Prove that M is a partially ordered set. 2. (a) Let f: { } C

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1 Nalada Ope Uivesity Aual Eamiatio - BSc Mathematics (Hoous), Pat-I Pape-I Time: Hs Full Maks: 8 Aswe ay five questios, selectig at least o fom each goup Goup 'A' (a) Defie a equivalece elatio Show that a elatio R o the set N N is a equidece elatio whe (a, b) R (c, d) iff a + d = b + c What do you mea by a patially odeed set? Let G be a goup ad M be the set of all sub goups of G Defie A Bto mea A B Pove that M is a patially odeed set (a) Let f: X Ybe such that f (A c ) = f ( A) fo all A X, show that f is oe oe { } C Pove that ay deumeable uio of deumeable sets is deumeable Goup 'B' (a) Itoduce Idempotet ad Nilpdet mati ad give thei eamples with suppot If A =, fid the value of A 4A + I, whee I is the uit mati with pope dimesio 4 (a) By givig the idea of Hemitia ad skew Hemitial matices, pove that evey squae mati is uiquely epessed as the sum of a Hemitia ad skew- Hemitia mati If A be a -owed squae mati, the show that (AdjA) A = A (AdjA) = A I, Whee A is the detemiat of A ad I is the uit mati of ode Goup 'C' 5 Defie HCF of two iteges which is deoted by (a, b) Pove that fo a, b Z thee eist iteges s ad t such that (a, b) = as + bt 6 (a) Defie a biay opeatio o a o empty set A Costuct two eamples (with poof) that o empty sets ae ot goups Pove that the set of esidue classes of iteges modullo a pime umbe fom additive abelia ad multicative abelia goups 7 (a) I a goup, evey elemet possesses uique ivese Pove this statemet Show that the ivese of the poduct of two elemets of a goup i the poduct of thei iveses i the evese ode Goup 'D' 8 (a) Descibe the elatios betwee oots ad Coefficiets of a th degee equatio Fid the coditio that the oots α, β, γ, δ of the equatio 4 + p + q + + s = should be coected by the elatio α β = γ δ 9 Solve the cubic equatio = by Cado's method Goup 'E' (a) State ad pove De Moive's theoem fo iteges i ( 4 + ) π Pove that ( i) = e (a) State ad establish the theoem of Gegoies seies π α If ta ( + iy) = Cosα + isiα, pove that y = log ta + 4

2 NALANDA OPEN UNIVERSITY BSc Mathematics (Hos) PART I, PAPER II Aual Eamiatio, Time : Hous Full Maks : 8 Aswe Five Questios i all, selectig at least Oe Questio fom each Goup All questios cay equal maks (a) State ad pove Rolle's Theoem If Cos y b = log a Goup 'A', the show that y ( + ) y + ² y = ³ + y³ (a) If u = ta u u, show that + y = siu + y y Evaluate Lim e log ² ( + ) (a) Fid the adius of cuvatue at ay poit (, θ ) fo the cuve = a Cosθ Detemie the equatio of asymptotes fo the cuve y + y ( y) + y ( y) + = Goup 'B' 4 Evaluate ay two of the followig itegals : (a) ² d ( ³ ) ( + ) d ( ) (c) d ² ( ) 5 (a) Deive the eductio fomula fo the itegal Si d Evaluate the defiite itegal log Si d 6 Evaluate + ² d dy + ² + y² π 7 Fid the suface aea ad volume of the solid of evolutio of the cuve y ( a ) a² ² = about its asymptote Goup 'C' 8 (a) Pove that fom a eteal poit, thee ae thee omals daw to the paabola y² = 4a Tace the paabola 9 ² 4y + 6y² 5 y + 5 = 9 (a) Fid the equatio of the omal at ay poit P ( α ) of a coic l = + e Cos θ A ellipse of semi-aes a ad b touches the ais of at the oigi Pove that the locus of the cete is ²y² + (y² a²) (y² b²) = Goup 'D' (a) Deive the equatio of taget plae to the sphee ² + y² + z² + u + vy + wz + d = Fid the coditio that the coe a ² + by² + cz² + fyz + gz + hy = has thee mutually pepedicula taget plaes (a) Fid the equatio of the ight cicula cylide whose ais is the lie ad adius a ² y² z² Deive the equatio of the omal to the ellipsoid + + = a² b² c² * * * α y β = = l m z at the poit (, y z ),

3 NALANDA OPEN UNIVERSITY BSc (Hos), PART I Mathematics (Subsidiay), Pape I Aual Eamiatio, Time : Hous Full Maks : 8 Aswe Five Questios i all, selectig at least Oe Questio fom each Goup All questios cay equal maks Goup 'A' (a) State ad pove Distibutive Laws of set theoy Let f : A B ad g : B C be two bijective maps The, pove that gof : A C is bijective ad ( ) f og gof = (a) Pove that i a goup, evey elemet possesses uique ivese Defie a cyclic goup ad show that evey cyclic goup is abelia (a) Show that the th oots of uity fom a geometic pogessio Si Si Evaluate Lim Si 4 (a) State ad pove the theoem o Gegoies seies a ib ab Pove that ta i log a ib =, whee a ad b ae eal quatities + a² b² Goup 'B' 5 (a) Date ad pove Cauchy's Root Test Let =, = +, = +,, + = +, Show that the sequece ( ) is Coveget havig limit of Covegece 6 (a) Fid the coditio that the cicles + y² + g + f y + c ad ² = ² + y² + g + fy + c =, itesect othogoally Show that the equatio ² + y² 4 + 5y + 4 = epesets a ellipse Hece, obtai its focii ad ecceticity 7 (a) Fid the coditio that the lie l + my + =, is tagetial to the coic a² + hy + by² + g + fy + c = Fid the equatio of pai of taget lies daw fom (, ) to ² + y² 4 + y + = Goup 'C' asi 8 (a) If d² y dy y = e, show that ( ² ) = a² y d² d State ad pove Maclaui's seies theoem 9 (a) Evaluate Lim e log ² ( + ) Fid the adius of cuvatue of the cuve ² y = a ( ² + y² ) at the poit ( a, a) ˆ ˆ ˆ ˆ ˆ ˆ d = w a b dt * * * (a) Pove that = ( i ) i + ( j) j + ( k)k If = a Cos wt + b Si wt, the show that ( )

4 NALANDA OPEN UNIVERSITY BSc Mathematics (Hos) PART II, PAPER III Aual Eamiatio, Time : Hous Full Maks : 8 Aswe ay Five Questios, selectig at least oe fom each Goup All questios cay equal maks Goup 'A' Use Dedekid cut o othewise pove that = 6 (a) State ad pove Cate Dedekid theoem o ode completeess theoem fo eal umbe system Apply Dedekid's theoem to deduce the theoem of least uppe boud (a) Show that if ad y be ay two positive eal umbes, the thee eists a positive itege such that > y Defie a ope set i R(set of eal umbes) ad give two eamples with poof 4 (a) State ad pove Bolzao-Weiestass theoem I = a b, I = a,, be a sequece of closed bouded itevals i R If [ ] [ ], b such that I I + fo each N ad ( b a ) = Lim The show that = cotais pecisely oe poit Goup 'B' 5 (a) Defie cotiuity of a fuctio o a ope iteval If a fuctio f is cotiuous o a closed ad bouded iteval [a, b], the pove that it attais its bouds o [a, b] Discuss cotiuity of f ( ) = e ( ) ad ( ) = 6 (a) Defie Beta fuctio ad evaluate a Beta fuctio Γ Fid the value of ( ) 7 (a) Pove that evey coveget sequece is bouded Test the covegece of the seies = + f at the oigi, > 8 (a) State ad eplai Cauchy's Codesatio Test ² ²4² ²4²6² Test the covegece of the seies ( > ) + + ² + ³ + to ² ²5² ²5²7² Goup 'C' 9 (a) Itoduce the cocept of a vecto space ad give a suitable eample with eplaatio If V be a vecto space ove the field F ad α, β V, the pove that β + ( α β ) = α (a) Defie a sub-space of a vecto space Pove that the set W of the elemets of the vecto space ( F ) ( + y, y, y ), whee, y F, is a sub=space of V ( F ) I V of the fom (a) Defie Eige values ad Eige vectos of a mati Pove that the Eige vectos coespodig to distict Eige values of a mati ae liealy idepedet I

5 Fid the Eige vectos of the mati A = * * *

6 NALANDA OPEN UNIVERSITY BSc Mathematics (Hos) PART II, PAPER IV Aual Eamiatio, Time : Hous Full Maks : 8 Aswe Five Questios i all, selectig at least Oe Questio fom each Goup All questios cay equal maks Goup 'A' (a) Solve ay two of the followig : y = + p + ap (ii) y = p + p p² (iii) (i) ( ) ² yp ² ( ² y² ) p y = Show that the system of paabola y ² = a( + a) is self-othogoal Solve the diffeetial equatios give below : d² y dy d² y dy (i) + y = ³ (ii) 5 + 6y = e d² d d² d d² y (a) Use the vaiatio of paametes to solve + a² y = Seca d² d Solve ² y dy ² 4 + ( 4² ) y = e d² d Goup 'B' 4 (a) Show that, [ a b, b c, c a ] = [ a, b, c ] If a, b, c be thee uit vectos, b ad c beig o-paallel ad such that b a ( b c ) = Fid the agle which a makes with b ad c 5 (a) Give the physical meaig of divegece ad cul of a vecto valued fuctio Pove the followig esults : (i) ( ) =, (ii) div = ³ 6 Evaluate F ds, whee S is the suface of the cube bouded by the plaes =, =, y =, ˆ S y =, z =, z = ad F = ²ˆ i 4yz ˆj + zkˆ Goup 'C' 7 (a) If all foces i a coplaa system ae otated about thei poits of applicatio though the same agle i thei ow plae, the pove that thei esultat passes though a fied poit i the body Thee foces P, Q, R act alog the sides of a tiagle fomed by the lies + y =, y = ad y = Fid the equatio of the lie of actio of the esultat foce 8 (a) State ad pove covese of the piciple of vitual wok The middle poits of opposite sides of a joited quadilateal ae coected by light T T' ods of legths l ad l' If T ad T' be tesios i thee ods, the pove that + = l l ' Goup 'D' 9 A paticle moves i a staight lie OA with a acceleatio which vaies as the distace fom a fied poit o i the staight lie ad is always away fom the fied poit If the paticle was iitially at a distace 'a' fom O ad pojected with the velocity V towads O Discuss the motio (a) Fomulate the tagetial ad omal velocities of a movig paticle i plae A paticle descibe a cateay ude a foce which acts paallel to its aias Fid the law of the foce ad the velocity at ay poit of the path (a) Deduce the diffeetial equatio of motio of a paticle movig i a cetal obit i the h² dp fom p = p³ d

7 Discuss the motio of a paticle movig ude ivese squae law ad show that the path is a comic sectio * * *

8 NALANDA OPEN UNIVERSITY BSc (Hos), PART II Mathematics (Subsidiay), PAPER II Aual Eamiatio, Time : Hous Full Maks : 8 Aswe Eight Questios i all, selectig at least Oe Questio fom each Goup All questios cay equal maks Goup 'A' Evaluate ay two of the followig : d a² ² (i) ( < a) (ii) + + d (iii) d ² ( ) (a) Evaluate Si d as the limit of sum Evaluate logsi d (a) Deive the eductio fomula fo Sec d Evaluate Lim ² ( ) ( ) { ( ) } ² + ² ² + ² ² + ² 4 (a) Fid the etie legth of the cadoid = a( + Cosθ ) Fid the aea betwee the cuve y ² ( a ) = ³ ad its asymptote ² 5 Solve ay two the diffeetial equatios as ude : (i) p ² p ( e + e ) + = (ii) y = ( + p) + ap² (iii) y = p + logp 6 What do you mea by othogoal tajectoy of a system? Fid the othogoal tajectoy of the family + a Cosθ 7 Detemie the geeal solutio of the followig equatios : (i) d² y dy d² y dy 7 + y = (ii) + y = Cos d² d d² d (iii) d² y + y = ² Cos d² Goup 'B' 8 (a) Show that the plae y + z + = touches the sphee ² + y² + z² 4 + z = ad hece fid the poit of cotact Fid the equatio of the ight cicula coe whose ais is the ais, vete the oigi ad semivetical agle π 9 Detemie the equatio of the ight cicula cylide whose ais is (a) By givig the idea of a cove set, pove that the set (, ) ² π y z = = ad adius 7 Goup 'C' S = { / + 6} is a cove set What is the idea of cove combiatio of vectos? Show that ay poit that ca be epessed as a cove combiatio of two poits i R, lies o the lie segmet joiig the two poits Wite shot otes o : (a) Neghbouhood of a poit Iteio poit (c) Bouday poit (d) Eteme poit (e) Ope set Goup 'D' Use vecto method to fid the esultat of a system of coplaa foces ad the deive the coditios of equilibium Thee foces each equal to P, act alog the sides of the tiagle ABC take i ode Fid the esultat foce ad equatio of lie of actio of the esultat ad hece deduce its distace fom A ad its poit of itesectio with BC Goup 'E' 4 (a) State ad pove the piciple of cosevatio of liea mometum Compoud two simple hamoic motios alog the same staight lies 5 (a) A paticle moves i a staight lie fom est ad a acceleatio which is popotioal to the distace fom a fied poit O i the staight lie ad is always away fom O Discuss the motio The eegy of a stetched elastic stig is equal to half the poduct of the tesio ad the etesio 6 (a) Fid the velocities ad acceleatios i the itisic co-odiates A paticle is pojected with a velocity ga so that it just cleas two walls of equal height a which ae at a distace a fom each othe Show that the time of passig betwee the walls is * * * a g

9 Nalada Ope Uivesity Aual Eamiatio - BSc Mathematics (Hoous), Pat-III Pape-V Time: Hs Full Maks: 8/75 Aswe ay five Questios, selectig at least oe questio fom each goup Goup-A (a) Defie ope set Pove that a sub-set G of a metic space is ope iff G is a uio of ope sphees Pove that the uio of ay family of ope sets i (, d) a metic space, is ope (a) Let (, d) be a metic space Pove that the itesectio of ay family of closed i X, is closed I a metic space, pove that each closed sphee is a closed set (a) Let ad y be metic spaces ad f a map of ito y Show that f is cotiuous iff f(a) f(a) fo evey sub-set A of X Let X ad Y be metic space ad f is a mappig of X ito Y if f be a costat mappig, the show that f is cotiuous 4 (a) I a topological space, give a chaacteizatio of cotiuity i tems of ope sets Show that evey sub-space of a Hausdoff space is Hausdoff Goup-B 5 (a) Pove that evey bouded mootoic iceasig fuctio defied o a closed iteval is Riema itegable Let f() be defied o [, ] by the coditio fo to be positive itege, the f() =(-) - whee + is R itegable ad f ()d = log4 6 (a) If f ad g ae two bouded ad R-itegable fuctios o [a, b], the show that f+g is R- b itegable o [a,b] ad f() + g()}d = f()d + a b { g()d Pove that π = α Goup-C 7 Discuss the Covegece of toα, α α α α α α α α fo diffeet values of α 8 (a) State ad pove Pigshiem's Theoem o double seies If a double seies is absolutely coveget, the show that it is also coveget, Goup-D 9 (a) Let N be a o-zeo omed liea space Show that N is Baach space iff f{: =} is complete Pove that a omal liea space is a Baach iff evey absolutely summable seies is summable (a) By takig R to be eal omed liea space give by a b a = +, whee =(, ), show that T is a Cotiuous liea tasfomatio T:R R defied by T(, )= Give defiitio ad eample of a ie poduce space with illustatio (a) Defie Hilbet space Show that l space) with ie poduct of two vectos =(,,, ) ad y=(y,y, y ) defied by (, y ) = { (,,, )/,,, C} i y i, is a Hilbet space, whee l = i= = Coside the liea space P[,] of all eal valued polyomials o [,] with the ie poduct (f,g)= f(t)g(t) dt whee f,g P[,] Show that it is a ie poduct space, but ot a Hilbet space

10 Nalada Ope Uivesity Aual Eamiatio - BSc Mathematics (Hoous), Pat-III Pape-VI Time: Hs Full Maks: 8/75 Aswe ay five Questios, selectig at least oe questio fom each goup Goup-A (a) Defie a automophism of a goup G Let G, the pove that the fuctio f defied by f ( g) = g fo g i G, is a automophism of G Let C(G) deotes the cete of a goup G ad I(G) be the set of ie automophisms o G The pove that C(G) I(G) (a) Defie a ig homomophism Let f : R T be a homomophism of a ig R oto a ig T The show that f is a isomophism iff Keel f={} Show by a eample that if I ad J ae ideals of a ig R, the I U J is ot a ideal of R (a) Show that ay ig ca be embedded i a ig with uity Defie Picipal ideal ig ad show that the ig of iteges, is a picipal ideal ig 4 I the ig F[], show that the picipal ideal geeated by the polyomial -s, whee s F, is both a pime ideal ad maimal ideal Goup-B 5 State ad pove Cato's Theoem 6 (a) Defie sum ad poduct of cadial umbes ad gibe eamples of each with suppotive agumets Pove that λ = C (symbols have thei usal meaigs) 7 (a) Defie maimal ad miimal elemets of patial odeed set Let A = {,4,5,8,9} be odeed by divides y Fid its maimal ad miimal elemets Itoduce the cocept of ode types ad costuct the poduct of two ode types Goup-C 8 (a) Defie a patitio of a set ad equivalece classes Show that the equivalece classes ae eithe disjoit o same Amog ay (+) iteges ot eceedig, show that thee must be a itege that divides oe of the othe ileges 9 (a) Eplai the coditios of Beoullis' tials Pove that P = ( + ) P Goup-D (a) Show that if f ( Z) = u(, y) + iν (, y) is diffeetiable at ay poit Z = + iy, the the patial deivatibs,,ν, ν eist ad satisfy Cachy-Riema diffeetial equataios u y If f (Z) is a aalytic fuctio of Z, show that (a) State ad pove Cachy's itegal fomula Z e dz Evaluate, whee C is the cicle Z = (Z + ) 4 y + y f(z ) = 4 f (Z) (a) Z + Fo the fuctio f ( Z) =, fid Taylo's seies valid i the eghbouhood of the poit Z + Z Z =i Eplai sigulaities of a fuctio What kid of sigulaity has the fuctio Cos at Z Z =?

11 Nalada Ope Uivesity Aual Eamiatio - BSc Mathematics (Hoous), Pat-III Pape-VII Time: Hs Full Maks: 8/75 Aswe ay five Questios, selectig at least oe questio fom each goup Goup-A (a) Defie a feasible solutio of a Liea Pogammig Poblem Pove that the set of all feasible solutio of a LPP fom a cove set Solve the followig LPP gaphically: Ma Z = Subject to: ; whee, (a) Itoduce degeeate ad o-degeeate basic solutio Obtai all basic solutio of the system: + + = 4, ad specify degeeate ad o-degeate solutios = Use simple method to solve the LPP Ma Z =, Subject to the costaits + 4, ad, Fom the iitial basic feasible solutio of the taspotatio poblem by usig mati miima method: W W W W 4 Capacity F F F Requiemet Goup-B 4 Fid a ecessay ad sufficiet coditio fo the itegability of the total diffeetial equatio Pd + Qdy + Rdz = 5 (a) Solve the diffeetial equatio z ( z y) d + z(z + )dy + ( + y)dz = Apply chapit's method to fid the complete itegal of p + qy = pq z z 6 (a) Solve the patial diffeetial equatio 5 = y y y Costuct the geeal solutio of the Lagage's liea equatio p(y z ) qy(z ) = z( y ), by fomig its auilliay equatios 7 (a) Use Moge's method to fid the complete solutio of the equatio 5ys + y t + (p + qy) = Fid the othogoal pojectio o the -z plae of the cuves which lie o the paaboloid z = +y ad satisfy the equatio dz=(+z)d+ydy Goup-C 8 (a) Fid the attactio of a uifom sphee at a eteal poit A fustum of a uifom thi hallow coe attacts a paticle placed at the Vete Show that the R attactio is π ρsiαcosαlog, whee R ad ae the adii of cicula eds, α the semi veticle agle ad ρ the suface desity of the coe 9 (a) State ad establish Laplace equatio i Catesia Co-odiates fom Pove that the half of the potetial of a uifom spheical shell at a eteal poit, is due to that potio of the sphee which is eae to Goup-D (a) Pove that the diffeece betwee pessues at two poits of a homogeeous fluid vaies as the depth of oe poit below the othe A hemi-spheical vessel filled with wate, is placed i a ivested positio o a hoizotal table Fid the esultat thust of the wate o the vessel (a) Fid the depth of the cete of pessue of a cicula aea of adius 'a' immesed vetically i a homogeeous liquid with its cete at a depth h below the fee suface A od of small coss sectio ad desity ρ has a small potio of a metal of weight th that of the od attached to oe etemity Pove that the od will float at a agle (icliatio) i a liquid of desity σ if ( + ) ρ = σ z

12 Nalada Ope Uivesity Aual Eamiatio - BSc Mathematics (Hoous), Pat-III Pape-VIII Time: Hs Full Maks: 8/75 Aswe ay five Questios (Calculato ot allowed) (a) Evaluate (i) ( )( )( ) (ii) a b e +, whee a ad b ae costats Deive Lagage's Itepolatio fomula ad apply it to fid log 656, whee log- 654=856, log 658=88, log 659=889 ad log 66=8 (a) Eplai divided diffeeces of a data ad pove that the value of ay divided diffeece is idepedet of the ode of agumets Epess f() = + i factoial otatios, the iteval of diffeecig beig uity (a) Itoduce Gauss Cetal diffeece fowad itepolatio fomula Fid the value of u 8 by Stilig fomula, whe u = 495, u 5 = 486, u =476, u 5 =4596 ad u 4 =446 4 (a) Deive Simpso's 8 ule fo umeical itegatio d By usig Weddle's ule, evaluate, by dividig the age ito eight equal + pats 5 Solve the followig diffeece equatios: (a) u + 4u+ + u = u u + u = Descibe Eule's method of solutio fo diffeetial equatio Hece, fid dy y appoimate value of y fo =, give that =, whe y= fo = d y + 7 (a) Use Gauss-Joda Method to solve the system of liea equatios: + + = 8, = ad = 6 (Take iitial coditio =, y=, z=) Eplai Gauss-Seidel Method fo costuctio of solutio of a system of liea equatios ad elaboate by meas of suitable eample 8 (a) Use Goup Relaatio Method to solve the followig system: + y + z + 4 =, y + z + = ad + y z + 45 = (Hit: Take iitial coditio =, y= ad z = ) Descibe Coveget Gauss-Seidel Method 9 (a) Apply aalytic method fo fidig oots of a equatio, based o Rolle's theoem ad demostate o + si = Solve 9 + = fo the oots lyig betwee = ad =4 by bisectio method (a) Discuss Newto-Raphso's Method to obtai appoimate value of oot of f()= Use sythetic divisio to solve f()= - -()+9999=, i the eghbouhood of = * * *