EXERCISE - 01 CHECK YOUR GRASP

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1 DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc () 5 n 5 c n( ) n 5 n 5. () (). Diffrniaing, w g d() c sinc i passs hrough (,) c 6 Hnc 6. cos Pu v hn v + dv v cos v dv cos v v + dv an v n + c an n + c As i passs hrough h poin (, ) so c an (n ) 6. ( + ) + I.F. an an + an an an ( ) an an + c 9. + d() n + c L I.F. + n 5 + n c so soluion is.. + c c an n + an n an + c an (c )

2 EXERCISE - BRAIN TEASERS. Givn k k + c Now a, so c. 5 + (as k 5 ) Whn, 6 min. 5. n c 6. + n c ( n c ) ( n c ) ( ) ( ) a + c( ) a a a n c + a c( a ) c k ( ) n + nc a (a ) a (a ) + (a + ) ( + a ) 7. f() nf() ( ) a a L u du f(u)du nf() f(u)du nf() f() n[f() + f'()] f() n f'() n n n n n n + nc n n n c c'. m + c n n I saisfis + m + m m c (m m) + (m c) This is an idni so m m m & c m So wo such sraigh lin ar possibl. EXERCISE - Tr u/ Fals :. d + + hn d and + Hnc is a soluion of h abov quaion. Now, hn d and + On puing his in quaion, also saisfis h quaion. 5. a + b + c d d a Hnc dr, dgr. a + b MISCELLANEOUS TYPE QUESTIONS Assrion & Rason :. A + n B Hnc dr is. So samn-i is fals & samn-ii is ru. Comprhnsion : # F rsrvoir A d A d A A k A A d A k A A A A k Similarl B B k so A B A B (k k ) log A, A B A A k

3 and a, A.5 B so (k k ) (k k ) log / and also log log. A, A k B so k k. L a boh h rsrvoirs hav sam quani of war, hn A B (k k ) / log /. Now A B (k k ) f() (k k ) f'() (k k ) (k k ) ln (k k ) f() is dcrasing. EXERCISE - [A] CONCEPTUAL SUBJECTIE EXERCISE. + + g + f + c Diffrniaion, w g, + ' + g + f' Again diffrniaing, + (') + '' + f''...(i) Again diffrniaing, (')'' + ''' + ''' + f''' ( ') ''' + ''' (from (i)) '' '' '( '') ''' ( ') 5. c + c + c... (). c + c + c... () d d 9c + c + c... () 7c + 8c + c... () Appl () 6 () + () 6 () d d Pu r sc, r an r... () and sin... () On diffrniaing (), rdr... () On diffrniaing (), cos d r sc d... (). 6. rdr Now r sc d r r dr r n(r + r sc d ) n(sc + an) + nc r + r c (sc + an ) + c sin sin cos sin cosc cos n an sin c n an c sin dm m dm m n m + c m k (a, m m ) k m m m m (a, m m ) n

4 8. A n n f() (f()) n (n ) ' n (n ) (n ) n C n (,f()) n C, (f(), f() ) n n /n. Pu X + h Y + k 9. h k h k dy X Y dx X Y Pu Y vx Xdv v v dx v h, k v dx dv ( v)( v) X dv n X c v v v n v n n X c v / Y X n X Y n c X Y X Y n X Y X Y c X Y (X Y) k ( + ) k( ) + + k k k + k k k n c(k ) k k k Pu a. so Pu so da da a a da a a da a a a v v + dv v v dv v v da (a ) + + a da a a dv v v ( v)dv v + v an v n + v n + c an a n a n + c an a n + a c whr a + ( ) ( ) ( ) I.F. / an c 6. ' b + b ' b ( + b ) ( b) b b log b log c b log log( + b) log( b) + logc ( + b)( b) c

5 .. b + ( c + b ) ( + b) b + k ( + b) cos cos Pu cos I.F. / cos sin c sin c Pu I.F. is. ( + ) c c + I.F. + ( ) n ( ) ( ) ( ) n n 6. Equaion of angn 5. Y (X ) whn Y, X a ± a a c ± a n c k 5. O Y ( ) ( ) an () n c ann c 5 7. Equaion of angn Y HG ± c P(, ) K J, a a + c X an n + nc (X ) n n. + c ( + ) + ( n) c disanc from igin Equaion of nmal Y (X )

6 Y X + disanc from igin Now ihr ( + ) ( ) n + v an v nc n Hnc soluion will b n + v + n ± an v + nc v k ± an v c ± an / 5 8. Inpu ra gm/min Afr im volum of ank 5 + ( ) Concnraion of sal in im Oupu ra m. gm/min. 5 m 5 gm/li 5 li. a Pu v v + dv v v v + dv v v dm m + 5 li/min m(5 + ) (5 + ) m(5 + ) c li/min v v dv v dv v n +v + an v n + nc ( a, m c ) m(5 + ) 5 ( + ) m 5 5 gm EXERCISE - [B] BRAIN STORMING SUBJECTIE EXERCISE. ( a ) du + Pu Q; dv + Pv Q d (u v) P(u v) d(u v) P u v n (u v) P P Q I.F. u v Q. + c u v u v u Q + c' [u saisfis i] u v u v u + k u v u v u + k (u v)... () ( b ) If u + v is a paricular soluion hn compar wih () k +, k + ( c ) If is a paricular soluion hn i saisfis () u + k (u v) u v u. Y ' A ais, X +. ' consan (X )

7 mid poin of PQ ', 8. f() F() + c f() F '() mid poin lis on l F() ' sin + cos. ( sin ) Pu I.F. sin (,)P. sin sin.sin cos ( sin ) + c + ( ). + c ( + ) + c, c + Q sin [Pu sin ] ( ) sin ( ) + c sin + c sin sin + c 6. f() > d I.F. ( f()) k f() + k [f() ] (k + ) (k ) + k k n k k +. + (k + ). k d (.k + ) g(). k + dcrass g() f(). k + f(). k + f(). k + f() A. k. sin + c. sin f() cos (sin ) Pu + P Q, z c. sin cos + P Q dz + z dz + z dz + z Q P + P z Q dz + z Q Q dz dz z Q Q n z + z + a Q Q( z)

8 EXERCISE - 5 [A] JEE-[MAIN] : PREIOUS YEAR QUESTIONS. v + dv v.v dv dv v v Pu v + dv v v log( + v ) log + c v log + log( + v ) logc log.. c log c v v v dv v logc c c + c ' c ' c c '... (i) pu in quaion (i) log '. Givn ( ) log( ) log + logc c log '.... (i) Equaion (i) passs hrough (, ) C ( ) C Pu in (i) ( ) ( + ) + ( ). Equaion of givn parabola is A + B whr A and B ar paramrs A d + This is h quaion of givn parabola dr, dgr 5. ( + ) ( an ) (+ ) + + an + Now I.F. soluion an an an ( ) an an ( ) an an + C + K 6. Givn famil of curvs is an an + C + a... () + ' a'... () Now pu h valu of a from () o in () + '.' + ( )' ( )' 7. + ( + ) + d ( ) ( ) log + c + log c 8. c( + c )... () 9. c + c c c ( + ) C Pu in quaion() ( ) Dgr dr Pu v, v + dv dv dv v log v log which is homognous qu. v v + dv v log v(logv + ) v vlogv + v v log c log(logv) log + logc

9 . Givn A + B Divid b B A B + B Diffrnia w.r. A +... (i) B Again Diffrnia w.r.. A B + d Pu A B d d d +... (ii) + in quaion (i) I hav scond dr and firs dgr.. L h cnr of circl is (h, ) and radius will b also h. quaion of circl ( h) + ( ) h h + h + h h +... (i) Equaion (i) passs hrough igin diffrniaing i w.r.. h+ h + pu in quaion (i) pu v, v + dv + v dv v + dv dv v log + c log+c... (i) Givn () log + c c pu (i) n +. Equaion of circl ( h) + ( ) 5...(i) Diffrnia w.r.. ( h) + ( ) ( h) ( ) pu in (i) ( ) + ( ) 5 ( ) (') + ( ) 5 c X. c...() c ' c c...() " c c c " c '...() Now () () ' c Pu in () ' ". ' " (') 5. cos (sin ) sin cos an sc an sc an sc an sc... () Pu in quaion () From quaion () & (), w g, +. an sc an I.F. log sc sc soluion of diffrnial quaion is :. sc sc. sc. + c sc an + c sc (an + c) 6. log + + c (), (log )?... ()

10 () log + + c c log 5..(log )? log + log + log 5 log + log + 7 d 7. k(t ) d K (T ) K T C A I C I K T I (T) KT T KT T I 8. Equaion of angn a (, ) is inrcp Accding o qusion n n + nc c Now a, c c Ingraing fac (I.F.) Gnral soluion is. / / c I / L I / pu I ( ) ( ) Gnral soluion is / / C + + C/ Pu, + + C/ C / + / dp(). P() 5 ingra dp P 9 n (P 9) C...() givn P 85 C n 5 from () ln (P 9) + n5 n P 9 5 P 9 n 5 a P n 9 5 n8. P. C /, P C p ( 5) 5 + 5

11 EXERCISE - 5 [B] JEE-[ADANCED] : PREIOUS YEAR QUESTIONS. L X b iniial populaion of h counr and Y b is iniial food producion. L h avrag consumpion b a unis. Thrf, food rquird iniiall ax. I is givn Y ax 9.9 ax... () L X b h populaion of h counr in ar. Thn dx ra of chang of populaion X. X dx X. ( + ) and () 5. Givn : liquid vapas a a ra propional o is surfac ara. dv S... () W know, volum of liquid r h and surfac ara r (of liquid in conac wih air) r h and S r... (). Ingraing dx X. log X. + c X A.. whr A c A, X X, hus X A X X. L Y b h food producion in ar. Thn Y Y.9aX (.) ( Y.9aX from ()) Food consumpion in h ar is ax.. Again f no food dfici, Y X.9 X a (.) > a X. (.). >.9 9 Taking log on boh sids, [n (.).] n n 9 n n 9 n(.). Thus h las ingral valus of h ar n, whn h counr bcoms slf sufficin, is h smalls n n 9 ingr grar han qual o n(.). I.F. soluion is ( + ) ( + ) ( + ) ( + ) + c givn () c + C 6. Also, an R H From () and (), r h... () r co and S r... () Subsiuing () in (), w g co. r. dr co dr K R Kr T whr T is rquird im afr which h con is mp. co ( R) K(T ) R co KT H KT (using ()) T H K cos ( ) sin log( + ) log( + sin) + c log( + ) + log( + sin) c h cos sin Givn () mans whn, log + log c logc c + sin r sin H

12 8. (a) v, v + dv This is homognous so pu v + dv. v (v) dv v v ( v )dv v v pu v givn () v v v v v dv v logv log + c c c Now pu, +. (b) d log dv v + log + c log + c log c Givn () + c c + Now ( ), + + ( ) ( + ) Bu >. (a) lim ƒ() ƒ() Appling L-hospial, w g ƒ '() ƒ() + ƒ '() ƒ() ( ) + ( b ) d ƒ() ƒ() c + Also ƒ() c ƒ() + + c ( + c) + Cnr ( c,); radius. sc sc + c () sc sc + 6 Now cos cos cos c 6 sc(sc 6 ) + 6 cos cos Hnc S(I) is ru and S(II) is fals.. (A) ( ) (B) 5 c,. ln I ( )( )( )( )( 5) Appling c I (5 )( )( )( )( ) I I.

13 (C) ƒ () cos + sin ƒ '() cos sin + cos cos ( sin + ) cos sin sign of ƒ ' () changs from v o +v whil ƒ () 5 passs hrough,. 6 6 (D) ƒ () an (sin + cos ) cos sin ƒ () > (sin cos) / / ( /, /) /. Givn ƒ() Tangn a poin P(, ) Y (X ) (, ) 5 / Now inrcp Y Givn ha, is a linar diffrnial quaion wih I.F. n n Hnc, soluion is. C C Givn ƒ() Subsiuing w g, C so 7 9 Now ƒ( ) 9 5. (a) (Bonus) (Commn : T h givn rla ion do s no hold f, hrf i is no an idn i. Hnc hr is a n r r i n givn qu s ion. T h c rc idn i mus b-) 6 ƒ() ƒ() 5, Now appling Nwon Libniz hm 6ƒ() ƒ'() + ƒ() ƒ() ƒ'() L ƒ() d C (whr C is consan) + C ƒ() + C Givn ƒ() C ƒ() + 6 (b) Givn (), g() g() L '() + (). g'() g() g'() '() + (() g()) g'() '() () g '() () () dg() I.F. g() g() d (g( )) g( ) g( ) g ( ) ( ). g( ).dg() g( ) g( ) g( ) (). g(). c pu + c c (). g() g() g() g() + () + () 6. an I.F. an sc cos Equaion rducs o.cos.sc.cos cos + C () + C cos () sc

14 ( (A) is crc) ( (C) is wrong) Also '() sc + sc an and '. 6 ' ( (D) is crc) 7. ƒ'() ƒ() < Mulipl boh sid b ƒ '() ƒ() d ƒ() Now, g() ƒ() g() is a dcrasing funcion. g() g ƒ() ƒ() < ƒ() / / / / ƒ() obviousl ƒ() is posiiv / ƒ() 8. sc L v dv v dv v v sc v ( (B) is wrong) cos v dv sinv n+ c sin n c passing hrough sin c c 6 sin n, 6 Paragraph f Qusion 9 and 9. (ƒ''() ƒ'() + ƒ()) D((ƒ'() ƒ()) ) D((ƒ'() ƒ() ) (ƒ'() ƒ()) is an incrasing funcion. As w know ha ƒ() has local minima a (ƒ'() ƒ()) a L F() (ƒ'() ƒ()) F() < in, (ƒ'() ƒ() < in, ƒ'() < ƒ() in, opion C. D( (ƒ'() ƒ()) (, ) D(D( ƒ()) (, ) D ( ƒ()) L F() ƒ() F''() > mans i is concav upward. F() (,) (,) F() F() F() < (, ) ƒ() < (, ) ƒ() < Opion D is possibl