F.Y. Diploma : Sem. II [AE/CH/FG/ME/PT/PG] Applied Mathematics

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1 F.Y. Diploma : Sem. II [AE/CH/FG/ME/PT/PG] Applied Mahemaics Prelim Quesio Paper Soluio Q. Aemp ay FIVE of he followig : [0] Q.(a) Defie Eve ad odd fucios. [] As.: A fucio f() is said o be eve fucio if f() f() A fucio f() is said o be odd fucio if f() f() Q. (b) If f() , fid f(o) + f(3) [] As.: Give f() f(0) (0) 3 3(0) f(3) (3) 3 3(3) f(0) + f(3) f(0) + f(3) 0 Q. (c) Fid if y e a [] As.: y e a Diff wr e d (a ) + a d () [use produc rule] e + a () e + a Q. (d) Evaluae si [] As.: si si si d () (cos ) (cos ). cos + cos cos + si + c Q. (e) Evaluae [] +cos As.: cos cos sec a + c Q. (f) Fid he area bouded by he curve y 3, ais ad he ordiaes, 3 [] b As.: A y a

2 Vialakar : F.Y. Diploma Applied Mahemaics sq. uis Q. (g) Three ubiased cois are ossed. Wha is he probabiliy of geig eacly wo [] heads. As.: Three ubiased (fair) cois are ossed. Sample space S {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} (S) 8 Le C Eve eacly wo heads {HHT, HTH, THH} (C) 3 The, C 3 P(C) S 8 P (eacly wo heads) 3 8 Q. Aemp ay THREE of he followig : [] Q.(a) Fid if + y y + 5y 6 0 a (, ) [] As.: Fid if + y y + 5y 6 0 a (, ) Give + y y + 5y 6 0 Diff w.r.. + y y() y y (y + 5) y + y y5 A (, ) () () () () 5 (,) 8 (,) Q. (b) If a cos 3, y b si 3, fid a π As.: a cos 3 y b si 3 diff. w.r.. diff. w.r.. d d d (a cos3 ) d d d (b si3 ) [] a d d (cos3 ) b d d (si )3 [ + mark] a 3(cos ) (si ) b 3 (si ) (cos ) (Chai Rule) (Chai Rule) - -

3 Prelim Quesio Paper Soluio 3a (cos ) si 3b (si ) cos By parameric fucio / d / d 3b (si ) cos 3a (cos ) si bsi acos bsi acos b( / ) a( / ) b a Q. (c) If I ad I be he curres ad R ad R be wo resisaces i parallel o he oal curre I I + I which is cosa. The he hea developed i a circui is give by H IR + IR. Show ha hea developed i a circui is J miimum if I R I R where R, R,, J are cosas. [] As.: Give H J (I R I R ) H I R (I I ) R J ( I I + I I I I ) Diff. w.r.. I dh di J (I R + (I I ) () R ) 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 Hea is miimum if I R I R Q. (d) A beam is be i he form of he curve y si si. fid he radius of [] curvaure a π As.: y si si cos cos si si Pu - 3 -

4 Vialakar : F.Y. Diploma Applied Mahemaics cos cos 0 () si si Radius of curvaure 3/ () 3/ uis Q.3 Aemp ay THREE of he followig : [] Q.3(a) Fid he equaio of age & ormal o he curve + 3y + y 5 a (, ) [] As.: + 3y + y y y y + y 0 + y 3y (3 + y) 3y 3y 3y A ad y m slope () 3() 3() () Equaio of age is y y m( ) y ( ) y + + y 0 Equaio of ormal is y y m ( ) 5 5 y ( ) y y y 0 Q.3 (b) If e y y prove ha log y log y -. [] As.: Give : e y y Takig logarihm y log y () Diff. w.r.. + log y () y - -

5 Prelim Quesio Paper 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) log y Q.3 (c) Fid if y si + (a ) [] As.: Le u si, v (a ) y u + v () Diff. w.r.. du + dv () u si Takig logarihm log u si log Diff. w.r.. du u si + (log ) (cos ) du u si log cos du si si log cos (3) v (a ) akig logarihm log v log (a ) diff. w.r.. dv V a sec + log (a ) () dv v sec log(a ) a dv sec (a ) log (a ) a () From (), (3), () si si (log ) (cos ) + (a ) sec log (a ) a - 5 -

6 Vialakar : F.Y. Diploma Applied Mahemaics - (a ) Q.3 (d) Evaluae : + [] 3 (a ) As.: I Pu a I 3 + c (a ) + c Q. Aemp ay THREE of he followig : [] e ( + ) Q.(a) Evaluae : cos ( e ) [] e ( ) As.: I cos (e ) Pu e e + e (e + e ) e ( + ) I cos sec a + c a (e ) + c Q. (b) Evaluae : 5+cos [] As.: Pu a cos I a c a 3 a + c 3-6 -

7 Prelim Quesio Paper Soluio - Q. (c) Evaluae :. a [] As.: Le I a Iegraig by Pars I a (u) I a d (a A (v) ) a a a a a a + c ( a + a ) + c I ( + ) a + c cos Q. (d) [] ( + si ) (3 + si ) As.: Pu si cos cos ( ) (3 ) ( ) (3 ) A + B 3 A (3 + ) + B ( + ) Pu A() A Pu 3 B() B 3 log ( + ) log (3 + ) + c log ( + si ) log (3 + si ) + c Q. (e) Evaluae: As.: Le I π/3 π/6 + co /3 /6 co /3 6 cos si [] - 7 -

8 Vialakar : F.Y. Diploma Applied Mahemaics I Usig propery I I /3 si /6 si cos () b f() a /3 /6 b a f(a b) si ( / ) si ( / ) cos( / ) /3 cos /6 cos si () Addig () ad () I /3 si cos cos si /6 I I I /3 /6 /3 / Q.5 Aemp ay TWO of he followig : [] Q.5(a) Fid he area bewee he parabola y ad y [6] As.: y y Pu y i y 6 ( 3 6) 0 0 or y Y y y A b (y y ) a 0 3/ 3 3/ sq. uis

9 Prelim Quesio Paper Soluio Q.5 (b) (i) Form a Differeial Equaio for y A si m + B cos m [3] 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 (b) (ii) Solve + y si [3] As.: Comparig wih py Q p, Q si Boh are fucios of oly. Give D.E. is Liear D.E. I.F. e p e 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 quaiy of charge of coulombs passes hrough a coducig wire durig [6] small ierval of ime sec is give by dq i, where i is curre i ampere. If i 0 si 00 ad ha q 0, 0. Fid he charge a ime As.: dq i dq 0 si 00 dq 0 si 00 Iegraig dq 0 si 00 q 0 ( cos00) 00 + c q cos00 + c 0 whe q 0, 0 0 cos 0 + c c 0 + c 0-9 -

10 Vialakar : F.Y. Diploma Applied Mahemaics c 0 cos00 q q ( cos 00 ) 0 coulombs Q.6 Aemp ay TWO of he followig : [] Q.6(a) (i) I samplig a large umber of pars maufacured by a machie, he mea [3] umber of defecive i a sample of 0 is. Ou of 000 such samples. How may would be epeced o coai aleas 3 defecive pars. As.: Usig Biomial Disribuio 0, mea (m) mea m p 0p p 0. 0 q p The probabiliy of aleas 3 defecives i a sample of 0 is {p [ 0] + p [ ] + p [ ]} P [ r] C r. p r. q r for Biomial Disribuio P [ 0] + p [ ] + p [ ] 0 c 0. (0.) 0 (0.9) c (0.) (0.9) c (0.) (0.9) P [Aleas 3 defecive] No. of samples N P Q.6 (a) (ii) Assumig ha i 0 idusrial accides are due o faigue. Fid he probabiliy ha eacly ou of 8 accides will be due o faigue. As.: p q p r Usig Biomial Disribuio p [ r] C r. p r. q r p [ ] 8 C. (0.) (0.8) 8 P [y ] 0.9 This is required probabiliy. [3] Q.6 (b) I a es of 000 elecric bulbs, i was foud ha he life of paricular make [6] was ormally disribued wih average life of 00 hours ad sadard deviaio of 60 hours. Esimae he umber of bulbs likely o bur for (a) bewee 90 hours ad 60 hours (b) more ha 50 hours Give A () 0.77, A (.83) 0.66 As.: 00, 60, N 000 Sadardized value of if Z For 90, z

11 Prelim Quesio Paper Soluio For z 60, z 60 For z (a) p [90 60] p [ < z < ] p [ < z < 0] + [0 < z < ] [0 < z < ] Required o. of bulbs NP (b) P [ > 50] p[z >.83] 0.5 p [0 < z <.83] Required o of bulbs NP approimaely Q.6 (c) The umber of road accides me wih by ai drivers follow Poisso Disibuio [6] wih mea. Ou of 5000 ais i he ciy. Fid umber of drivers. (i) who do o mee wih accide (ii) who me wih a accide more ha 3 imes give : e As.: Give mea For poiso Disribuio m r e.m p r r! 0,,, (i) P [who do o mee wih a accide] p [ 0] 0 e. 0! No. of drivers who do o mee wih a accide N P drivers (ii) P [who me wih a accide more ha 3 imes] p [ > 3] p [ 3] {p (0) + p () + p () + p (3)} 0 3 e. e. e. e. 0!!! 3! No. of drives N P drivers - -