# MATHEMATICS IV 2 MARKS. 5 2 = e 3, 4

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1 MATHEMATICS IV MARKS. If c epesents cicle with dius 6, find the vlue of c. R 9 f c ; g, f 6 9 c 6 c c. Find the eccenticit of the hpeol Eqution of the hpeol Hee, nd + e + e 5 e 5 e. Find the distnce etween the two points in plne whose pol coodintes e (, /6 (, /) Distnce etween the points P, 6 nd Q, is Hint: cos 5 cos (6 5) PQ cos( ) () () ()()cos cos 6 cos 5 + sin 6 sin cos5 6 units. Stte the Simpson s ule fo numeicl Integtion of function f() ove the intevl [, ] dividing [, ] into n su-intevls. Let P {, + h, + h,... n + nh }, whee h (n is n even intege) e ptitions of [, ]. n Let,,,... n e the vlues of f() t,,,... n espectivel. h f() d n 5... n... n. 5. Find the vete nd focus of the pol Eqution of the pol X ( ) 6 Comping with ( h) ( + k) We get h, k, 6 Coodintes of the focus e (h, k + ) Vete of the pol (h, k),., (, ) 6. Find the cente nd eccenticit of the hpeol Eqution of the hpeol is ( + 8) 6( + ) 6 9( ) 6( + + ) ( + ) 6( + ) ( ) ( ) 6 9 ( h) ( k) Compe with, we get h, k 6, 9 Cente of the hpeol (h, k) (, ) Eccenticit e Tking the oigin s the pole nd the positive -is s the initil, convet into pol fom. Eqution of the cuve in Ctesin fom is Put sin, cos, we get ( sin ) ( cos )

2 cos sin cot cosec is the eqution in pol fom. 8. Find e d. e d e d 9. Evlute: Put 9 t d (). e d d d.e () e d.e e + c e ( ) + c 8 d. 8 Then 9 8 d dt (o) 8 d 9 dt Now, 8 d 8 dt 9 9.tn (t) C tn ( ) C t Evlute: Let f(i) Then I / cos 5/ ( ) d. 5 / 5 / sin ( ) cos ( ) / 5 / cos () d... () 5 / 5 / sin () cos () / 5 / cos d 5 / 5 / sin cos I f() d f( ) d / 5 / sin d... () 5 / 5 / sin sin Adding () nd () I / / 5 / 5 / cos () sin () d 5 / 5 / 5 / 5 / sin () cos () sin () sin () / / 5 / 5 / cos sin / 5 / 5 / d d [] sin cos I I / 5 / cos d. 5 / 5 / sin cos. Stte Tpezoidl ule fo Numeicl integtion of function f(), ove the intevl [, ] dividing [, ] into n su intevls. Let P {, + h, + h,... n + h } Whee h, e pticin of [, ] dividing [, ] into n equl pts. n Let,,... n e the vlues of f() t,,... n espectivel. Then Tpezoidl ule h f() d [( + n ) + ( n )]. d. If the eqution epesents hpeol, find the limits of c (whee c is n el constnt). 9 c 5 c Given eqution is 9 c 5 c This eqution epesents hpeol, then 9 c > nd 5 c < 9 > c nd 5 < c 5 < c < 9. Find the length of the pependicul fom the pole to the stight line 6 cos + sin.

3 Eqution of the given line in the pol fom is 6 (i.e.,) 6 cos sin cos + sin 6 cos cos 6 Pependicul distnce fom pole to the given lines is p 6 units.. Evlute: ( )e cos (e ) d Put. e t Then [. e + e ()] d dt e ( + ) d dt Now e ( ) dt d sec cos (e ) (t) dt tn (t) + c tn (e ) + c cos t 5. Evlute: ( ) d, >, >,,. ( ) ( ) + ( ) + ( ) d d d d d d d Hint: d c, >, log c c c log log log log log log log log 6. Clculte d Simpson s ule dividing [, ] into 6 equls intevls. f(),, n 6 h 6 n 6 6 X f() B Simpson s ule f() d h [ + ( n ) + ( n ) + n ] h ( + 6 ) + ( ) + ( + ) [(8 + 8) + ( ) + ( + )] (6 8 ) Otin the pmetic equtions of the cicle epesented Cente of the cicle (h, k) (, ) nd dius () h k c The pmetic equtions of cicle with cente (h, k) nd dius () e given h + cos ; k + sin, whee < (i.e.,) + cos ; + sin 8. Find the eqution of tngents to the ellipse + 8, which e pependicul to + + Slope of the given line + + is The tngent is pependicul to the line + + Slope of the tngent is m Eqution of the tngent is m m... () Eqution of the ellipse is, Hee, 8, m 8 Fom () equtions of tngents e () () 8 9. If the eccenticit of the hpeol is 5, find the eccenticit of its conjugte hpeol.

4 Eqution of the hpeol is e e Eqution of the conjugte hpeol is... () e 6, () Find the length of the ltus ectum nd eccenticit of the conic ( + 7 cos ) 8 Eqution of the conic is ( + 7 cos ) cos cos Comping with + e cos ; we get l, e 7 Length of the ltus ectum l Eccenticit e 7.. If e cos, pove tht + If e cos( + c), then n n e cos c ntn Put,, c nd n, then e cos e cos ( + ) e cos + d. Evlute: e cos d, R cosh sinh cosh () sinh () cosh sinh cosh() sinh() d e cos[ + tn ()] Hint: cos h () sin h () [(cos h() sin h())] d cos h() d sin h() d sin h cos h + c. Find the e cut off etween nd The given pol meet, the X is t A(, ) nd Y is t P(, ) nd Q(, ) The pol is smmeticl ut X is Requied e Ae of OAP P(, ) 8 6 ( ) d 8. sq. units. O A X Q(, ). Find the eqution of the hpeol with eccenticit, focus (, ) nd the coesponding diecti is +. S(, ) is the focus

5 Eqution of the diectid is + e P(, ) is n point on the hpeol. Dwn PM pependicul to the diecti PM 5 B Def SP e. PM SP e.pm ( ) + ( ) ( ) 5 5( ) ( ) Eqution of the hpeol is M + + P(, ) S(, ) n 5. Otin: ( ) d, n Let I n ( ) d f() d f( ) d n I n n n ( ) d ( ) d ( ) d n n n n. n (n ) (n ) n n n n (n )(n ) n I. (n )(n ) 6. Find the e ounded +, X-is nd the odintes nd. Ae equied is A d 8 7 ( ) d sq. units 8 6 Y X X Y 7. Find the eqution of the cicle whose etemities of the dimete e (, ) nd (, 5). Hee (, ) (, ), (, ) (, 5) Eqution of the cicle hving A(, ) nd B(, ) s ends of the dimete is ( )( ) + ( )( ) The eqution of the equied cicle is ( )( ) + ( )( 5) (i.e.,) Show tht the line + is tngent to the pol 6. Also find the point of contct. Eqution of the line is () +... () is eqution of tngent in tems of m slope 6 Hee 6 Comping equtions () nd () we get m ; m Hee m + is tngent to 6 () Point of contct is,, (,) m m. 9. Find the pole of the line 6 with espect to the ellipse +. Eqution of the ellipse is +, Hee, Pole of the line 6 is n m,, l, m 6, n n

6 m ( 6) 7, n ( ) n ( ) Pole (7, ).. Find the e of the tingle fomed the points (, ), 5 5, nd,. 6 The e of the tingle fomed (, ), (, ) nd (, ) is sin( ) Requied e of the tingle fomed (, ), 5 5, nd, 6 is 5 ( 5)()sin ().sin(7 ) ()( ) 6 sq. units.. Find the n th deivtive of e + 5,. Let e + 5. Then n D n [e + 5. ] (e + 5 ) n + n C (e + 5 ) n ( ) + n C (e + 5 ) n ( ) n. (e + 5 ) + (n) n. e + 5 (n)(n ) () +. n e + 5 () e + 5. n [9 + 6n + (n)(n )].. Find the e ounded sin, is, nd. Y Requied e sin d cos cos + cos ( ) + + sq. units tn (). d on R {}. tn () d d tn d (tn ). d d tn. d d tn d tn n n C. Evlute: / sin d X - / sin( ), if We hve sin sin(), if / / / sin d sin d sin d sin( ) d sin() d / / / / sin() d sin d cos / cos cos cos cos cos / + 5. Evlute: d d Hint: 6 f () d f() C f() d Find the e enclosed etween the cuves,.

7 Given equtions e... ()... () Eliminting, we get X ( ) (o) (o) Points of intesection e O(, ), A(, ) Ae of the egion A 8 ( ) d. sq. units A(, ) X 7. If point P is moving such tht the lengths of tngents dwn fom P to + + nd e in the tio :, then find the eqution of the locus of P. Let P(, ) e n point on the locus. PT, PT e the lengths of tngents fom P to the given cicles. PT Then we hve PT (i.e.,) PT PT 8 B squing on oth sides 8 (o) 6 6 Locus of P(, ) is ( + ) + 8 / 8. Evlute: 6 sin cos d / sin cos d Using Simpson s ule find ppoimtel d tking equl intevls. Let,,, n h.5 n P {,.5,.5,.75, } i B Simpson s Rule h d [( + ) + ( + ) + ].5.5 d. 7 [5 + (.65) + (.5)] 8. [5 + ( ) + (.5)]. Show tht the line l + m + n is noml to the cicle S, if nd onl if gl + mf n. Eqution of the cicle is S (i.e.,) + + g + f + c The line l + m + n is noml to the cicle S, if the cente of the cicle ( g, f) lies on l + m + n l( g) + m( f) + n gl + mf n. Find the vlue of k, if the lines + + nd + k e conjugte with espect to the pol 8 The pole of the line + + w..t the pol n m () () is P,, P(6, 6) The lines e conjugte, P(6, 6) lies on + k

8 6 ( 6) + k k. Find the eqution of the hpeol efeed to its es s es of co-odintes whose distnce etween foci is 6 nd eccenticit is. Given e e 6 e (e ) ( ) Eqution of the hpeol is i.e.,.. Find the e cut off etween, +. The points of intesection of nd + e otined fom + ( )( ), Point of intesection e (, ), (, ) Requied e of the egion A ( ) d (9 8 ) sq. units. X Y O X. Evlute: cos + sin + d Put tn Then d Now dt t d cos sin t ; sin t t nd cos t dt dt t t t ( t) t t log + t + C log tn C Y e (+ ) 5. Find d on R { } ( + ) e ( ) e ( ) d ( ) ( ) ( ) e e d e. C C ( ). d Hint: e [f() + f()] d e. f() + C 6. Evlute: log d + Let f() log f( ) log log f() f( ) f() is odd log d. MARKS. Find the midpoint of the chod of the cicle intecepted the line. Eqution to the cicle is its cente is (, ) Eqution of the chod AB is... () Let M e the midpoint of AB

9 CM AB Slope of AB Slope of CM C B Eqution of CM is + ( ) + g + o Solve X we get co-odintes of M 7 9 /5, /5, M, 5 5. A M. Show tht the cicles c ; c touch if +. c Given the cicles e c... () c... () the eqution of dicl is of () nd (_ is The cicles () nd () touch ech othe if the dicl is is common tngent. Pependicul distnce fom cente (, ) to is equl to dius. c c. ( c) ( + ) c( + ) c.. Show tht the poles of the tngents to the cicle ( h) + with espect to the cicle + lie on the cuve ( h) ( + ). Let P( ) e the pole The pol of P() ) w..t. the cicle + is +... () It is tngent to the cicle ( h) +, then pependicul distnce fom cente (h, ) dius h squing oth sides we get (h ) Hence locus e ( h) ( + )... Find the eqution to the cicle which psses though the points (, ), (6, 5) nd whose cente lies on the line + 6. Let the eqution of the cicle e + + g + f + c it psses though the points (, ) (6, 5) + + g() + f() + c 8g + f + c 7... () nd g(6) + f(5) + c g + f + c 6... () cente ( g, f) lies on + 6 ( g) + ( f) 6 g + f () Fom () nd () 8g + f + c 7 g + f + c 6 Solving we get g, f, c 5 Hence equied eqution of cicle e Show tht the e of the tingle fomed the tngent t ( ) to the cicle + with the coodinte is is sq. units. Eqution of the tngent to the cicle X + t ( ) is +... () Since e of tingle fomed the line + + c c With co-odintes is is Ae of the tingle fomed the tngent + ( ) with co-odinte is is

10 Ae sq. units. 6. Find the eqution of the cicle which pss though the oigin nd is othogonl to the given two cicles + 6 ; Let the eqution of the cicle e + + g + f + c... () Since it psses though oigin c () cuts the cicle + 6 nd othogonll. g( ) + f( ) c... () g( 8) + f( ) c +... () Solving () nd () we get g 7/, f 8/9 Requied eqution e Find the vlue of C, if the cicle nd + + c cut ech othe othogonll nd + + c cut ech othe othogonll. gg + ff c + c ( 8)() 6( ) + c 6 + c c If the pol of p with espect to + touches the pol. Show tht the locus of p is + Let P(, ) e n point on the locus. Then pol of P(, ) w..t the cicle + is S + (o) +... () It touches the pol c m () Hence locus of p(, ) is + 9. If the pol of the point p w..t. the pol touches the cicle +. Find the locus of the point P. Let P(, ) e n point on the locus. The pol of P w..t. the pol is ( + )... () It touches the cicle +, then d Hence locus of P(, ) is.. Show tht the eqution of the chod joining the points P() nd Q() on the ellipse + is cos + sin cos. P() P( cos, sin ) Q() Q( cos, sin ) e two points on the ellipse Eqution of the chod joining the points P() nd Q() is (sin sin ) ( sin ) ( cos ) (cos cos ) sin sin cos. ( cos ) sin.sin cos sin cos o cos sin cos.

11 . Find the eqution of the noml t P() on the ellipse cos, sin d d sin, d d cos d tn d d d tn (slope of Noml) +. Eqution of Noml e sin tn ( cos ) cos sin cos sin. sin cos ( ) sin cos sin cos o is equied eqution of Noml. cos sin. Find the eccenticit, foci nd length of the ltus ectum of the given ellipse ; e foci ( e, ) 8, 7, Length of ltus ectum (6) If l + m + n epesents the eqution of the noml to the hpeol ( + ). m n Eqution of the hpeol is Eqution of the noml is sec tn +... () But eqution of the given line l + m + n... () Comping () nd () we get sec nd tn n m n m n ( ). m n (sec tn ) sec mtn n, show tht. If the lines l + m + n, l + m + n e conjugte with espect to the hpeol then pove tht ll mm nn. Two stight lines e side to e conjugte, if pole of line w..t. hpeol lies on the othe l + m + n... () l + m + n... () Pole of l + m + n w..t. hpeol Since the lines e conjugte is n m m n n n ll + mm + nn ll mm nn which is equied condition. m ; n, 5. Show tht the locus of point of intesection of the tngents of hpeol which e inclined t 5 to ech othe is ( + ) ( + ). Eqution of pi of tngents is S.S S Let P( ) e locus points S:

12 ( ).( ) ( ) Now enging eqution we get [ ] + [ ] + + ( ) tn h (ngle etween tngents) h tn ( + ) (h ) ( )( ) [ + ] [ + ] Hence locus e ( + ) ( + ). 6. If PSQ is focl chod of conic whose focus is S nd the length of the semi-ltus ectum is l, then show tht +. SP SQ Let the eqution of the conic e e cos. Let P(, ), Q( + ) e the ends of the focl chod e cos nd e cos( + ) + e cos e cos + + e cos 7. Pove tht the eqution cos + sin epesents cicle. Given cos + sin Put cos sin which is cicle. 8. Pove tht the cicle cos ( ) nd sin ( ) e othogonl. Eqution of the fist cicle is cos ( ) Its cente A, Eqution of the second cuve is sin( ) cos ( ) cos Its cente B, These two cicles psses though the pole O OA + OB AB + cos( ) cos AB OA + OB Hence cicles cut othogonll. cos 9. Find the equtions of stight lines pssing though the point, nd pllel nd pependicul to the given stight line cos + sin. Let the eqution of line pllel to cos sin e k cos sin It psses though the point k cos sin ( ) k () k,

13 Hence eqution of line is cos sin Let the eqution of the line to cos sin is R cos sin it psses though the point, k cos sin k Hence the eqution of line is cos sin. If cos (m log ), then show tht + + m nd hence deduce tht n+ + (n + ) n + + (m + n ) n. cos (m log ) sin(m log ). m m. sin(m log ) Diffeentiting w..t.. +. m cos (m log ). m + m cos(m log ) m + + m Diffeentiting n times w.. to, leinitz s theoem ( ) n + ( ) n + m n + ( ) n. + nc ( ) n () + nc. ( ) n ). + [( ) n. + nc.( ) n ] + m n n(n ). n + + n n n + n + + n + m n. n + + (n + ) n + + (n n + n + m ) n. n + + (n + ) n + + (m + n ) n. sin + cos. Find: sin + cos d. Let N A d (D) + 8(D) d sin + cos A( cos sin ) + B( sin + cos )... () Eqution coeffs. of sin nd cos, we get A + B A + B Solve, we get A,B Fom () sin + cos (cos sin) 8 (sin cos ) 5 5 sin cos (cos sin) 8 d d d sin cos 5 sin cos 5 log sin cos 8 () c 5 5 d. Evlute: 5 + sin d I d 5 sin dt t Put tn t d,sin t t dt dt dt dt. 8t t 5 5t 8t t t 6 t 5 t 5 t 5tn dt t tn c tn c.. Evlute: 6 + d 8 Put t d dt. d dt.

14 8 t (t ) d. dt dt 6 t t dt dt t [t tn (t)] + c 8 d [ tn ( )] + c 6 d. Find: + sin + cos Put tn t Then d dt t t nd sin t dt t d t t sin cos t ( t ) t nd cos t dt dt log tn c t t t t. 5. Evlute: + d I d d d 9..sin + C 9 6. If n is positive intege nd I n tn n d, show tht I n I n tn n d tn n. tn d tn n (sec ) d tn n sec d tn n d I 5 tn 5 d n tn I n n tn tn tn I tn tn I + tn d tn tn + log sec + c. (tn) n n I n nd hence evlute I 5. log(+ ) 7. Find: d + I I / log( ) d log( tn ).sec d putting tn tn / log( tn ) d / / tn log tn d log d tn / [log log( tn ) log. I I log I 8 log

15 / 8. If I n I n I n n tn d, then pove tht I n + I n / / / n n n tn d tn.tn d tn (sec ) d / / n n tn sec d tn d / I n + I n I I I / n tn I n n tn d I n / I d I. / nd lso evlute: n tn d. n- 9. If I n sec n sec tn n 6 d, then show tht I n + I n nd find the vlue of n n sec d. I n sec n d sec n. sec d d (sec n ). sec d d sec n sec d d sec n. tn (n ) sec n sec. tn d sec n tn (n )I n + (n )I n (n )I n sec n tn + (n )I n I n n sec tn (n ) n n I n sec d sec d 5 5 / / 6 / sec d Solve: ( + ) d + ( + ) d ( + )d + ( + ) d d d d d ln ln C. C. /. Solve: ( ) d + 5 d ( ) d + 5 d d 5 d I.F. e d d e d ln e 5 5. d d 5 C 5( ) + c.

16 . Solve: e d + e d e d + e d e. e d + e.e d e.d e d e d e d e e c. 7 MARKS. Find the eqution of the cicle which psses though the oigin nd elongs to the coil sstem of which (, ), (, ) e limiting points. Eqution of the cicle of which (, ) s limiting point is ( ) + ( ) i.e., () Eqution of the cicle of which (, ) s limiting point is ( ) + ( ) () Rdicl is of () nd () is 6 + Eqution of the coil sstem will e 5 + L ( + + 5) + (6 + ) It psses though the oigin ( + + 5) + ( + ) ¼ Hence the eqution of the equied cicle is ( + + 5) + (6 + ) o ( + ) 7 Definition: A sstem of cicles of which eve pi of cicles hs the sme dicl is is clled sstem of coil cicles o coil sstem. Eqution of the coil sstem in its simplest fom: Let + + g + f + c epesents the memes of the coil sstem fo diffeent vlue of g, f nd c Let the line of centes e the -is nd the common dicl is e the -is. The centes of ll cicles of the sstem lie on the -is its odintes e zeo (i.e. f ). Let the equtions of n two cicles e of the sstem X + + g + c... () X + + g + c... () Rdicl is of () nd () is (g g ) + (c c ) Rdicl is lies on -is c c o c c Let c c c The cicles () nd () ecomes X + + g + c... () X + + g + c... () Simill n othe cicles of the sstem with () nd () will e + + g + c Hence the eqution of coil sstem of cicles in simplest fom is c whee is pmete. C is constnt.. If (, ) is limiting point of the coil sstem of which is meme, find the othe limiting point. Eqution of the cicle coesponding to the limiting point (, ) is ( ) + ( + ) X () Anothe cicle of the sstem is The dicl is of the sstem is S S (i.e) Eqution of the coil sstem is S + L ( ) + ( ) whee is pmete. Its cente (, ) Rdius ( ) (9 ) Fo limiting point dius o Othe limiting point is (, ).. Find the eqution of the cicle cutting the two cicles nd othogonll nd touching the line + Let eqution of the cicle cuts the given cicle othogonll is + + g + f + c it cuts othogonll g() + f() c + 8 g c () it cuts othogonll g( ) + f() c + 8 8g c () Fom () nd () g c 9 it touches the line + d f 8f g f (f ) g t c f 8

17 f o f 8 Equtions of the cicles e + 8 nd Find the limiting points of the coil sstem detemined the cicles nd Eqution of the cicles e nd Rdicl is of the given cicles is 5 5 Eqution of the coil sstem is 5 + L ( + + 6) + ( 5) + ( ) ( + ) 5 its cente is c(, + ) Rdius ( ) ( ) ( 5 ) Fo limiting points, dius ( + ) ( + ), Hence limiting points e L ( +, ) (, ); L ( +, ) (, ). 5. If the two cicles pssing though the points (, ), (, ) nd touching the line m + c cut ech othe othogonll, then show tht c ( + m ). Let the eqution of n cicle e + + g + f + k... () Given () psses though the points(, ), (, ) + f + k... () f + k... () Solving () nd () : f nd k Given () touches the line m + c o m + c m( g) c g k m ( mg + c) (m + )(g + ) ( k ) g + cgm + (m + ) c. Let g g e the oots of the qudtic in g. g g ( + m ) c... () The equtions of the two cicles e + + g... (5) nd + + g... (6) (5) nd (6) cut ech othe othogonll if (g g + ) g g ( + m ) c o ( + m ) c 6. Show tht the locus of poles of ll the tngents to the pol with espect to the pol is the pol. Let P(, ) e the pole. The pol of P(, ) w..t the pol is ( + ).... () It is tngent to the pol c (o). m Hence the locus of P(, ) is. 7. Deive the eqution of the pol in its stndd fom. S e the locus nd ZZ e the Z diection of the pol. A is the mid point of SZ. Let SA AZ Tke A s the oigin nd AS s -is AS co-odintes of S e (, ). Tke AY s -is A(, ). Suppose P(, ) e n point on the pol M Dw PM to ZZ Dw PN to -is PM ZN ZA + AN + Fom the definition of the pol Z SP PM SP PM ( ) + ( + ) ( + ) ( ) : This is the stndd eqution of the pol. P N A (, ) S 8. Fine the condition tht the line m + c is tngent to the pol.

18 Let P(, ) e point on the pol the line m + c meets the pol t P. Eqution of the tngent t P(, ) is ( + ) +... () But the line is m + c () nd () epesent the sme line m c m + c P(, ) c nd m m P(, ) lies on the pol m c m c which is the equied condition. m A 9. Tngents e dwn to the pol t points whose scisse e in the tio k :. Pove tht the intesect on the cuve (k / + k / ). Let the tngents t R(t ) nd Q(t ) on the pol intesect t P) ) Eqution of the tngent t R(t ) is t + t... () Eqution of the tngent t Q(t ) is t + t... () The point of intesection of () nd () is (t t, (t + t )) t t... () (t + t )... () Given scisse of R(t ) nd Q(t ) e in the tio k : t : t k : t kt Now eliminting t, t, t fom () () (5) We get the locus of P(, ) Fom () t t (kt ) t B (5); t k Fom () (t + t ) t t tt k k k k B using (6) [(k) / + k / ] Locus of P(, ) is (k / + k / ).. If P is point on the ellipse + Let S nd S e the foci of the ellipse P e n point on the ellipse PM nd PM e pependiculs fom P to SP SP diectices l nd l espectivel. Fom definition of the ellipse e, PM PM e SP e(pm) SP e(pm) SP + SP e(pm + PM) e(mm) e (distnce etween the two diectices) e e Mjo is of stndd ellipse. SP + SP length of mjo is, which is constnt fo given ellipse.. Find: d., show tht SP + SP (constnt) whee S nd S e the foci. ( )( ) d d ( + )( + ) d ( ) ( ) d + c. Evlute: d + ( )d d I M Z l A S B B S A l M Z

19 d ln d ln ( ) ( ) ln tn c. cos. Evlute: cos + sin d I cos d cos sin Numeto A(denominto) + B d d (denominto) cos A( cos + sin ) + B( sin + cos ) A + B, A B solving them we get A (cos sin) ( sin cos ) 5 5 d cos sin ;B 5 5. d sin cos d log cos + sin + c. 5 5 cos sin 5 5 sin. Show tht: d + cos. I I I sin d ( )sin( ) d cos cos ( ) I sin d I cos ( sin) d tn cos cos / 5. If I n I n I n n tn d, then pove tht I n + I n / / / n n n n tn d tn.tn d tn (sec ) d / / n n tn sec d tn d / I n + I n I I I I / n tn I n n tn d I n / Io d. / / nd lso evlute: tn d. m n m Show tht I m, n sin cos d I m, n nd hence find the vlue of m + n sin cos d. I m, n / / m n m n sin cos sin.sin.cos d

20 / / m n d m n sin sin.cos d (sin ) sin.cos dd d / m cos n () m cos n sin (). (m ) sin cos. d n n + m n / / sin n n.cos.cos d m n m m sin cos ( cos ) d Im,n Im,n n n m m I m, n I m, n n n m I m, n I n m,n I 7 5 sin.cos d f() d f( ) d 7 5 sin ( )cos ( ) d I I 7. Find the e of the egion ounded the ellipse Eqution of the ellipse is + nd deduce the e of the cicle. B (, ) Y Ae of the egion ounded the ellipse (Ae egion OAB) d d A (, ) A (, ) sin Cicle e, when B 8. Show tht the e ounded, + nd -is... () Ae of shded potion A OAB + A ABC d d ( ) d d, + nd -is is ( ) ( ) sin sin sq. units sq. units O sq. units. A C 9. Solve the diffeentil eqution ( + ) d d + ( + ) ( + ) d d + ( + ) d d

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