Vidyalankar. 1. (a) Y = a cos dy d = a 3 cos2 ( sin ) x = a sin dx d = a 3 sin2 cos slope = dy dx. dx = y. cos. sin. 3a sin cos = cot at = 4 = 1
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1 . (). (b) Vilnkr S.Y. Diplom : Sem. III [AE/CE/CH/CM/CO/CR/CS/CW/DE/EE/EP/IF/EJ/EN/ET/EV/EX/IC/IE/IS/ ME/MU/PG/PT/PS/CD/CV/ED/EI/FE/IU/MH/MI] Applied Mhemics Prelim Quesion Pper Soluion Y cos d cos ( sin ) sin d sin cos slope d d cos sin sin cos co co / y differenie w.r.. y y y. y ( ) y cos sin Vilnkr y y y y / /SY/Pre_Pp/Mech/Mhs_Soln
2 Vilnkr : S.Y. Diplom Mhs III. (c). (d). (e) Rdius of curvure is, I logloglog Pu log (log ) log d I d log () + C log [log (log )] + C I I log( ) log e e / / (/) (/) (/) / e log log () e e e e e () e e e. e e e C 9 9 e e e C 9 7 / y / Vilnkr y /SY/Pre_Pp/Mech/Mhs_Soln
3 Prelim Quesion Pper Soluion. (f). (g). (h). (i) Order Degree Are y sq. unis Y e e LHS e e RHS y log y e log e e log e e LHS RHS 5 5 I ( )( )( ) A B C ( )( )( ) A ( + ) ( + ) + B ( + ) ( + ) + C ( + ) ( + ) pu A (+) () Vilnkr A pu B () () B pu c () () C /SY/Pre_Pp/Mech/Mhs_Soln
4 Vilnkr : S.Y. Diplom Mhs III. (j) I ( ) log( ) log( ) log( ) C log log( ) log C I log C ( ) / n n / (sec ) / n. (k) Whie blls 6 Blck blls 9 Blue blls 5 One bll is drwn n(s) C Probbiliy h drwn bll is blck bll A n(a) 9 C 9! P(A) n(a) 9 C 8!! n(s) C! 9!!. (l) Probbiliy o ge si when one die is ossed 6 Probbiliy of si occurring in 8 oss 8 6 y Slope, m. () Vilnkr (,) () ( ) 8 m slope of ngen 9 /SY/Pre_Pp/Mech/Mhs_Soln
5 Equion of ngen is (y y ) m ( ) (y ) ( ) y 8 + y Slope of norml, m Equion of norml is (y y ) m ( ) (y ) ( ) y y 8 m / / /. (b) Given curve, y differen w.r.. / / y / / y / / y differen gin w.r.. y y / / / / / / y / / / y / / y / / / / y / / y / / / / y / y y / / / / / Prelim Quesion Pper Soluion y y y / / / / / y / / / / / / / Vilnkr / / / y / (y)..(proved) /SY/Pre_Pp/Mech/Mhs_Soln 5
6 Vilnkr : S.Y. Diplom Mhs III. (c). (d) y 9 7 differen w.r for ereme vlues, 8 or for 6 I is minim Y min 9 7 for 6 I is mim Ym cos I sin cos / sin / cos / sin / cos/ sin/ cos / sin / cos / sin / cos / sin / cos / sin / divide numeror & Denominor by cos / n / n / n / I n / n /n/ Vilnkr n log sec c log sec c / 6 /SY/Pre_Pp/Mech/Mhs_Soln
7 Prelim Quesion Pper Soluion. (e). (f). () seccosec I log (n ) pu log (n ) sec d n sec cosec d d I log + c log [log (n )] + c I 5 cos pu n d cos d I d 55 5 d 9 d n log c log c n I I co Vilnkr sin (i) sin cos sin / sin cos f() f( ) cos (ii) cos sin /SY/Pre_Pp/Mech/Mhs_Soln 7
8 Vilnkr : S.Y. Diplom Mhs III dding (i) & (ii) sin cos I sin cos cos sin I sin cos sin cos d when I I n I. (b) cos I sin pu sin cos d n () n (). (c) The equion of prbols re y 8 (i) 8y (ii) From (ii), y ( 5), 5 8 y 8 y 8 The required Are is 8y Vilnkr (8, 8) y 8 8 /SY/Pre_Pp/Mech/Mhs_Soln
9 . (d) A (Are under prbol y 8 8 Prelim Quesion Pper Soluion 8 ) (Are under prbol 6 Sq. unis cos y n n y cos cos sec y n sec Compring wih py Q P sec Q n sec p sec n Generl soluion is p p ye e Q C n n ye e nsec c pu n sec d e d c e d e d d c e e dc e e c n n n y e ne e c y y y y pu y v dv V 8y ). (e) Vilnkr /SY/Pre_Pp/Mech/Mhs_Soln 9
10 Vilnkr : S.Y. Diplom Mhs III. (f) V + dv v v (v) ( v ) dv v V v dv vvv v (v ) dv v dv V V Inegring boh sides, v dv v () v v log v log c ( ) log v log c logv y log y + c y c ( y) pu + y v dv + dv Vilnkr dv v dv v v dv v v v v dv /SY/Pre_Pp/Mech/Mhs_Soln
11 Prelim Quesion Pper Soluion. (). (b) v dv v dv v Inegring boh sides dv v v V n c y ( + y) n c I 5 5 (i) b b f() f( b ) I 5..(ii) dding (i) & (ii) 5 I 5 5 I I I 5 5 Vilnkr ( ) I sin cos pu n d /SY/Pre_Pp/Mech/Mhs_Soln
12 Vilnkr : S.Y. Diplom Mhs III. (c) sin ; cos d I d ( ) 6 d 6 d 699 d n log n P (, ) Q (, 5) R (, ) re he verices of ringle Equion of PQ is y 5 ( ) ( ) y + 5 (i) Equion of QR is y 5 5 ( ) y 5 (ii) Equion of PR is y ( ) ( ) y 5 (iii) c ( ) log c ( ) Are of PQR A (region PLOQP) + A(region OMRQO) A (region PLMRP) 5 5 A ( 5) (5 ) P(,) Q(,5) L m R(,) Vilnkr 5 sq. unis /SY/Pre_Pp/Mech/Mhs_Soln
13 Prelim Quesion Pper Soluion. (d). (e). (f) y y ( ) y Inegring boh he sides y n () n () + n (y) c n (y) c y e y y y y y m e y y N y y m y + y N y + y y m N y Equion is ec. Generl soluion is m N(erm free from ) c y consn e y y o c e y y c y sin (log ) cos(log) L.H.S. ( sin (log )) cos(log) sin (log ) cos(log ) Vilnkr sin(log ) cos(log ) cos (log ) y sin (log ) cos (log ) + cos (log ) + sin (log ) RHS /SY/Pre_Pp/Mech/Mhs_Soln
14 Vilnkr : S.Y. Diplom Mhs III 5. () Any person is seleced rndom p (rice eer) p (Non rice eer) 5. (b) le, A none rice eer ou of A one rice eer ou of A wo rice eer ou of A hree rice eer ou of p(a ) c c 9 p(a ) c c 8 pa c c 7 pa c c p A p A p A p A Required probbiliy m n i n ifi i fi % (ppro) c c c c Vilnkr m.7 m r e (m) p( r) r!.7 r e (.7) r! /SY/Pre_Pp/Mech/Mhs_Soln
15 Prelim Quesion Pper Soluion 5. (c) 5. (d) 5. (e) n p. m np. m r e (m) p ( r) r! (i) re defecive e () p( )!.8 (ii) A les re defecive (more hn re defecive) p ( ) p( > ) [p ( ) + p( ) + p( )] e () e () e ()!!! e e 5. I e ( ) cos (e ) pu e e e d e ( + ) d I d cos n() + c e c n sec d I ( ) n ( ) n () n ( ) Vilnkr /SY/Pre_Pp/Mech/Mhs_Soln 5
16 Vilnkr : S.Y. Diplom Mhs III 5. (f) 6. () y pu + y v dv + dv v dv v dv v v v v dv v dv v v dv v dv v Inegring dv v v n (v) c n ( y) c + y y n 8 y c P (A).8 P (B).6 P (A B).9 P (A B) P (A) + P (B) P (A B) P (A B) P (A) + P (B) P (A B) P (A B).5 P(AB) P (A / B) P(B).5.6 Vilnkr P (A / B) /SY/Pre_Pp/Mech/Mhs_Soln
17 6. (b) 8 is divided ino wo prs le one pr be & second be (8 ). le, y (8 ) y 8 8 for ereme vlue 8 for If is mim. one pr, second pr, (c) [cos + sin ] y [sin cos ] [ sin + cos + sin ] d cos [ cos + sin cos ] d sin / d / d sin cos n. (i) d sec sec / d sec cos sec Prelim Quesion Pper Soluion Vilnkr /SY/Pre_Pp/Mech/Mhs_Soln 7
18 Vilnkr : S.Y. Diplom Mhs III 6. (d) n sec sec sec sec sec (proved) I n ( ) pu n d n n I d n n d sec sin d cos cos sind sind sind ' d I ( cos ) + cos d I cos + sin + c n. cos (n ) + sin (n ) + C n P. m np. m r e (m) P( r) r! 6. (e) Vilnkr 8 /SY/Pre_Pp/Mech/Mhs_Soln
19 Prelim Quesion Pper Soluion 6. (f) (i) ecly e () P ( )!.89 (ii) mos P( ) P( ) + P( ) + P( ) e () e () e ()!!! e.996 P(rge hi) 5 P P(rge no hi) 5 5 q n P( r) n C r (p) r (q) nr (i) rge is ecly hi 5 imes 5 5 P( 5) C5 5 5 (ii) les shos hi he rge r P( ) C r 5 5 r r Vilnkr /SY/Pre_Pp/Mech/Mhs_Soln 9
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