Nalanda Open University

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1 B.Sc. Mathematics (Honous), Pat-I Pape-I Answe any Five questions, selecting at least one fom each goup. All questions cay equal maks. Goup - A. If A, B, C, D ae any sets then pove that (a) A BC D AC BD AD BC A BC D AC BD. State and pove fundamental theoem on equivalence elation.. What do you mean by patial ode elation and total ode elation and well odeed set? Give one eample of each. 4. Let f : A B and g : B C and both f and g ae one to one and onto mappings. Then pove that (i) gf : A C is one to one and onto (ii) (gf) - = f - g - 5. (a) Define a Lattice, complete Lattice and set an eample of Lattice which is not complete Lattice Pove that an infinite union of denumeable sets is denumeable. Goup - B 6. (a) Define a goup and show that the fou fouth oots of unity namely, -, i, -i fom a goup with espect to multiplication. Pove that G = {0,,,, 4, 5} is a finite abelian goup of ode 6 with espect to addition modulo (a) If a goup G has fou elements then show that is must be abelian. Pove that a goup G is abelian if b - a - ba = e a, b G 8. (a) If H and H ae subgoups of a goup G then show that H H is also a subgoup of G. Pove that the ode of evey element of a finite goup is a diviso of the ode of the goup. Goup - C 9. (a) If A and B ae two non-singula matices of the same ode then pove that (AB) - = B - A - Find the invese of the mati Solve the following system of linea equations by mati method. + y + z = 6 + y z = -5 y + z =. 8 6 Find the eigen values and eigen vectos of the mati A Goup - D. Find the condition that the equation 4 p q + + s = 0 may have its oots in aithmetical pogession and solve the equation = 0.. Solve 4 6 = 0 by Eule s method. 4. State and pove De-Moive s theoem. Eamination Pogamme, 07 (Revised) B.Sc (Pat I) All Honous Subjects Ecept Home Science, Geogaphy & Statistics Honous Date Papes. Time Eamination Cente 0//07 (Hons) P-I.0 to 6.0 pm, Patna //07 (Hons) P-II.0 to 6.0 pm, Patna 4//07 Rastabhsha-00 o Hindi+Udu 00.0 to 6.0 pm, Patna 7//07 Math (Sub) P-I 8.00 to.00 am, Patna 8//07 Geogaphy (Sub) P-I 8.00 to.00 am, Patna 9//07 Chemisty (Sub) P-I 8.00 to.00 am, Patna 0//07 Home Science (Sub)-P I 8.00 to.00 am, Patna //07 Zoology (Sub) P-I 8.00 to.00 am, Patna 0/4/07 Physics (Sub) P-I 8.00 to.00 am, Patna 04/4/07 Botany (Sub) P-I 8.00 to.00 am, Patna 05/4/07 Statistics (Sub) P-I 8.00 to.00 am, Patna

2 B.Sc. Mathematics (Honous), Pat-I Pape-II Answe any Five questions, selecting at least one fom each goup. All questions cay equal maks.. (a) If y sin( a b) then find yn Goup - A a If y e, then pove that ( - ) y n+ (n + ) y n+ (n +a )yn = 0. (a) State and pove Taylo s Theoem. asin 4 Pove that e Lt. e log( ). (a) Evaluate 0 sin Lt. sin 0 u u u 4. (a) If u log( y z yz) then show the y z ( y z ) If the nomal at any point to the cuve y a make an angle with the -ais then show that its equation is y cos sin a cos d 5. (a) find the adius of cuvatue fo the pedal cuve p=f() o pove that, symbols dp have usual meaning. Find the asymptotes to the cue ( y )( y ) 9y Goup - B 6. Evaluate any two of the following: (a) 5 d d d 4 (c) a 9 d 7. (a) Show that a cos b sin ab Show that log(sin ) d log Lt. n 8. (a) Evaluate n... n n n n Obtain the eduction fomula fo cos m sin nd 9. (a) Find the aea of the loop of the cuve y ( a) ( a) 0 0. Find the suface of the solid obtained by evolving the cuve a cos about the initial line. Goup C l. (a) Find the pola equation of the conic in the fom e cos l Find the pola equation of tangent at any point of it to the covic e cos. (a) Deduce the equation of a plane in the fom l my nz p a b c Pove that 0 epesents a pai of plains. y z z y. (a) Find the equation to the pependicula fom the oigin to the line 4y 4z 7 0 ; y 5z 0 also find the co-odinates of the foot of pependicula. Find the image of the point (,,) by the plane y z 4. (a) Find the equation of the sphee though the cicle y z 9, y 4z 5 and the point (,,) Find the pole of the plane l my nz p with espect to the sphee y z a

3 B.Sc. Mathematics (Subsidiay), Pat-I Pape-I Answe any Five questions, selecting atleast One question fom each goup. All questions cay equals maks. Goup - A. If {Ai} i I be an abitay indeed family of sets then show that: ' (a) A i Ai ' A i Ai ' ii ii ii ii. (a) Pove that A( B C) ( AB) ( AC) Pove that A( B C) ( AB) ( AC). (a) Define an equivalance elation and give two eamples of it. If f : X Y, A Y, B Y then show the f ( A B) f '( A) f ( B) 4. (a) Show that the set G={, w, w } whee w is an imaginay cube oot of unity is a goup with espect to multiplication. Pove that the ode of evey element of a finite goup is finite and is less than o equal to the ode of the goup. 5. (a) Define an abelian goup and show that if a goup G has fou elements it must be abelian. Pove that a necessay and sufficient condition fo a non-empty subset H of goup G to be a sub goup of G is that a H, b H ab H. Whee b - is the invege of b in G 6. (a) If f is a homomophism of goup G into a goup G then pove that (i) f(e)=e whee e is the identity of G and e is the identity of G (ii) (a - ) = [f(a)] - a G (iii) If the ode of a G is finite then ode of f(a) is the diviso of the ode of a. If f is a homomophism of a goup G into a goup G with kenel K = { G : f() = e is the identity of G } then show that K is a nomal subgoup of G. Goup - B 7. (a) Etact all the oots of 7 + = 0 using De-Moive s Theoem. Decompose log( i ) into eal and imaginay pats. 8. (a) State and pove Cauchy geneal pinciple of convegence of a eal sequence. Show that the sequence (an) whee a n ' n 4n n is convegent. 9. (a) Pove that the seies p p p p 4... is convegent fo p > and divegnet fo p. Test the convegence of the seies whose n th tem is n n Goup - C 0. Find the condition unde which a geneal equation of second degee a hy by g fy c 0 epesents an ellipse. l. Deduce the pola equation of a conic in the fom e cos.. (a) State and pove Eule s theoem on Homogeneous functions of two vaiables. f f If f (, y) cos y y cos then pove that y y

4 . (a) Pove that the adius of cuvatue fo the pola cuve ) ( f is given by ) Find the maimum value of 4. (a) Give the geometical meaning of scala tiple poduct of vectos. Pove that. ). ( ). ( ) ( c b a b c a bc a

5 B.Sc. Mathematics (Honous), Pat-II Pape-III Answe any five Questions, selecting at least one question fom each goup. Goup-A. (a) State and pove theoem of geatest lowe bound. State and pove theoem of least uppe bound.. (a) State and pove Dedekind s theoem fo eal numbes. State and pove fundamental theoem of classical analysis.. Pove that between two distinct eal numbes thee lie infinity of ationals and infinity of iationals Goup-B 4. (a) Pove that evey convegent squence is bounded. Pove that a monotonic inceasing sequence tends to its bound. 5. (a) Pove that the sequence defined by u =, u n+= un conveges to. Show that the sequence (a n) defined by a = 7, a n+= 7 an conveges to a positive oot of the equation 7 = (a) Test the convegence of the seies p p p p Test the convegence of the seies whose n th tem is n n 7. (a) Test the convegence of the seies n ; fo all > 0 n n State and pove Couchy s n th oot test fo convegence of an infinite seies. 8. (a) State and pove Raabe s test fo convegence of the positive tem seies u n. ( n )( n ) Test the convegence of the seies ( n )( n 4) Goup-C 9. (a) Pove that is W and W ae finite dimensional subspaces of a vecto space V, then W +W is a finite dimensional and dim.w + dim.w = dim (W W ) + dim(w + W ) 0. (a) 4 6 Find the eigen values and eigen vectos of the mati A 5 Find the ank of the mati ****** Eamination Pogamme, 07 (Bachelo of Science (Pat-II) All Subjects Ecept B.Sc Geogaphy & Home Science (Honous) Date Pape Time Name of Eamination Cente //07 HONOURS PAPER III 8.00 to.00 am, Patna 5//07 HONOURS PAPER IV 8.00 to.00 am, Patna 7//07 Hindi 00 o U 50+Hn to.00 am, Patna 0//07 (SUB.) (Botany - II) 8.00 to.00 am, Patna 0//07 (SUB.) (Mathematics - II) 8.00 to.00 am, Patna 04//07 (SUB.) (Chemisty - II) 8.00 to.00 am, Patna 06//07 (SUB.) (Physics - II) 8.00 to.00 am, Patna 07//07 (SUB.) (Zoology - II) 8.00 to.00 am, Patna 09//07 (SUB.) (Geogaphy - II) 8.00 to.00 am, Patna //07 (SUB.) (Home Science- II) 8.00 to.00 am, Patna 6//07 (SUB) Statistics-II) 8.00 to.00 am, Patna

6 B.Sc. Mathematics (Honous), Pat-II Pape-IV Answe any five Questions, selecting at least one qu estion fom each goup. Goup-A. (a) Solve any two of the following diffeential equations (a) (p-y)(-py)=p (-a)p + (-y) p - y=0. (a) y Pove that the system of confocal conic is self othogonal. a b Eamine the equation p + py + a = 0 fo singula solutions. d y dy 4. (a) Solve 4 y (use method of change of vaiable) d d d y Using method of vaiation of paamete solve a y sec a OR solve d d y y (using method of vaiation of paamete) d e 4. (a) Pove that Show that [ a b Goup-B a ( b c ) ( a. c ) b ( a. b) c b c c a] [ a b c ] 5. (a) d d v d u Pove that ( u v) u v dt dt dt Find a unit tangent vecto to any point on the space cuve = a cost, y = a sint, z = bt whee a, b ae constants and t is time. 6. (a) Pove the ( u v) u v Pove that.( u v) v.( u) u.( v) Goup-C 7. State and pove the necessay and sufficient condition fo equilibium of a system of co-plana foces. 8. (a) State and pove the necessay and sufficient condition of the pinciple of vitual wok. Goup-D 9. Deive the tangential and nomal acceleation in pola co-odinates. 0. If in a simple Hamonic motion u, v, w be the velocities at distance a, b, c fom a fied point on the staight line which is not the cente of foce, show that the peiodic time T is given by the equation u v w 4 ( b c)( c a)( a b) T a b c. ******

7 B.Sc. Mathematics (Subsidiay), Pat-II Pape-II Answe any Eight questions, selecting atleast one fom each goup. All questions cay equal maks. Goup-A. Evaluate any two of the following: ( ) (a) e d ( ) d d (c) sin ( cos ). Evaluate any two sin (a) d sin cos 0 0 d ( a cos b sin ) (c) log(tan 0 ) d. (a) n Evaluate Lt.... n n n n n m n Find the eduction fomula fo cos sin d 4. (a) Find the aea between the cuve y ( a ) ( a ) and its asymptote. 5. Find the peimete of the loop of the cuve 9ay ( a)( 5a) 6. Find the volume fomed by the evolution of the loop of the cuve y ( a ) ( a ) about the - ais. 7. Solve the following diffeential equations : (a) y p ap y p 4 p 8. Solve the following diffeential equations: d y y dy (a) 4y y e d d d d Goup B 9. Find the length and the equation of the shotest distance between two skew lines y z m n and y m z n 0. Find the volume of the Tetahedon the co-odinates of whose vetices ae given.. (a) Define a conve set and a hype plane and pove that a hype plane is a conve set. Pove that the intesection of a finite numbe of conve sets is a conves set.. State and pove pinciple of vitual wok. Goup C. Find the necessay and sufficient condition fo the euqilibium of a system of co-plana foces. 4. (a) What do you mean by S.H.M., deive an epession fo time peiod. Discuss the motion of a body unde invese squae law in details. ******

8 B.Sc. Mathematics (Honous), Pat-III Pape-V Answe any five questions, selecting at least one question fom each goup. All questions cay equal maks. Goup 'A'. (a) State and pove MinKowski inequality.. If d is a metic fo X, show that the function defined by d* : X X R d(, y) defined as d * is also a metic fo X. d(, y). If, q such that and a,b ae eal numbes such that p q p q y a b p a b a 0, b 0 then pove that ab o a b p q p q 4. (a) Pove that in a metic space (, d) each open sphee is an open set. 5. (a) Pove that evey metic space is T-space. Pove that evey metic space is fist countable. Goup 'B' 6. What do you mean by a Hausdoff space. Show that evey discete topological space is Hausdoff. 7. Let (X,T) be a topological space and A and B ae subsets of X. If A denotes closue of A then show that (a) ( A B) A B ( A B) A B (c) A A Goup 'C' 8. State and pove Dabou s theoem. 9. Pove that if a bounded function f is R-integable ove [a,b] and M and m ae bounds of f then m ( b a) f ( ) d M( b a) if b a. b a Goup 'D' 0. Test the covegence of the seies p m nlog n(log log n). Show that the sum of the seies... is half the sum of the seies Eamination Pogamme-07 B.Sc (Pat III) Botany, Chemisty, Mathematics, Physics & Zoology Honous Date Papes Time Eamination Cente 4//07 Honous Pape V 8 to AM, Patna 6//07 Honous Pape VI 8 to AM, Patna 8//07 Honous Pape VII 8 to AM, Patna 0//07 Honous Pape VIII 8 to AM, Patna //07 Pape XV (Geneal Studies ) 8 to AM, Patna q

9 B.Sc. Mathematics (Honous), Pat-III Pape-VI Answe any five questions, selecting at least one question fom each goup. All questions cay equal maks. Goup 'A'. (a) Define a ing homo mophism. Let f:r R be a homomophism of ing R onto a ing R then show that f is a isomophism iff kenel f={0} Show by an eample that if I and I ae ideals of a ing R then I, UI is note an ideal of R.. (a) Show that any ing can be embedded in a ing with unity. Define a pincipal ideal ing and show that the ing of integes is a pincipal ideal ing.. (a) define an automophism of a goup G. Let G then pove that the function f defined by f (g) = - g fo g in G is an automophism of G. Let G be a goup. Then fo any element g in G, pove that Co(g) is a Sub goup of G. Goup 'B' 4. (a) Give eamples of two polynomials f() and g() such that deg (fg) < deg (f) + deg (g) Pove that the set of all polynomials in Z[] with constant tem O is pime ideal in Z[]. 5. (a) What is a pime field? pove that Q the set of all national numbes is a pime field. Let R[] be the set of all polynomials whee a 0,a,a,..., am R and am 0 as well, m is a non-negaive intege. If R is a commutative ing with unity then show that R [] is also a commutative ing with unity. Goup 'C' 6. State and pove Canto's theoem. 7. (a) Pove that C (Symbols have thei usual meaning) Fo any thee Cadinal numbes,, pove that (i) (ii) ( ) (iii) ( ) 8. (a) Let X be any non-empty set. Then show that Cad (P(X) is wee PX is the powe set of X. Intoducee the Concept of ode types and Constuct the poduct of two ode types. Goup 'D' 9. What ae the neassay and sufficient Condition fo diffeentiability of a Complet valued function. 0. Show that the function f(z) = y is not analytic at the oigin, although Cauchy Riemann diffeential equations ae satisfied.. (a) State and pove Cauchy integal fomula. Evaluate e z dz whee C is the cicle Z = 4 C ( z )

10 B.Sc. Mathematics (Honous), Pat-III (Gaphpape may be supplied) Pape-VII Answe any five questions, selecting at least one question fom each goup. All questions cay equal maks. Goup 'A'. Use simple method to solve the following L.P.P. Min: Subject to = 7, =, j 0; j =,,, 4, 5, 6. Solve the following L.P.P. gaphically Minimize z = + Subject to 6, + 4 8, -6,, 0. (a) Define a cove set, the subset of R n and show that the finite intesection of conve sets is cove set. Pove that evey hypeplane is conve. Goup 'B' 4. d dy t (a) Solve 4 y t, 5y e dt dt d dy dz Solve. y z y z 5. Solve (a) (y+z)p + (z+)q = +y pz qz = z + (+y) 6. Using Chapit s method to solve (p +q ) = pz Goup 'C' 7. Find the attaction of a unifom sphee at an etenal point of it. 8. State and pove Laplace theoem in Catesian fom. Goup 'D' 9. (a) Find the depth of the cente of pessue of a tiangle immesed in a liquid with the vete in the suface and base hoizontal. Find the depth of cente of pessue of a cicula aea of adius a immesed vetically in a homogeneous fluid. 0. (a) A od of small coss-section and density has a small potion of metal of weight n th that of the od attached to one etemity. Pove that the od will float at an engle in a liquid of density if ( n ) n. Pove that the diffeence of pessues at two points of a homogeneous fluid vaies as the depth of one point below the othe.

11 B.Sc. Mathematics (Honous), Pat-III Pape-VIII Answe any Five questions. All questions cay equal maks.. (a) Descibe Eule s method of solution fo diffeential equation and hence dy y find appoimate value of y fo =0,. Given that when y= fo d y = 0.. (a) Use Gauss-Jodan s method to solve the system of linea equations + + =8, + +4 =0 and =6 taking initial condition =0, y=0, z=0. Apply analytic method fo finding oots of an equation based on Rolle s theoem and demonstate on sin 0 4. (a) Deive Simpson s th ule fo numeical integation. 8 0 d By using Weddle s ule evaluate Use goup elaation method to solve the following system of equations -0+y+4z+4=0, -0y+z+0=0, +y-0z+45=0 (Take initial condition =0, y=0, z=0) 6. (a) Discuss Newton-Raphson s method to obtain appoimate value of oot of f()=0 Use synthetic division to solve f()= - -(.00) =0 in the neighboughood of =. 7. (a) Eplain the meaning of the opeatos E and and show that E and ae commutative with espect to vaiables. Evaluate (i) (-)(-)(-) and (ii) n (e a+b ) whee a and b ae constant. 8. (a) Descibe Newton-Gegoy fomula fo backwad intepolation. 9. (a) Descibe Picad s method of successive appoimation. Apply Runge-Kutta method fo the solution of fist ode diffeential equation. 0. (a) Eplain Gauss s method of elimination fo the solution of a system of m equations in m vaiables. Solve the following system of equations..,... 0,

Question Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if

Question Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if Subject: Mathematics-I Question Bank Section A T T. Find the value of fo which the matix A = T T has ank one. T T i. Is the matix A = i is skew-hemitian matix. i. alculate the invese of the matix = 5 7