NBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian?

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Lawrence Ross
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1 NBHM QUESTION 7 NBHM QUESTION 7 NBHM QUESTION 7 Sectio : Algebra Q Let G be a group of order Which of the followig coditios imply that G is abelia? 5 36 Q Which of the followig subgroups are ecesarily ormal subgroups? The kerel of a group homomorphism The ceter of a group The subgroup cosistig of all matrices with postive determiat i the group of all ivertible matrices with real etries (uder matri multiplicatio) Q3 List all the uits i the rig of Gaussia itegers Q4 List all possible values ocurig as deg f (degree of f) where f is a irreduciable polyomial i [] Q5 Write dow a irreduciable polyomial of degree 3 over the field F 3 of three elemets Q6 Let A ( a ij ) be a matri with real etries Let A ij be the cofactor of the etry a ij of A Let A ( A ij ) be the matri of cofactors What is the rak of A uder the followig coditios: The rak of A is? The rak of A is? Q7 Let A be a matri with comple etries which is ot a diagoal matri Pick out the cases whe A is diagoalizable A is idempotet A is ilpotet A is uitary Q8 For, let M( ) deote the rig of all matrices with real etries Which of the followig statemets are true? If A M() is ilpotet ad o-zero the there eists a matri B M() such that B A If A M ( ),, is symmetric ad positive defiite, the there eists a symmetric matgri B M( ), such that B A If A M ( ),, is symmetri, the there eists a symmetric matri B M ( ) such that 3 B A Q9 Which of the followig matrices are o-sigular? I A where A is a skew-symmetri real matri, Every skew-symmetric o-zero real 5 5 matri Every skew-symmetri o-zero real Q Let V be a real fiite-dimesioal vector space ad f ad g o-zero liear fuctioals o V Assume that ker(f) ker (g) Pick out the true statemets ker ( f ) ker( g) k ker( g)/ ker( f ) for some k such that k There eists a costat c such that g cf Sectio : Aalysis Q I each of the followig cases, state whether the series is absolutely coverget, codioally coverget (ie coverget but ot absolutely coverget) or diverget ( ) 3 ( ) log ( ) e Q Determie the iterval of covergece of the series: ( ) ( ) Q3 What is the cardiality of the followig set? A { f C [,] : f (), f (), f '( t)) for all t [,]} wwwaadistituteorg

2 NBHM QUESTION 7 Q4 Evaluate: lim k wwwaadistituteorg k cos where deotes the largest iteger ot eceedig Q5 Which of the followig improper itegrals are coverget? d d 5 6 d Q6 Which of the followig series coverge uiformly? ( a cost b si t) over the iterval [, ] where a ad b e cos over the iterval ], [ ( ) over the iterval [-, ] Q7 Let f C[, ] Determie the cases where the give coditio implies that f f ( ) d for all itegers f ( )cos d for all itegers f ( )si d for all itegers Q8 Let C be the circle defied by z 3 i the comple plae, described i the aticlockwise directio Evaluate: z z dz C z Q9 Pick out the true statemets: Let f ad g be aalytic i the disc z ad let f g o the iterval [,] the f g If f is a o-costat polyomial with comple coefficiets the it ca be factorized ito (ot ecessarily distict) Liear factors There eists a o-costat aalytic fuctio i the disc z which assumes oly real values Q Let be a ope ad coected set ad let f : be aalytic fuctio Pick out the true statemets; f is bouded if is bouded f is bouded oly if is bouded f is boude if, ad oly if, is bouded Sectio 3: Topology Q I each of the followig, f is assumed to be cotiuous Pick out the cases whe f caot be oto f : [,] f : [,] [,] f : [,] Q Cosider the set of all matrices with real etries idetified with, edowed with its usual topology Pick out the true statemets The subset of all ivertible matrices is coected The subset of all ivertible matrices is dese The subset of all orthogoal matrices is compact Q3 Pick out the fuctios that are uiformly cotiuous o the give domai f ( ) o the iterval ], [ f ( ) o f ( ) si o Q4 Let (X, d) be a metric space ad let A ad B be subets of X Defie d( A, B) if{ d( a, b) : a A, b B } Pick out the true statemets If A ad B are disjoit, the d( A, B) If A ad B are closed ad disjoit, the d( A, B) If A ad B are compact ad disjoit, the d( A, B) Q5 Pick out the sets that are homeomorphic to the set {(, y) : y } {(, y) : y } {(, y) : y } {(, y) : y } Q6 Let ( X, d ), i,,3, be the metric spaces where i i X X X3 C{,] ad d ( f, g) sup f ( ) g( ) t[,] 3 d ( f, g) f ( ) g( ) d) d ( f, g) ( f ( ) g( ) d) Let id be the idetity map of C[,] oto itself Pick out the true statemets

3 NBHM QUESTION 7 id : X X is cotiuous id : X X is cotiuous id : X3 X is cotiuous Q7 Pick out the compact sets { z, z ) : z z } The uit sphere i, the space of all square summable real seqeuces, with its usual metric d({ i },{ yi }) i yi i The closure of the uit ball of C [,] i C[,] Q8 Let f : S be ay cotiuous map, where S is the uit circle i the plae let A {(, y) S S : y, f ( ) f ( y)} Is A o-empty? If the aswer is yes, is it fiite, coutable or ucoutable? Q9 Let f : S be ay cotiuous map, where S is the uit circle i the plae Let A {(, y) S S : y, f ( ) f ( y)} Is A o-empty? Q Let f C [,] such that f ( t ) ad f '( t ) for all t [,] Let A { t [,] : f ( t) t } Is A o-empty? If the aswer is yes what is its cardiality? Sectio 4: Applied mathematics Q Let u be a smooth fuctio defied o the ball cetered at the origi ad of radius a i 3 Assume that u u u y z through- u out the ball Compute: ds S where S is the sphere with cetre at the origi ad radius a u ad deotes the outer ormal derivative of u o S Q Cosider a homogeeous fluid movig with velocity u i space Write dow the equatio which epresses the priciple of coservatio of mass Q3 Let C be the equatorial circle o the uit sphere i 3 ad let be the uit taget vector to C take i the aticlockwise sese Compute: F ds where F(, y, z) i yj zk C Q4 Determie the value of the least possible positive umber such that the followig problem has a o-trivial solutio: u "( ) u( ), u '() u '() Q5 A pedulum of mass m ad leght is pulled to a agle from the vertical ad released from rest Write dow the differetial equatio satisfied by the agle ( t) made by the pedulum with the vertical at time t, usig the priciple of coservatio of eergy (If s is the arc leght measured from the vertical positio, the the ds velocity v is give by v ) dt Q6 Fid d Alembert s solutio to the problem: u t u, t u(,) u (,) t Q7 So lve: Mi imize z 3, such that ad such that 3, ad 3 Q8 Cosider the iterative scheme B c for, where B is a real N N matri ad N c The scheme is said to be coverget if the seqeuce { } of iterates coverges for every choice of iitial vector Pick out the true statemets The scheme is coverget if, ad oly if the spectral radius of B is < The scheme is coverget if, ad oly if, for some matri orm we have B < The scheme is coverget if ad oly if B has a eigevalue such that Q9 Write dow the Laplace trasform L[ f ]( p ) of the fuctio f ( ) si a, wher a Q What is the ecessary ad sufficiet coditio for the problem to admit a solutio? wwwaadistituteorg

4 NBHM QUESTION 7 u f i u g o where is a bouded domai with boudary, is the Laplace operator, f ad g are give u smooth fuctios ad ormal derivative of u deotes the outer Sectio: 5 Miscellaeous Q Fid the area of the polygo whose vertices are the th roots of uity i the comple, plae whe 3 Q Defie p ( t) cos( cos t) for t [,] Epress p4 ( t ) as a polyomial i t Q3 What is the probability that a poit (, y ) chose at radom i the rectagle [,] [,] is such that y? Q4 A ur cotais four white balls ad two red balls A ball is draw at radom ad is replaced i the ur each time What is the probability that after twp sucessive draws, both balls draw are white? Q5 Let ABC be a triagle i the plae such that BC is perpedicular to AC Let a,b,c be the legth of BC, AC ad AB respectively Suppose that a,b,c are itegers ad have o commo divisor other tha Which of the followig statemets are ecessarily true? Either a or b is eve iteger The area of the triagle ABC is a eve iteger Either a or b is divisible by 3 Q6 What are the last two digits i the usual decimal represetatio of 4 3? Q7 Fid the umber of itegers less tha 36 ad prime to it Q Let V be a four dimesio vector space over the field F 3 of three elemets Fid the umber of distict oe-dimesioal subspaces of V Sectio : algebra Keys 7 a all 3, i 4, 5 eg 3 ( ay polyomial of degree 3, for which, ad are ot roots (mod 3)) 6 ; 7 a,c 8 b,c 9 a,c a,c Sectio : Aalysis coditioaly coverget; diverget; absolutely coverget ], ] 3 4 / 5 a,c 6 a 7 all 8 8 i 9 a,b oe Sectio 3: Toplogy a,b b,c 3c 4 c 5 b 6 a,c 7 c 8 Yes; ucoutable 9 Yes Yes Q8 Let be a positive iteger Give a eample of a seqeuce of cosecutive composite umbers Q9 For a poit P (, y) i the plae, write f ( P ) a by, where a ad b are give real umbers Let f ( A) f ( B) Let C be a poit ot o the lie joiig A ad B ad let C be the replectio of C with respect to this lie If f ( C) 5, fid f ( C ') wwwaadistituteorg 3

5 NBHM QUESTION 7 Sectio 4: Applied Mathematics a div v d dt 6 u(, t) t g(cos cos ) 7 mi z 4 at poit (8 /7,4/7)(Either data ca be accepted as full aswer ) 8 a,b 9 L[ f ]( p) a /( a p ) fd gds Sectio 5: Miscellaeous si 4 8t 8t 3 /3 4 4/9 5 all Eample; ( )!,,( )! ( ) wwwaadistituteorg

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure