Regn. No. North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: ,

Size: px

Start display at page:

Download "Regn. No. North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: ,"

Eric Bryan
5 years ago
Views:

1 1. Section-A contains 0 Multiple Choice Questions (MCQ). Each question has 4 choices (a), (b), (c) and (d), for its answer, out of which ONLY ONE is correct. From Q.1 to Q.10 carries 1 Marks and Q.11 to Q.0 carries Marks each.. Section-B contains 10 Multiple Select Questions(MSQ). Each question has 4 choices (a), (b), (c) and (d) for its answer, out of which ONE or MORE than ONE is/are correct. For each correct answer you will be awarded marks. 4. Section -C contains 0 Numerical Answer Type (NAT) questions. From Q.41 to Q.50 carries 1 Mark each and Q.51 to Q.60 carries Marks each. For each NAT type question, the value of answer in between 0 to In all sections, questions not attempted will result in zero mark. In Section A (MCQ), wrong answer will result in negative marks. For all 1 mark questions, 1/ marks will be deducted for each wrong answer. For all marks questions, / marks will be deducted for each wrong answer. In Section B (MSQ),there is no negative and no partial marking provisions. There is no negative marking in Section C (NAT) as well. Regn. No. -5, Mall Road, G.T.B. Nagar, Opp. G.T.B. Nagar Metro Station Gate No., Delhi Delhi DOWNLOAD CAREER ENDEAVOUR APP E : info@careerendeavour.in South Delhi : 8-A/11, Jia Sarai, w : ww Near-IIT w.careerendeavour.in Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

2 SECTION-A [Multiple Choice Questions (MCQ)] Q. 1 Q. 10 carry one mark each. 1. Let A ( aij ) 4 matri with real entries such that the space of all solutions of the linear system. 1 A 4 is given by 1 t t : t 4t Then Rank(A) =? (a) 4 (b) (c) (d) 1. The integrating factor of the differential equation log dy y log d is (a) e (b) log (c) log log (d). Let f be a real valued function satisfying f f a c a for some 0 and c 0. Consider S 1 : for 1, f is continuous at a. S : for 1, f is differentiable at a. Then (a) Only S 1 is true (b) only S is true (c) Both are true (d) None of them is true 4. Evaluate the tripple integral T y d dy dz, where T is the Region bounded by the surfaces = y y, 4 z y and z y. (a) (b) (c) (d) If ( ) ( )( ) where ( ) 0 f g a g a and g is continuous at a, then (a) f is increasing near a if g( a) 0 (b) f is decreasing near a if g( a) 0 (c) f is increasing near a if g( a) 0 (d) None of these 6. Number of elements of order 0 in U(100) are (a) 8 (b) 1 (c) 16 (d) 0 South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

3 7. Let [ ] be the vector space of all real polynomials over field. Which of the following subsets of [ ] is not a subspace. (a) { p( ) : p(1) 0} (b) { p( ) : p(0) p(0) 0}; p( ) is derivative of p( ) (c) { p( ) : p( ) p(1 ) } (d) { p( ) : p( ) has a real root at } 8. Let y( t ) be the solution of the initial values problem y y y 0, y(0) b, y(0) 0, for which value of b is lim y( t) 0? t (a) There is no value for b with the limit 0 (b) 0 (c) 1 (d) 1 9. The angle between the surfaces ln z y 1 and y z at the point (1, 1, 1) is (a) (b) (c) 4 (d) None 10. Let P be a polynomial in the real variable of degree 7. Then P n lim is n (a) 5 (b) 7 (c) (d) 0 n Q. 11 Q. 0 carry two marks each. 11. Let R + be multiplications group of all postive reals and R be a addifine group of all reals and f : R R be a map given by f log R, (a) f is not a homomorphism (b) f is an isomorphism (c) f is a homomorphism but not onto (d) f is a homomorphism but not one-one 1. Let V be the vector space of dimension and F be a finite field with 4 elements and let V be the vector space over field F then number of distinct basis of V is (a) 100 (b) 00 (c) 00 (d) None 1. If t is a solution of 1 t d t dt dt and 0 1, then 1 is equal to (a) 1 6 (b) 1 6 (c) (d) South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

4 14. Let, y y ;, 0,0 f y y 0 ;, 0,0 y Consider [S 1 ] f has directional derivatives at (0, 0) in every direction. [S ] f is differentiable at (0, 0). Then (a) Only S 1 is true (b) Only S is true (c) Both S 1 and S is true (d) None of them is true 15. Evaluate the integral cos R,0,,,, and 0,. y y d dy where R is rhombus with successive vertices at (a) 4 (b) (c) (d) Let A and B be n n real matrices such that AB = BA = 0 and A + B is invertible, then which of the following is not always true? (a) rank( A) rank( B) (b) rank( A) rank( B) n (c) nullity( A) nullity( B) n (d) A B is invertible Let f ( ) be differentiable on the interval (0, ) such that f (1) 1 and 0. Then f ( ) is t f ( ) f ( t) lim 1, for each t t (a) (b) (c) 1 (d) For a linear transformation T :, Kernel of T is {(,... ) ; ; 0}, then dimension of Range of T is (a) 5 (b) 4 (c) 6 (d) 19. The line integral of ˆ ˆ v i yj z kˆ over the straight line path from ( 1,, ) to (,, 5) is (a) 6 (b) 9 0. Consider the sequence nn1 (c) 5 defined recursivily as 1 1 (d) None 1 n, n sup [0, n1) :sin 0 Then limsup n equals (a) (b) 0 (c) 1 (d) 1 1. The solution of the equation b c ac bac in K 4 is where K 4 is a abelian group of order 4 (a) a (b) b (c) c (d) None of these South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

5 5 R. y d dy, over the rectangle 0,1] [0, R, where [.] is greatest integer function of. (a) 0 (b) 1 (c) (d). The directional devivative of iˆ 4 ˆj 5k ˆ is f, y, z y 4yz z at the point (1,, ) in the direction of (a) (b) 5 14 (c) 7 15 (d) None 4. The sum of the series k1 k k is (a) log (b) 1 (c) e (d) e 5. Number of onto homomorphism from Z 8 to Z 6 is / are (a) 1 (b) (c) (d) 0 6. Let A {( aij ) 5 5 s.t. aij i j i & j } then (a) A is diagonalizable (b) det (A) = 15 (c) A is nilpotant (d) Rank (A) = 5 7. dy Consider the initial value problem f (, y), d y(0) 1, for which of the following choices of function f should we not epect a unique solution? (a) f (, y) y (b) f (, y) y y e (c) f (, y) y 1/ (d) f (, y) (1 y) 4/5 8. Let 4 4 y y if (, y) (0,0) 4 4 f (, y) y. Which of the following is true? 0 if (, y) (0,0) (a) f (0,0) f (0,0) y y (b) (c) f( h,0) does not eist for any real h (d) f (, y) f (, y) (, y) y (, y) (0,0) y lim f (, y) For the set ; m, n, the element 1 where u is n m u (a) Both an element in the set and a limit point of the set. (b) Neither an element in the set nor a limit point. (c) An element in the set but not a limit point. (d) A limit point of the set, but not an element in the set. 0. The arc of the parabola y from (1, 1) to (, 4) is rotated above the y-ais. The area of resulting surface is 6 (a) (b) (c) (d) South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

6 6 SECTION-B [Multiple Select Questions (MSQ)] Q. 1 Q. 40 carry two marks each. 1. If 1 f ( ) [ ], then ([.] denotes the greatest integer function) (a) f ( ) is discontinuous at 1, 10, 15 n (b) f ( ) is continuous at, where n is any integer (c) / 0 1 f ( ) d (d) lim f ( ). Let f and g be two functions defined on an Interval I. Then the maimum and minimum function of f and g defined for all I by ma f, g ma{ f ( ), g( )} and min f, g min f, g respectively. Then which of the following statement is/are True? (a) ma, min, f g f g f g for all I. (b) ma, ma, f h g h f g h for any function h defined on I. (c) min, min, f h g h f g h for any function h defined on I. (d) min, ma, f g f g for all I.. 0 ; c if Q or 0 f 1 ; p if, p, q and p, q are co-prime q q Then, (a) f is continuous on [0, 1]. (b) f is continuous only at irrationals. (c) f is discontinuous only at rationals. (d) f is integrable on [0, 1] and f d South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

7 7 4. Let 4 T : be the linear tranformation given by T (, y, z, t) ( y z t, z t, y z t) then (a) {(1,1,1),(0,1, )} form a basis for Range space of T (b) {(,1, 1,0),(1,,0,1)} form a basis of Ker T (c) Rank T = (d) Nullity T = 5. f for (a) (b) f is continuous. 1, then f is strictly increasing on [1, ). (c) f is integrable on every closed subset of (d) None of the above. 1,. 6. Let an log 4 1 n n and bn then, log n n for all n (a) Both (b) Both (c) (d) Both a n and b n are convergent in. a n and b n are divergent in. a n is increasing and b n is a decreasing sequence. a n and b n are increasing. 7. Choose incorrect statements: (a) There is an isomorphism from ( Q, ) to ( Q *, ) where Q* Q 0 (b) Aut G Aut G and G 1 is infinite group then G is also infinite. 1 (c) If Aut G Aut G and G 1 is finite then G is finite 1 (d) G1 G then Aut( G1 ) Aut( G ) (where symbol denote isomorphism) 8. Let f : be a continuous function with period p 0. Then g( ) f ( t) dt is a (a) constant function (c) continuous function but not differentiable p (b) continuous function (d) neither continuous nor differentiable South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

8 9. Let the vector field ˆ ˆ F ( ay cz) i y( b cz) j ( ay c ) kˆ. The integral F dr 0 for any piecewise smooth closed curve C if (a) b c a (b) a c b (c) y 0 (d) z Let us consider a vector field F y z iˆ y ˆj z k ˆ Then, (a) F is conservative (b) The total work done in moving a particle along a closed path is 0. (c) The work done in moving a particle from ( 1,, 1) to (,, 4) is 87. (d) None of the above. C 8 SECTION-C [Numerical Answer Type (NAT)] Q. 41 Q. 50 carry one mark each. 41. The work done by the force F y iˆ y ˆj in moving a particle along a closed path c containing the curves y 0, y 16 and y in the first and fourth quadrants is. 4. Let log 0, y 0 u v where is. y u e and v e y. Then 4. The number of non-isomorphic groups of order 6 is. 44. Evaluate 45. lim (, y) (0,0) D y da, where D is the region bounded by y and y 1 is. cos y Let ˆ ˆ F zi y j z kˆ and S is the boundry of the region bounded by the paraboloid z y and the plane 4 S z y then F nˆ ds 47. If dy y( d y dy), y(1) 1 and y( ) 0. Then, y( ) is equal to South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

9 9 48. Let A, then trace A 4 is 49. lim 4 sec tan 1 cos 4 is. 50. Let P be a non-zero polynomial of degree n. The radius of convergence of the power series is. n0 P n n Q. 51 Q. 60 carry two marks each. 51. The volume of the solid that lies above the cone. z y and below the sphere y z z is 5. Let H 10 be the space of matrices with entries in satisfying aij ars whenever i j r s then dim H 10 is 5. Let y ( ) and y ( ) be solutions of y y (sin ) y 0 such that (0) 1, (0) 1 and (0) 1, (0). Then the value of Wronskian w(, ) at 1 is 54. In a non-abelian group the elements a has order 108 then the order of a 4 is 55. Let f : be a continuous function s.t. f, if f then f 4. dy 56. Given that y( ) is a solution of the differential equation 5 y (4 y) with (0) d y. When then the solution y increases to, where is. 1/ n 57. Find lim n!. 58. Let S be the set of non-null diagonalizable matrices such that A k = 0 for positive integer k ( k n), then number of elements is set S is 59. Let c be the trace of the cone z y intersected by the plane z = 4 and S in the surface of the cone below z = 4, then y d dy z dz c. 60. The smallest number n for which a group of order n for which Lagrange's theorem is not true is. END OF THE QUESTION PAPER South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

10 10 SPACE FOR ROUGH WORK South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

11 IIT-JAM MATHEMATICS TEST SERIES - 6 (Full Length Test Series - ) Time : Hours Date : 8-01-018 M.M. : 100 ANSWER KEY SECTION-A [Multiple Choice Questions (MCQ)] 1. (c). (b). (c) 4. (a) 5. (c) 6.

(b) SECTION-B [Multiple Select Questions (MSQ)] 1. (a, c). (a,b,c,d). (b,c,d) 4. (a,b,c,d) 5. (a,b,c) 6. (a,c) 7. (a,b) 8. (a,b) 9. (a,c) 40. (a,b,c) SECTION-C [Numerical Answer Type (NAT)] 41. (67.

11 11 IIT-JAM MATHEMATICS TEST SERIES - 6 (Full Length Test Series - ) Time : Hours Date : M.M. : 100 ANSWER KEY SECTION-A [Multiple Choice Questions (MCQ)] 1. (c). (b). (c) 4. (a) 5. (c) 6. (c) 7. (a) 8. (c) 9. (d) 10. (d) 11. (b) 1. (d) 1. (a) 14. (a) 15. (c) 16. (a) 17. (a) 18. (a) 19. (b) 0. (b) 1. (d). (c). (d) 4. (c) 5. (d) 6. (a) 7. (d) 8. (a) 9. (a) 0. (b) SECTION-B [Multiple Select Questions (MSQ)] 1. (a, c). (a,b,c,d). (b,c,d) 4. (a,b,c,d) 5. (a,b,c) 6. (a,c) 7. (a,b) 8. (a,b) 9. (a,c) 40. (a,b,c) SECTION-C [Numerical Answer Type (NAT)] 41. (67.8) 4. (1) 4. () 44. (.1) 45. (0) 46. (8) 47. () 48. (4) 49. (0.5) 50. (1) 51. (0.15) 5. (19) 5. (.718) 54. (18) 55. () 56. (4) 57. (1) 58. (0) 59. (19) 60. (1) South Delhi : 8-A/11, Jia Sarai, Near-IIT Hauz Khas, New Delhi-16, Ph : , North Delhi : -5, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. ), Delhi-09, Ph: ,

Regn. No. North Delhi : 33-35, Mall Road, G.T.B. Nagar (Opp. Metro Gate No. 3), Delhi-09, Ph: ,

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