M Mrks : Q Write the principl vlue of CBSE ALL INDIA EXAMINATION [Set With Solutions] SECTION A tn ( ) cot ( ) Time Allowed : Hours Sol tn ( ) cot ( ) tn tn cot cot cot cot [Rnge of tn :,, cot : ], [ 5 5 cot cot Q Write the vlue of tn sin cos Sol Let Y tn sin cos tn sin cos cos tn sin tn sin tn tn tn tn Y Q For wht vlue of, is the mtri skew-smmetric mtri? Sol Let A Since A is skew-smmetric mtri, which mens A = A T B equlit of mtrices, Q If mtri A nd A ka, then write the vlue of k Sol Given A nd A ka AA ka k k k k A ka AA kaa [Post multipling both the sides b A I k I k T For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) Q5 Write the differentil eqution representing the fmil of curves m, where m is n rbitrr constnt Sol We hve m (i) d d m d [B using eq (i) This is the required differentil eqution Differentiting wrt both the sides, d m Q If A ij is the cofctor of the element ij of the determinnt Sol A We hve 5 5 7 5 A 5 ( ) 5( 5) 5 5 7 then, write the vlue of ie, A Q7 P nd Q re two points with position vectors b nd b respectivel Write the position vector of point R which divides the line segment PQ in the rtio : eternll Sol Given OP b, OQ b Also R divides PQ in : eternll (OQ) (OP) ( ) ( b b OR ) ie, OR b Q8 Find, if for unit vector, ( )( ) 5 Sol Given ( )( ) 5 5 5 () 5 5 Q9 Find the length of the perpendiculr drwn from the origin to the plne z Sol The length of the perpendiculr drwn from the origin (, ) to the plne z is : () () () p ( ) A B Cz D [Using p units A B C ie, p units units 9 Q The mone to be spent for the welfre of the emploees of firm is proportionl to the rte of chnge of its totl revenue (mrginl revenue) If the totl revenue (in `) received from the sle of units of product is given b R() = 5, find the mrginl revenue, when 5, nd write which vlue does the question indicte Sol We hve totl revenue R() = 5 So, Mrginl Revenue = Rte of chnge of totl revenue function d ie, ( 5) Mrginl Revenue when 5 (5) Vlue indicted : More is the revenue generted b the firm, more will be the mone spent for the welfre of emploees For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) SECTION B Q Consider f : R [, ) given b f ( ) Show tht f is invertible with the inverse f of f given b f ( ), where R + is the set of ll non-negtive rel numbers Sol Given f : R [, ) defined s f ( ) For one-one : Let, R We hve f ( ) f ( ) ie, ( )( ) ( ) [ s, R So, f is one-one For onto : Let [, ) nd f ( ) [ R Clerl, R for ll [, ) Thus, for ech [, ) there eists R f is onto function Since f is one-one nd onto function, so it is bijective function nd hence it is invertible Also f : [, ) R is given s f ( ) Q 7 Show tht : tn sin OR Solve : cos(tn ) sin cot Sol LHS: Let Y = tn sin Put 9 7 (i) sin sin cos sin sin Now Y = tn cos sin sin sin sin cos sin 7 cos sin [B (i) 7 RHS [Hence Proved OR We hve cos(tn ) sin cot cos cot sin cot [tn cot sin(cot ) sin cot cot cot For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) Q Using properties of determinnts, prove tht : Sol Q If Sol LHS: Let ( ) ( ) ( )( ) 9 ( ) [Appling C C C C [Tking common from C [Appling R R R, R R R [Tking common from R & R ( ) [Epnding long C ( ) ( ) 9 ( ) RHS [Hence Proved e, prove tht d ( log ) log Given e Tking log on both the sides, log( ) log( e ) log ( )log e [log e [Differentiting wrt both the sides log d d ( log ) ( log ) d d d ( log ) log d ( log ) Q5 Differentite sin with respect to () Sol Let sin sin () () tnθ sin tn θ, where tnθ θ tn ( log ) d ( log ) d ( log ) [Hence Proved log sin ( ) sin (sin θ) θ tn [Diff wrt both the sides d d (tn ) d d ( ) ie, (log ) ( ) For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) d Hence, log (log ) k k, if Q Find the vlue of k, for which f ( ) is continuous t, if d OR If cos θ nd sin θ, then find the vlue of t θ = Sol Since f is continuous t so, lim f ( ) f () (i) Now, () f () () And, LHL (t ) : (ii) lim f ( ) lim k k k k k k lim k k ( k) ( k) lim k k k k lim k k k (iii) B (i), (ii) nd (iii), we hve : k OR We hve cos θ nd sin θ Differentiting wrt θ, d = dθ dθ ( cos θ) cos θsin θ (i) And, d d = dθ dθ ( sin θ) sin θcosθ (ii) d d dθ So, sin θcosθ tnθ [Using (i) & (ii) dθ ( cos θsin θ) d d d dθ Now, ( tnθ) sec θ = [Diff wrt both the sides d sec θ ( cos θsinθ) [B using (i) d sec θ sinθ d sec So, 9 7 t θ= sin Q7 cos cos Evlute: OR Evlute: cos cos Sol cos cos Let I ( cos ) ( cos ) cos cos cos cos For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge 5
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) (cos cos ) (cos cos )(cos cos ) cos cos cos cos (cos cos ) (sin cos ) C OR Let I Put t in st integrl so, ( ) dt dt I t t ( ) ( ) ie, I log ( ) ( ) C I log C Q8 Evlute : 5 ( ) Sol 5 Let I 5 5 5 ( ) 5 ( ) 5 Put t so, 5 dt dt I = 5 ( t ) t 5 t t dt 5 5 ie, I = log t log t C 5 = log log C 5 5 So, I = log log C 5 Q9 Evlute : sin e Sol Let I (i) [Using sin e sin e sin( ) sin e I sin e (ii) e Adding (i) nd (ii), I sin sin sin e e e f ( ) f ( ) I I I Q If ˆi ˆj 7kˆ nd b 5i ˆ ˆj kˆ, then find the vlue of, so tht perpendiculr vectors Sol We hve ˆi ˆj 7kˆ nd b 5i ˆ ˆj kˆ b ˆi ˆj 7kˆ 5i ˆ ˆj kˆ = i ˆ ˆj (7 )kˆ nd b ˆi ˆj 7kˆ 5iˆ ˆj kˆ = i ˆ (7 )kˆ Given ( b) ( b) tht implies ( b)( b) i ˆ ˆj (7 )k ˆ i ˆ (7 )kˆ (9 ) b nd b re For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) 5 5 Hence, the vlue of is 5 nd 5 Q Show tht the lines r i ˆ ˆj k+ ˆ (i ˆ ˆj k) ˆ nd r 5i ˆ ˆj + (i ˆ ˆj k) ˆ re intersecting Hence find their point of intersection OR Find the vector eqution of the plne through the point (,, ) nd (,, ) nd perpendiculr to the plne z Sol Let the lines re l : r i ˆ ˆj k+ ˆ (i ˆ ˆj k) ˆ nd l : r 5i ˆ ˆj + (i ˆ ˆj k) ˆ Coordintes of n rndom point on l re P(,, ) nd on l re Q( 5,, ) If the lines l nd l intersect then, the must hve common point on them ie, P nd Q must coincide for some vlues of nd Now, 5 (i), (ii) nd (iii) Solving (i) nd (ii), we get :, Substituting these vlues in LHS of (iii), ( ) ( ) RHS So, the given lines intersect ech other Now, point of intersection is, P(,, ) OR Let the dr s of norml vector to the plne be A, B, C Then the eqution of plne through (,, ) is A( ) B( ) C( z ) (i) And (,, ) lies on plne (i) so, A( ) B( ) C( ) A B 5C (ii) Also it is given tht plne (i) is perpendiculr to z So, A B C (iii) [Using b b c c A B C A B C Solving (ii) nd (iii), 8 5 8 7 Substituting the proportionte vlues of A, B nd C in (i), we get : 8( ) 7( ) ( z ) ie, 8 7 z 9 vector eqution of plne is r (8i ˆ 7j ˆ k) ˆ 9 Q The probbilit of two students A nd B coming to the school in time re /7 nd 5/7 respectivel Assuming tht the events, A coming in time nd B coming in time re independent, find the probbilit of onl one of them coming to the school in time Write t lest one dvntge of coming to school in time Sol We hve, P(A) = Probbilit of A coming in time = 7 nd, P(B) = Probbilit of B coming in time = 5 7 Probbilit of onl one of them coming to the school in time = P( A B ) + P( A B ) = P(A) P( B ) + P( A ) P(B) = 5 5 7 7 7 7 [ P(E ) = P(E) = 9 Advntge of coming to school in time : Not missing the morning prers nd/or missing n topic in first period of the clss SECTION C Q Find the re of the gretest rectngle tht cn be inscribed in n ellipse b OR Find the equtions of tngents to the curve 8, which pss through the point (/, ) For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge 7
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) Sol Given eqution of ellipse (i) b Let ABCD be the rectngle inscribed in the ellipse Let the length of the rectngle be AB = p, nd width be AD = q such tht mjor is intersects the sides AD nd BC t F nd F respectivel Since A(p, q) lies on the ellipse (i) so, p q p q b (ii) b Y ( p, q)b A(p, q) X F O F X Now, re of rectngle A = (p) (q) ie, A p q ( p, q)c D(p, q) p Assume A f (p) p b [B using (ii) Y b b b f (p) p p f (p) p p nd f (p) p b For points of locl mim & minim, f (p) p p p, p b f b [Vlue p is rejected f (p) is mimum t p which mens A is mimum t p Hence A is lso mimum t p And mimum re is, A b b Squnits OR Given curve is 8 (i) Let the point of contct on (i) be P(, ) So, 8 (A) Differentiting (i) wrt, d d d (ii) t P Eqution of tngent t P is : ( ) ( ) ( ) (iii) As tngent (iii) psses through (/, ) so, ( ) (B) B (A) nd (B), 8, [B replcing the vlue of in (A) Hence points of contct re (, ) eqution of tngents re : ( ) ()( ) [B (iii) nd, ( )( ) ()( ) Q Find the re of the region bounded b the prbol nd Sol, if Given (i) nd, (ii), if Curve (i) is smmetricl bout -is since it contins even power of Solving (i) nd (ii), we get : Cse I : If, ( ), So, points of intersections re (, ), (, ) For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge 8
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) Cse II : If, ( ), So, points of intersections re (, ), (, ) Now, required re r(oabco) ( ii ) ( i) ( ) Required Are Squnits Q5 Find the prticulr solution of the differentil eqution Sol Given (tn ) d ( ) (tn ) ( ) (tn ) tn ( ) d d It is cler tht this is liner differentil eqution of the form P( ) Q( ) d Here P( ), Integrting Fctor tn Q( ) d e So, solution is given b, tn t e e t dt tn t t e t e e C tn Given tht,, we get : So, the required solution is : e tn tn tn e e d Put tn tn d e t e t dt ( t) e t dt dt dt tn tn e e (tn ) C tn tn e (tn ) e C e d, given tht, For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge 9 C tn tn d t dt Q Find the eqution of the plne pssing through the line of intersection of the plnes r (i ˆ j) ˆ nd r (i ˆ ˆj k) ˆ, whose perpendiculr distnce from origin is unit OR Find the vector eqution of the line pssing through the point (,, ) nd prllel to the plnes r (i ˆ ˆj + k) ˆ 5 nd r (i ˆ ˆj + k) ˆ Sol The eqution of the plne pssing through the line of intersection of the plnes r (i ˆ j) ˆ nd r (i ˆ ˆj k) ˆ is r [(i ˆ j) ˆ (i ˆ ˆj k)] ˆ () [Using r [ n n ] d d ie, r[( )i ˆ ( )j ˆ k)] ˆ (i) Now distnce of (i) from (, ) is, (i ˆ ˆj k)[( ˆ )i ˆ ( )j ˆ k)] ˆ Y ( ) ( ) ( ) A C B (, ) O (, ) X
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) ( ) ( ) ( ) Substituting the vlue of in (i), we get : r [i ˆ ˆj k] ˆ ie, r [i ˆ ˆj k)] ˆ nd, r [ i ˆ ˆj k] ˆ ie, r [i ˆ ˆj k] ˆ OR Required line is prllel to the plnes r (i ˆ ˆj + k) ˆ 5 nd r (i ˆ ˆj + k) ˆ Tht implies, the line will be perpendiculr to the normls of these plnes ˆi ˆj kˆ So, line will be prllel to (i ˆ ˆj + k) ˆ (i ˆ ˆj + k) ˆ i ˆ 5ˆj + kˆ Now given point on the required line is (,, ) so, ˆi ˆj + kˆ So eqution of line is : r ˆi ˆj + k ˆ + (5ˆj + kˆ i) ˆ Q7 In hocke mtch, both tems A nd B scored sme number of gols up to the end of the gme, so to decide the winner, the referee sked both the cptins to throw die lterntivel nd decided tht the tem, whose cptin gets si first, will be declred the winner If the cptin of tem A ws sked to strt, find their respective probbilities of winning the mtch nd stte whether the decision of the referee ws fir or not Sol Let A : tem A is declred winner, B : tem B is declred winner Let E : die shows si on the throw Clerl P(E) = /, P(E ) = P(E) = / = 5/ If cptin of tem A strts then he m get si in st throw or rd throw or 5 th throw nd so on P(A) = P(E) + P(E E E) + P(E E E E E) + 5 5 = [Using S 5, sum of infinite GP r ie, P(A) = And P(B) = P(A) = 5 The decision of referee wsn t fir since tem A hs more chnces of being declred winner despite the fct tht both the tems hd secured sme number of gols Q8 A mnufcturer considers tht men nd women workers re equll efficient nd so he ps them t the sme rte He hs nd 7 units of workers (mle nd femle) nd cpitl respectivel, which he uses to produce two tpes of goods A nd B To produce one unit of A, workers nd units of cpitl re required while workers nd unit of cpitl is required to produce one unit of B If A nd B re priced t ` nd ` per unit respectivel, how should he use his resources to mimize the totl revenue? Form the bove s n LPP nd solve grphicll Do ou gree with this view of the mnufcturer tht men nd women workers re equll efficient nd so should be pid t the sme rte? Sol Let the number of the goods of tpe A nd B produced be respectivel nd To mimize, Z = `( + ) Subject to the constrints: (i) nd, 7 (ii);, Considering the equtions corresponding to the inequtions (i) nd (ii), we hve: 7 5 7/ 7 For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) Tke the testing points s (, ) for (i), we hve: () (), which is true Tke the testing points s (, ) for (ii), we hve: () () 7 7, which is true The shded region OACBO s shown in the given figure is the fesible region, which is bounded B The coordintes of the corner points of the C fesible region re A(8, ), B(, ), C(, ) nd O(, ) So, Vlue of Z t A(7/, ) = `7/ Vlue of Z t B(, ) = ` Vlue of Z t C(, 8) = ` Vlue of Z t O(, ) = ` The mimum vlue of Z is ` which occurs t = nd = 8 O A X Thus, the fctor must produce units nd 8 units of the goods of tpe A nd B respectivel The mimum obtined profit erned b the fctor b producing these items is ` Yes, we gree with the view of mnufcturer tht men nd women workers re equll efficient nd so should be pid t the sme rte Q9 The mngement committee of residentil colon decided to wrd some of its members (s ) for honest, some (s ) for helping others nd some (s z) for supervising the workers to keep the colon net nd clen The sum of ll the wrdees is Three times the sum of wrdees for coopertion nd supervision dded to two times the number of wrdees for honest is If the sum of the number of wrdees for honest nd supervision is twice the number of wrdees for helping others, using mtrices find the number of wrdees of ech ctegor Aprt from these vlues, nmel, honest, coopertion nd supervision, suggest one more vlue which the mngement of the colon must include for wrds Sol Let, nd z be the number of wrdees for honest, coopertion nd supervision respectivel Acc to question, z, ( z) ie z, z ie z Clerl z B using mtri method: A, B nd, X z Since AX B A (AX) = A B [Pre-multipling b (A A)X = A B (I) X = A B [ IA = A = AI X = A B (i) Now, A ( ) ( ) ( ) A A is non-singulr nd hence, it is invertible ie, Consider C ij be the cofctors of element ij in mtri A, we hve C 9, C, C 7 C, C, C C, C, C Y 7 A both sides A eists For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) T 9 7 9 9 So, dja A ( dja) A 7 7 Now b (i), we hve X = A B 9 8 99 So, X = = 7 8 99 5 = z 5 B equlit of mtrices, we hve:,, z 5 the number of wrdees for honest, coopertion nd supervision respectivel re, nd 5 Another vlue which the mngement cn include m be regulrit nd sincerit CBSE ALL INDIA EXAMINATION [Set With Solutions] Q9 If mtri A nd A = pa, then write the vlue of p Sol Given A nd A pa 8 8 p p 8 8 8 8 p p 8 8 p p [B equlit of mtrices, p Q A nd B re two points with position vectors b nd b respectivel Write the position vector of point P which divides the line segment AB internll in the rtio : Sol We hve OA b nd OB b Since P divides AB in : internll so, OP OB OA (b ) ( b) ie, OP = Q9 If Sol d log ( log ) e, prove tht Given e Tking log on both the sides, log( ) log( e ) log( ) ( )log( e) ( ) [loge = ie, d d log log d d d ( log ) ( ) ( log ) ( log ) [Differentiting wrt both the sides ( log )() d ( log ) d log [Hence Proved ( log ) For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) Q Evlute : ( 8) Sol Let I ( 8) Put 8 t dt ( 8) dt ie, I dt ( t 8) t 8 t 8 t log t 8 log t C I log C 8 sin Q Evlute : cos sin Sol Let I (i) [Using cos ( )sin( ) I ( )sin f ( ) f ( ) sin sin cos ( ) I cos cos cos (ii) sin Adding (i) nd (ii), I cos Put cos t sin dt Also when t cos nd t cos dt dt I t tn t t ie, I tn () tn ( ) I Q If p 5i ˆ ˆj kˆ nd q ˆi j ˆ 5kˆ, then find the vlue of, so tht perpendiculr vectors Sol We hve p 5i ˆ ˆj kˆ nd q ˆi j ˆ 5kˆ So, p q 5i ˆ ˆj kˆ + ˆi ˆj 5kˆ = i ˆ ( ) ˆj 8kˆ And, p q 5i ˆ ˆj kˆ ˆi j ˆ 5kˆ = i ˆ ( )j ˆ kˆ Since p q nd p q re perpendiculr vectors so, ( p q)( p q) i ˆ ( ) ˆj 8k ˆ i ˆ ( ) ˆj kˆ ( 9) p q nd p q re Q8 Find the re of the region {(, ) : nd } using method of integrtion Sol We hve {(, ) : nd } Consider (i), (ii) Curve (ii) represents circle centered t (, ) hving rdius of Solving (i) nd (ii), ( 8 )( ) nd 8 [which is rejected, s it doesn t stisf (i)] So,, nd point of intersections re (, ) Y A(, ) Now required re r(oapmo) ( i) ( ii) O P(, ) X ( ) M(, ) For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) / ( ) sin / / ( ) ( ) sin ( ) sin () sin ( ) 8 8 ( ) Squnits Q9 Show tht the differentil eqution sin d is homogeneous Find the prticulr solution of this differentil eqution, given tht when d Sol We hve sin d sin f (, ) (s) (i) k Put k & k in (i), f ( k, k) ksin k k sin k f (, ) k k So, it is cler tht the given diff eq is homogenous diff eq of degree d dv Now, let v in (i) So, v dv v dv v sin v v sin v v cosec v dv cotv log C cot log C Since given tht when so, cot log C C required solution is cot log + CBSE ALL INDIA EXAMINATION [Set With Solutions] Q9 If mtri A nd A = A, then write the vlue of Sol Given A nd A A 8 8 8 8 8 8 8 8 [B equlit of mtrices, 8 For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) Q L nd M re two points with position vectors b nd b respectivel Write the position vector of point N which divides the line segment LM in the rtio : eternll Sol We hve OL b nd OM b Since N divides LM in : eternll so, OM ON OL ( b) ( b) ie, OP = 5b Q9 Using vectors, find the re of the tringle ABC, whose vertices re A(,,), B(,,) nd C(,5, ) Sol Given vertices of ABC re A(,,), B(,,) nd C(,5, ) So, AB OB OA (i ˆ ˆj k) ˆ (iˆ ˆj k) ˆ = ˆi ˆj kˆ nd, AC OC OA (iˆ 5ˆj k) ˆ (i ˆ ˆj k) ˆ = iˆ ˆj kˆ Now ˆi ˆj kˆ AB AC 9i ˆ 7ˆj kˆ AB AC 9 7 7 r(abc) ABAC 7 Squnits Q Evlute : ( ) Sol Let I ( ) Put t dt ( ) dt ie, I dt ( t ) t t t log t log t C I log C Q If sin( ) sin cos( ), prove tht Sol Given sin( ) sin cos( ) d sin ( ) sin cos( ) sin sin( ) d cos( ) ie, sin [Differentiting wrt both the sides d d sin( ) d d sin( ) cos( ) cos( ) sin( ) d d sin d [sin( )] sin( )sin( ) cos( )cos( ) sin ( ) cos ( ) sin sin d [sin( )] [sin( )] sin d sin ( ) Q Using properties of determinnts, prove tht Sol LHS: Let z z z z z d sin ( ) [Hence Proved sin z z ( z)( z z) z z z [Appling C C C C For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge 5
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) z z z z z z z z ( z) z z z z z ( z) z z z z z z ( z) z z ( z) ( z)( z ) ( z)( z) ( z)( z z) [Tking z common from C [Appling R R R, R R R [Epnding long C ( z)( z z) RHS [Hence Proved Q8 Find the re of the region {(, ) : nd 9} using method of integrtion Sol We hve {(, ) : nd 9} Consider (i), 9 (ii) Curve (ii) represents circle centered t (, ) hving rdius of / units Solving (i) nd (ii), ( ) 9 9 ( 9)( ) nd So, 9 [which is rejected, s it doesn t stisf (i)],, nd point of intersections re Now required re r(oapmo) / / ( i) ( ii) / For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge / / / / / 9 sin 8 8 / 9 9 sin 9 9 sin () sin 9 9 sin 9 9 sin 8 9 9 sin cos Squnits / / / / Y 9 A, O P(/, ) X M(/, )
CBSE Annul Em Pper All Indi (Mthemtics) B OP Gupt (+9-9558) Q9 Find the prticulr solution of the differentil eqution d when Sol We hve cot cot, ( ) d It is cler tht this is liner differentil eqution of the form P( ) Q( ) d Here P( ) cot, Q( ) cot Integrting Fctor cot d e log sin e sin d [Appling B prts in nd integrl So, solution is given b, sin sin cot sin sin d cos d d d sin sin d cos d ( ) cos d d sin sin d sin sin d Given tht when, we get : sin sin C C So, the required solution is : sin sin ie, ( )sin cot cot, ( ), given tht sin sin C Der Student/Techer, I would urge ou for little fvour Plese notif me bout n error(s) ou notice in this (or other Mths) work It would be beneficil for ll the future lerners of Mths like us An constructive criticism will be well cknowledged Plese find below m contct info when ou decide to offer me our vluble suggestions I m looking forwrd for response Also I would wish if ou inform our friends/students bout m efforts for Mths so tht the m lso benefit Let s lern Mths with smile :-) For n clrifiction(s), plese contct : MthsGuru OP Gupt [Mths (Hons), E & C Engg, Indir Awrd Winner] @theopgupt Contct Nos : +9-95 5 8 +9-978 8 Mil me t : theopgupt@gmilcom Officil Web-pge : wwwtheopguptcom For more stuffs on Mths, plese visit t : wwwtheopguptcom Pge 7