied Mhemics 0 cos 0 + c c 0 + c 0 c 0 cos00 q q ( cos 00 ) 0 couloms Q.6 Aemp y TWO of he followig : [] Q.6() (i) Usig Trpezoidl rulepproime vlue of give y, clcule he [] 0 0 y 0..7 As.: Usig Trpezoidl Rule Y h/ (y 0 + (y + y + y ) + y ) Y / (0 + ( ) + ) Y / (0 + (.6) + ) Y 5.6 Soluio By Trpezoidl Rule is Q.6 () (ii) Usig Simpso s oehird rule evlue y usig he followig le. 5 y As.: 5 y Usig Simpsos Rule Y h/ (y 0 + (y + y ) + (y ) + y ) Y / (0 + ( ) + (70) + 00) Y / (0 + (0) + (70) + 00) Y Soluio By Simpso s / Rule is [] Q.6 () Usig Simpso s rd rule fid he re uder he curve y si from 0 o [6] kig π 6 s he commo wih of he srip. Compre he resul wih he ec re. As.: f () [ mrks] - 0 -

11 Prelim Quesio Pper Soluio Usig Simpsos / Rule Y h/ (y 0 + (y + y + y 5 ) + (y + y ) + y 6 ) Y 0.56/ (0 + ( ) + ( ) + 0) Y 0.56/ (0 + () + (.7) + 0) Y Soluio By Simpso s / Rule is By cul iegrio : 0 0 si cos (cos cos 0) (cos 0 cos ) () Boh vlues re sme upo hree deciml plces. Q.6 (c) Usig Simpso s 8 h rule o fid As.: f () e y kig seve ordies. [6] Usig Simpso s 8 Rule h/8 (y 0 + (y ) + (y + y + y + y 5 ) + y 6 ) 0./8 ( + ( ) ) 0./8 ( + (0.99) + (.587) ) Soluio By Simpso s /8 Rule is