. 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 Marks each. 3. Sectio-B cotais 0 Multiple Select Questios(MSQ). Each questio has 4 choices (a), (b), (c) ad (d) for its aswer, out of which ONE or MORE tha ONE is/are correct. For each correct aswer you will be awarded marks. 4. Sectio -C cotais 0 Numerical Aswer Type (NAT) questios. From Q.4 to Q.50 carries Mark each ad Q.5 to Q.60 carries Marks each. For each NAT type questio, the value of aswer i betwee 0 to 9. 5. I all sectios, questios ot attempted will result i zero mark. I Sectio A (MCQ), wrog aswer will result i egative marks. For all mark questios, /3 marks will be deducted for each wrog aswer. For all marks questios, /3 marks will be deducted for each wrog aswer. I Sectio B (MSQ),there is o egative ad o partial markig provisios. There is o egative markig i Sectio C (NAT) as well. Reg. No. 33-35, Mall Road, G.T.B. Nagar, Opp. G.T.B. Nagar Metro Statio Gate No. 3, Delhi-0 009 Delhi-0 06 DOWNLOAD CAREER ENDEAVOUR APP: E : ifo@careeredeavour.i South Delhi : 8-A/, Jia Sarai, w : ww Near-IIT w.careeredeavour.i Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
SECTION-A [Multiple Choice Questios (MCQ)] Q. Q. 0 carry oe mark each.. Which of the followig series coverges? (a) (b) e (c) 00 log (d) 3!. Let G be permutatio group o a set A of 6 symbols. The the order of proper ormal subgroup of G is (a) 0 (b) 40 (c) 360 (d) 70 3. If e f ( x) f ( x) si x x x the / e f ( x) dx (a) (b) (c) 0 (d) Noe 4. Let x..., 3 6 0 ( ). The lim( x ) is (a) 3 (b) 9 (c) 8 (d) zero 5. Let f be a smooth vector valued fuctio of a real variable. Cosider two statemets S : divcurl f 0 S : grad div f 0 The (a) both S ad S are true (b) both S ad S are false (c) S is true but S is false (d) S is false but S is true dy 6. The differetial equatio of the family of trajectories of the family of curve give by F x, y, 0 dx is dy (a) F x, y, 0 dx dy (c) F x, y, 0 dx dx (b) F x, y, 0 dy dx (d) F x, y, 0 dy 7. Let S { x : x } the the closure of S is (a) Empty set (b) Sigleto set (c) Equivalet to the set of atural umbers (d) Ucoutable set South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
8. Suppose f : is a odd ad differetiable fuctio. The for every x0, f ( x0 ) is equal to (a) f ( x0 ) (b) f ( x0 ) (c) zero (d) Noe i 9. be the -cycles (,, 3, 4, 5, 6, 7, 8, 9, 0,, ) for which positive iteger i is also a cycle? (a) 3 (b) 6 (c) 9 (d) 0. The limit of sequece, ( a ), where (a) a... (b) zero (c) does ot exist (d) 3 Q. Q. 30 carry two marks each. 3. If x y is a itegratig factor of (6 y axy) dx (6 xy bx ) dy 0, a, b the. If f : (a) 3a 5b 0 (b) a b 0 (c) 3a 5b 0 (d) a b 0 3 is give by f ( x) x x f () xf () f (3) for all x i, the f () f () is (a) f (0) (b) f (0) (c) f (0) (d) f (0) 3. The area bouded by x y 5, 4 y 4 x ad x 0 i the first quadrat (a) (b) (c) 3 (d) 4 4. The geeral solutio of 3 d y d y dy 3 dx dx dx 3 3 y 0 is (a) (c) c cx c3x e x (b) ( 3 ) c c x c x e x (d) x c cx c3x e ( c cx c3x ) e x 5. Let G r s r s rs sr {,,,, } the (a) G is group of order atmost 6 for ay (b) G is group of order 6 oly if 3 (c) G is group of order, for each,,3,... (d) G is o-abelia for ay 6. Let,, 3 deote the eigevalues of the matrix 0 0 A 0 cost si t 0 si t cos t If 3, the the set of possible values of t, t, is (a) Empty Set (b) 4 (c), 4 4 (d), 3 3 South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
x 7. f dr?, where ˆ y f i ˆj x y x y C (a, 0) to (0, a) ad c be the quarter circular path x y a from (a) a (b) a (c) a (d) a 4 8. The sum of the series, is... 3 (a) e (b) log e (c) (d) 9. Which of the followig statemets is true? (a) f : D, where D, be a uiformly cotiuous fuctio, the f is bouded (b) The fuctio f :[0,] defied as si x 0 x f ( x) x x 0 is ot uiformly cotiuous over [0, ] (c) f :[,] defied by x if x 0 f ( x) x 0 if x 0 is ot uiformly cotiuous o [, ] (d) Noe of these 0. Suppose that L y y" a y ' a y b x, where a, a are costats ad b x is a cotiuous fuctio o 0 x. The cosider the statemets I. If b x is bouded o [0, ), the every solutio of L y b x b x as x, the every solutio of L y b x is bouded o [0, ). II. If 0 teds to 0 as x. (a) oly I is correct (b) oly II is correct (c) Both are correct (d) Neither of them is correct. A object moves i the force field f y ˆ i ( x ) yj ˆ. How much work is performed o the object moves from (, 0) couter clockwise alog the elliptical path x 4y 4 to (0, ), the back to (, 0) alog the lie segmet joiig the two poits (a) 0 (b) (c) (d) Noe. Which of the followig is correct (a) Let G S be the permutatio group o symbols the for all f, g G, ( f g) f g (b) Let G S ad let 3 A f G f { : Idetity} the A is a sub-group of G (c) Let G S4 let B the B cotais exactly 7 elemets 4 { S4 : Idetity but idetity} (d) A 3 has o self iverse elemet South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
3. Cosider the followig xy I. lim x, y, x y ad II. 3 3 x y lim x, y, x y The (a) Oly I exist (b) Oly II exist (c) Both limit exist (d) Neither of them exist 4. Let y( x ) be a o-trivial solutio of the secod order liear differetial equatio d y dy c ky 0 dx dx 5 where c 0, k 0 ad c k, the (a) y( x) as x (b) y( x) 0 as x (c) lim y( x) x exists ad is fiite (d) Noe of these 0 0 5. Let A ad B 0 3 The (a) there exist a matrix C such that A BC CB (b) there is o marix C such that A = BC (c) there exist a matrix C such that A = BC but A CB (d) there is o matrix C such that A = CB 6. D e (4x 5 y ) da? where D is the elliptical disc x y 5 4 is (a) 5 0 ( e ) (b) 0 ( ) 5 e (c) 5 0 ( e ) (d) 5 e 0 ( ) 7. The radius of covergece of the series a z, where 0, a a 3 a for, is (a) zero (b) 3 (c) 3 (d) 8. Let px px x 0 f ( x) x x 0 x x. If f ( x ) is cotiuous i the iterval [, ], the p equals (a) (b) (c) (d) South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
6 9. Number of elemets of order p i Z where p ad q are distict prime is p q (a) p (b) p (c) p (d) oe of these 30. Let P, P ad P 3 deote, respectively, the plaes defied by a x b y c z a x b y c z a x b y c z 3 3 3 3 It is give that P, P, P 3 itersect exactly at oe poit whe 3, 3 ad 3 4 the the plaes (a) do ot have ay commo poit of itersectio. (b) itersect at a uique poit (c) itersect alog a straight lie (d) itersect alog a plae, If ow SECTION-B [Multiple Select Questios (MSQ)] Q. 3 Q. 40 carry two marks each. 3. Let V be the vector space of all 3 real matrices ad W be the vector space of all real matrices. The (a) There is a oe-oe liear trasformatio from V to W (b) Kerel of ay liear trasformatio from V to W is o-trivial (c) There is a isomorphism from V to W (d) There is a oto liear trasformatio from W to V 3. Cosider the followig statemets ad choose correct (a) Every ifiite ad bouded set must has a limit poit (b) Ay fiite set caot have a limit poit (c) Ay ifiite but ubouded set caot have a limit poit (d) Every closed ad bouded set has a limit poit x ( t ) dt, x 33. Let f ( x). The 0 5x, x (a) f ( x ) is ot cotiuous at x (b) f ( x ) is cotiuous but ot differetiable at x (c) f ( x ) is differetiable everywhere (d) The right derivative of f ( x ) at x does ot exist South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
7 34. Let I ydx xdy the x y C (a) I = 0 alog ay closed path c (b) I alog ay closed path c (c) I = 0 alog ay closed path c ot eclosig the origi (d) I alog ay closed path c eclosig the origi 35. Let " ' L y y a y a y where a, a are costats, ad Let The which of the followig is/are correct? p r deote its characterstic polyomial. (a) If A, are costats ad P 0, the x x Be, where B is costat is solutio of L y x Ae. (b) If x ad x are solutios of L y b x of L y b x b x. ad L y b x (c) If a 0 ad a, where is +ve costat, the every solutio approaches to ifiity for large value of x. (d) Noe of these 36. Choose the correct statemet / s (a) Every o-abelia group has a o-trivial abelia sub-group (b) Every o-trivial abelia group has a cyclic sub-group (c) The smallest order for a group to have a o-abelia proper sub-group is, the x x is solutio (d) There exist a group cotaiig elemets a ad b s.t. o( a) o( b) ad o( ab) 3 37. Let f :, a b. Which of the followig statemets is/are true? (a) if f is cotiuous ad ijective, the f is mootoe. (b) if f is differetiable ad f ' x 0 for all x a, b (c) if f is differetiable ad f ' a 0 f ' b, the f is ijective., the there is c a, b (d) if f is differetiable ad f ' a d f ' b 38. Let f xyiˆ x ˆj be a vector valued fuctio, the (a) f dr 0 alog ay closed path c C, the there is a c a, b (b) f dr alog ay smooth path coectig A(0, 0) to B(, ) C (c) f is coservative (d) f is irrotatioal x of cos such that f ' c 0. L y A x such that f ' c d. South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
8 39. If 0; x f ( x) ; otherwise,,3,... o [0, ] the (a) f ( x ) is cotiuous o [0, ] (b) f ( x ) is discotiuous o [0, ] at coutable may poits (c) f ( x ) is discotiuous at ucoutable may poits (d) 0 f ( x) dx 40. Which of the followig statemets is/are true (a) { A M ( ) : trace( A) 0} is subspace of M ( ) T (b) { A M ( ) : A A 0} is subspace of M ( ) T (c) { A M ( ) : A A 0} is subspace of M ( ) (d) { x : Ax 0} is subspace of SECTION-C [Numerical Aswer Type (NAT)] Q. 4 Q. 50 carry oe mark each. 4. The umber of permutatio i S 6 which are cojugate to () (34) are 4. The extremum value of the fuctio z xy over the plae x y is 43. The sum of the series, 44. Compute D x y dxdy; x y 4 D is the triagular regio bouded by the lie x y ad the coordiate axis th 45. Let A be the matrix whose i, j etry is i j for all i, j,,. The rak of A is. 46. Let 3 3 x y f ( x, y), 99 98 99 x y x y the f y (0,) is South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
dy x y 47. The solutio of the iitial value problem e, y() at x is dx 48. Let S 3 be the group of all permutatio with 3 symbols the the umber of elemets i S 3 that satisfy the equatio x e (where e is idetity) is 49. If f ( x ) is a cotiuous fuctio defied for x 3 ad f ( x) x [,3] (where is the set of all ratioal umber). If f () the f (.53) is 50. The volume of the solid i the first octat that is bouded by the cylider x y y, the half coe z x y ad xy plae is 9 Q. 5 Q. 60 carry two marks each. 5. Let 0 0 0 0 0 0 P 0 0 0 0 0 0, the rak of 4 P is. x 5. The set of limit poits of : x is x 53. The miimum value of the fuctio f ( x, y) xy x y over its stadard domai is 54. Let si x d e ( f ( x)) ; x 0. If dx x 4 si x e dx f ( k) f () the k = x 55. Let V be a vector space over of dimesio 7 ad T : V be a o-zero liear trasformatio. Let W be a liear subsapce of V such that V ker T W Where ker T deotes the ull space of T. The dimesio of W is. 56. The smallest order for a group to have a o-abelia proper sub-group is 57. Cosider the vector field f ( ax y a) iˆ ˆj ( x y) kˆ, where a is a costat. If f curl f 0 the the value of a is South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
58. Compute the flux itegral f ds ˆ, where f xyiˆ zj ˆ ( x y) kˆ S 0 ad S be the triagular surface cut off from the plae x y z by the coordiate plaes (Assumig ˆ is the uit outward ormal) 59. Suppose that f : at c Iff is differetiable at c ad f c 0. Suppose that g x f x is differetiable f ' c. 60. If y 3e x e x x is the solutio of the iitial value problem y 4 x, y(0) 4 dx ad dy, dx x 0 d y where,, the is END OF THE QUESTION PAPER South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
SPACE FOR ROUGH WORK South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655
IIT-JAM MATHEMATICS TEST SERIES - 5 (Full Legth Test Series - ) Time : 3 Hours Date : -0-08 M.M. : 00 ANSWER KEY SECTION-A [Multiple Choice Questios (MCQ)]. (d). (c) 3. (c) 4. (b) 5. (c) 6. (d) 7. (d) 8. (a) 9. (d) 0. (a). (a). (b) 3. (d) 4. (c) 5. (a) 6. (b) 7. (d) 8. (c) 9. (c) 0. (c). (b). (d) 3. (b) 4. (a) 5. (a) 6. (c) 7. (b) 8. (b) 9. (b) 30. (b) SECTION-B [Multiple Select Questios (MSQ)] 3. (b,d) 3. (a,b) 33. (a,d) 34. (c,d) 35. (a,b,c) 36. (a,b,d) 37. (a,b,c,d) 38. (a,b,c,d) 39. (b,d) 40. (a,b,c,d) SECTION-C [Numerical Aswer Type (NAT)] 4. (45) 4. (0.5) 43. () 44. (0.) 45. () 46. ( 96) 47. ( ) 48. (4) 49. () 50. (.77) 5. (4) 5. (,) 53. (3) 54. (6) 55. () 56. () 57. () 58. (0.54) 59. (0) 60. ( ) South Delhi : 8-A/, Jia Sarai, Near-IIT Hauz Khas, New Delhi-6, Ph : 0-685008, 686009 North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: 0-654644, 656655