PAPER : IIT-JAM 2010

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Joan Pierce
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1 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 the covergece of a real sequece a? (a) a a 0 as (b) a a is coverget (c) a is coverget (d) The sequeces a, ad a a are coverget. The value of G l( ) x y dx dy, where G {( x, y) ; x y e } is x y (a) (b) (c) (d) 4. The umber of elemets of S 5 (the symmetric group o 5 letters) which are their ow iverses equals (a) 0 (b) (c) 5 (d) 6 4. Let S be a ifiite subset of such that S. Which of the followig statemets is true? (a) S must have a limit poit which belogs to (b) S must have a limit poit which belogs to \ (c) S caot be a closed set i (d) \S must have a limit poit which belogs to S 5. Let f :, 4 be a uiformly cotiuous fuctio ad let a Let x a f a ad y f a (a) Both ad (b) (c) 6. Let x y must be Cauchy sequeces i x must be a Cauchy sequece i but y y must be a Cauchy sequece i but x (d) Neither or x y eeds to be a Cauchy sequece i F xyz e iˆ z e ˆj ye k x x x ˆ a be a Cauchy sequece i (, )., for all N. Which of the followig statemet is true? eed ot be a Cauchy sequece i eed ot be a Cauchy sequece i be the gradiet of a scalar fuctio. The value of F. dr alog the orieted path L from (0, 0, 0) to (, 0, ) ad the to (,, ) is (a) 0 (b) e (c) e (d) e 7. Let F xyiˆ yj ˆ yzkˆ deote the force field o a particle traversig the path L from (0, 0, 0) to (,, ) alog the curve of itersectio of the cylider y x ad the plae z x. The work doe by F is L (a) 0 (b) 4 (c) (d)

2 8. Let X be the rig of real polyomials i the variables X. The umber of ideals i the quotiet rig X /(X X+) is (a) (b) (c) 4 (d) 6 9. Cosider the differetial equatio solutio y( x ) teds to dy ay by dx, where a, b > 0 ad y(0) = y 0. As x, the (a) 0 (b) a/b (c) b/a (d) y 0 0. Cosider the differetial equatio ( x y ) dx (x y ) dy 0. Which of the followig statemets is true? (a) The differetial equatio is liear (b) The differetial equatio is exact (c) e x+y is a itegratig factor of the differetial equatio (d) A suitable substitutio trasforms the differetiable equatio to the variables separable form. Let T : T (5,6) is be a liear trasformatio such that T (,) (,) ad T (0,) (,4). The (a) (6, ) (b) ( 6, ) (c) (, 6) (d) (, 6). The umber of matrices over (the field with three elemets) with determiat is (a) 4 (b) 60 (c) 0 (d) 0. The radius of covergece of the power series 0 az, where 0, a a a for N, is (a) 0 (b) (c) (d) 4. Let T : be the liear trasformatio whose matrix with respect to stadard basis {e, e, e } of is The T 0 0 (a) maps the subspace spaed by e ad e ito itself (b) has distict eigevalues (c) has eigevectors that spa (d) has a o-zero ull space

3 5. Let T : be the liear trasformatio whose matrix with respect to the stadard basis of is 0 a b a 0 c b c, where a, b, c are real umbers ot all zero. The T 0 (a) is oe to oe (b) is oto (c) does ot map ay lie passes through the origi oto itself (d) has rak 6. (a) Obtai the geeral solutio of the followig system of differetial equatios: dx x y dt, (9) dy 4x y e dt t (b) Fid the curve passig through,0 equatio ( y ) dx (x ta y) dy 0. (6) ad havig slope at ( x, y ) give by differetial 7. (a) Fid the volume of the regio i the first octat bouded by the surfaces x = 0, y = x, y = x, z = 0 ad z = x. (6) (b) Suppose f : for all x, y. is a o-costat cotiuous fuctio satisfyig f ( x y) f ( x) f ( y) (i) Show that f ( x) 0 for all x. (ii) Show that f ( x) 0 for all x. x (iii) Show that there exists such that f ( x) for all x. (9) 8. (a) Let f ( x) ad g( x ) be real valued fuctios cotiuous i [a, b], differetiable i (a, b) ad let g( x) 0 for all x ( a, b). Show that there exists c ( a, b) such that f ( c) f ( a) f ( c). (9) g( b) g( c) g( c) (b) Let 0 4 ad let {a } be a sequece of positive real umbers satisfyig a a a ( ) for. Prove that lim a exists ad determie this limit. (6) 9. Let G be a ope subset of. (a) If 0 G, the show that H { xy : x, y G } is a ope subset of. (9) (b) If 0G ad if x y G for all x, y G, the show that G = (6)

4 4 0. Let p x be a o-costat polyomial with real coefficiets such that 0 p x for all x. Defie f x p x for all x. Prove that (i) For each 0, there exists a > 0 such that f x for all x R satisfyig x a, ad (ii) f : is a uiformly cotiuous fuctio. (5). (a) Let M ( K) ad m( k ) deote respectively the absolute maximum ad the absolute miimum values of i the closed iterval [ 0, ]. Fid all the real values of k for which x 9x x k M ( k) m ( k). (6) (b) Let 0, ;,, ad for,,. Prove that, for (i) ( ) (ii) a b a b Deduce that lim for ay a, b. (9). (a) Let f ( x, y) x xy y, 0, 0, 4. Fid sufficiet coditios o (, ) such that (0, 0) is (9) (i) a poit of local maxima of f ( x, y) (ii) a poit of local miima of f ( x, y) (iii) a saddle poit of f ( x, y) (b) Fid the derivative of vector 6i ˆ ˆj k ˆ f x, y, z 7x x z z 8y at the poit A = (,, 0) alog the uit. What is the uit vector alog which f decreases most rapidly at A? Also, fid 7 the rate of this decreases. (6). Usig u x e, trasform the differetial equatio d y dy x 4x y cos x dx dx to a secod order differetial equatio with costat coefficiets. Obtai the geeral solutio of the trasformed differetial equatio. (5)

5 5 4. Let G be a group ad let A(G) deote the set of all automorphism of G, i.e. all oe-to-oe, oto, group homomorphisms from G to G. A automorphism f : G G of the form, f x axa x G (for some a G ) is called a ier automorphism. Let I(G) deote the set of all ier automorphism of G. (a) Show that A(G) is a group uder compositio of fuctios ad that I(G) is a ormal subgroup of A(G). (9) (b) Show that I(G) is isomorphic to G/Z(G), where : for all Z G g G xg gx x G is the cetre of G. (6) 5. (a) Give a example of a liear trasformatio T : such that ( ) (b) Let V be a real -dimesioal vector space ad let T : V satisfyig T ( v) v for all v V. T v v for all v. (6) V be a liear trasformatio (i) Show that is eve. (ii) Use T to make V ito complex vector space such that the multiplicatio by complex umbers exteds the multiplicatio by real umbers. (iii) Show that, with respect to the complex vector space structure o V obtaied i (ii), T : V V is a complex liear trasformatio. (9) 6. Let W be the regio bouded by the plaes x 0, y 0, y, z 0 ad x z 6. Let S be the boudary of this regio. Usig Gauss divergece theorem, evaluate S F. ds ˆ, where ˆ F xyi yz ˆj xzkˆ ad ˆ is the outward uit ormal vector to S. (5) 7. (a) Usig strokes theorem evaluate the lie itegral ˆ of L ˆ ˆ yi zj xk. dr, where L is the itersectio x y z ad x y 0 traversed i the clockwise directio whe viewed from the poit (,, 0). (9) (b) Chage the order of itegratio i the itegral x f ( x, y) dy dx. (6) 0 x 8. I a group G, x G is said to be cojugate to y G, writte x ~ y, if there exists z G such that x zyz. (a) Show that ~ is a equivalece relatio o G. Show that a subgroup N of G is a ormal subgroup of G if ad oly if N is a uio of equivalece classes of ~. (6) (b) Cosider the group of all o-sigular real matrices uder matrix multiplicatio. Show

6 6 that ~ (i.e. the two matrices are cojugate). (9) 9. Let S deote the commutative rig of all cotiuous real valued fuctios o [0, ], uder poitwise additio ad multiplicatio. For a 0,, let M f S f a a 0. (a) Show that M a is a ideal i S. (6) (b) Show that M a is a maximal ideal i S. (9) END

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Regn. No. North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: , . Sectio-A cotais 30 Multiple Choice Questios (MCQ). Each questio has 4 choices (a), (b), (c) ad (d), for its aswer, out of which ONLY ONE is correct. From Q. to Q.0 carries Marks ad Q. to Q.30 carries