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IIT JAM 2012 : Mathematics

40 pages, 41 questions, 12 questions with responses, 12 total responses,

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A A 2012- MA 2012- MA Test Paper Code: MA Time: 3 Hours Maximum Marks: 300 INSTRUCTIONS 1. This question-cum-answer booklet has 36 pages and has 29 questions. Please ensure that the copy of the question-cum-answer booklet you have received contains all the questions. 2. Write your Registration Number, Name and the name of the Test Centre in the appropriate space provided on the right side. 3. Write the answers to the objective questions against each Question Number in the Answer Table for Objective Questions, provided on Page 7. Do not write anything else on this page. 4. Each objective question has 4 choices for its answer: (A), (B), (C) and (D). Only ONE of them is the correct answer. There will be negative marking for wrong answers to objective questions. The following marking scheme for objective questions shall be used: (a) For each correct answer, you will be awarded 6 (Six) marks. (b) For each wrong answer, you will be awarded -2 (Negative two) mark. (c) Multiple answers to a question will be treated as a wrong answer. (d) For each un-attempted question, you will be awarded 0 (Zero) mark. (e) Negative marks for objective part will be carried over to total marks. 5. Answer the subjective question only in the space provided after each question. 6. Do not write more than one answer for the same question. In case you attempt a subjective question more than once, please cancel the answer(s) you consider wrong. Otherwise, the answer appearing last only will be evaluated. 7. All answers must be written in blue/black/blueblack ink only. Sketch pen, pencil or ink of any other colour should not be used. 8. All rough work should be done in the space provided and scored out finally. 9. No supplementary sheets will be provided to the candidates. 10. Clip board, log tables, slide rule, calculator, cellular phone and electronic gadgets in any form are NOT allowed. 11. The question-cum-answer booklet must be returned in its entirety to the Invigilator before leaving the examination hall. Do not remove any page from this booklet. 12. Refer to special instructions/useful data on the reverse. MA- i / 36 READ INSTRUCTIONS ON THE LEFT SIDE OF THIS PAGE CAREFULLY REGISTRATION NUMBER Name: Test Centre: Do not write your Registration Number or Name anywhere else in this question-cum-answer booklet. I have read all the instructions and shall abide by them. ... Signature of the Candidate I have verified the information filled by the Candidate above. ... Signature of the Invigilator A Special Instructions/ Useful Data : The set of all natural numbers, that is, the set of all positive integers 1, 2, 3, : The set of all integers : The set of all rational numbers : The set of all real numbers {e1 , e2 , , en } : The standard basis of the real vector space f , f : First and second derivatives respectively of a real function f f x ( a, b), f y ( a, b) : Partial derivatives with respect to x and y respectively of f : 2 at (a, b) R S : Product ring of rings R, S with componentwise operations of addition and multiplication MA- ii / 36 n A Q.1 IMPORTANT NOTE FOR CANDIDATES Questions 1-15 (objective questions) carry six marks each and questions 16-29 (subjective questions) carry fifteen marks each. Write the answers to the objective questions in the Answer Table for Objective Questions provided on page 7 only. Let { xn } be the sequence + 1, 1, + 2, 2, + 3, 3, + 4, 4,... . If yn = then the sequence { yn } is (A) (B) (C) (D) Q.2 + xn for all n , monotonic NOT bounded bounded but NOT convergent convergent The number of distinct real roots of the equation x 9 + x 7 + x 5 + x 3 + x + 1 = 0 is (A) Q.3 x1 + x2 + n If f : 1 (B) 2 3 (C) 5 (D) 9 is defined by x3 f ( x, y ) = x 2 + y 4 0 if ( x, y ) ( 0, 0 ) , if ( x, y ) = ( 0, 0 ) , then (A) f x ( 0, 0 ) = 0 and f y ( 0, 0 ) = 0 (B) f x ( 0, 0 ) = 1 and f y ( 0, 0 ) = 0 (C) f x ( 0, 0 ) = 0 and f y ( 0, 0 ) = 1 (D) f x ( 0, 0 ) = 1 and f y ( 0, 0 ) = 1 1 Q.4 The value of (A) 1 90 z y xy z =0 y =0 x = 0 (B) 2 z 3 dx dy dz is 1 50 (C) MA- 1 / 36 1 45 (D) 1 10 A Q.5 The differential equation (1 + x 2 y 3 + x 2 y 2 ) dx + ( 2 + x 3 y 2 + x 3 y ) dy = 0 is exact if equals (A) Q.6 1 2 (D) (C) e3x (B) y 3 a 2 + b2 c2 = 0 a 2 + b2 + c 2 = 0 (B) (D) Consider the quotient group 2 is element + in 3 3 (D) e3 y ) to the cone z = x 2 + y 2 , then a 2 b2 + c2 = 0 a 2 + b2 + c2 = 0 of the additive group of rational numbers. The order of the (B) 3 (C) 5 (D) 6 Which one of the following is TRUE ? (A) The characteristic of the ring 6 is 6 (B) The ring 6 has a zero divisor (C) The characteristic of the ring ( 6 ) 6 (D) The ring 6 6 Q.10 2 ( (A) 2 Q.9 (C) is the unit normal vector at 1,1, 2 + bj + ck For c > 0, if ai (A) (C) Q.8 3 2 An integrating factor for the differential equation ( 2 xy + 3 x 2 y + 6 y 3 ) dx + ( x 2 + 6 y 2 ) dy = 0 is (A) x3 Q.7 (B) is zero is an integral domain Let W be a vector space over and let T : 6 W be a linear transformation such that S = {Te2 , Te4 , Te6 } spans W . Which one of the following must be TRUE ? (A) S is a basis of W (B) T ( 6 ) W (C) (D) {Te1 , Te3 , Te5 } spans W ker (T ) contains more than one element MA- 2 / 36 A Q.11 Consider the following subspace of 3 : W = ( x, y, z ) 3 2 x + 2 y + z = 0, 3x + 3 y 2 z = 0, x + y 3 z = 0 . { } The dimension of W is (A) 0 Q.12 (B) 1 (C) 2 Let P be a 4 4 matrix whose determinant is 10. The determinant of the matrix (A) 810 (B) 30 (C) 30 Q.13 If the power series an x n converges for x = 3, then the series n =0 (A) (B) (C) (D) Q.14 3P (D) 810 (D) [ 1,1] a x n =0 n n converges absolutely for x = 2 converges but not absolutely for x = 1 converges but not absolutely for x = 1 diverges for x = 2 x If Y = 1 + x (A) Q.15 (D) 3 ( 1,1) x , then the set of all limit points of Y is ( 1,1] (B) If C is a smooth curve in 3 (C) [0,1] from (0, 0, 0) to (2,1, 1) , then the value of ( 2 xy + z ) dx + ( z + x ) dy + ( x + y ) dz 2 C is (A) 1 (B) 0 (C) MA- 3 / 36 1 (D) 2 is A Space for rough work MA- 4 / 36 A Space for rough work MA- 5 / 36 A Space for rough work MA- 6 / 36 A Answer Table for Objective Questions Write the Code of your chosen answer only in the Answer column against each Question Number. Do not write anything else on this page. Question Answer Number Do not write in this column 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 FOR EVALUATION ONLY Number of Correct Answers Marks (+) Number of Incorrect Answers Marks ( ) Total Marks in Questions 1-15 MA- 7 / 36 ( ) A Q.16 (a) Examine whether the following series is convergent: n! . 1 3 5 (2 n 1) n =1 (6) (b) For each x , let [ x ] denote the greatest integer less than or equal to x. Further, for a fixed (0,1), define an = sequence {an } converges to . 1 [ n ] + n 2 n for all n . Show that the n (9) MA- 8 / 36 A MA- 9 / 36 A x2 Q.17 (a) Evaluate lim x 0 (b) For a, b 0 4 + t 3 dt x2 . (6) with a < b, let f : [ a, b ] be continuous on [ a, b ] and twice differentiable on ( a, b ) . Further, assume that the graph of f intersects the straight line segment joining the points ( a, f ( a ) ) and ( b, f ( b ) ) at a point ( c, f ( c ) ) a < c < b. Show that there exists a real number ( a, b ) such that f ( ) = 0. MA-10 / 36 for (9) A MA-11 / 36 A Q.18 (a) Show that the point (0,0) is neither a point of local minimum nor a point of local maximum for the function f : 2 given by f ( x, y ) = 3x 4 4 x 2 y + y 2 for ( x, y ) (b) Find 2 . (6) all the critical points of the function f : 2 given by f ( x, y ) = x + y 3x 12 y + 40 for ( x, y ) . Also, examine whether the function f attains a local maximum or a local minimum at each of these critical points. (9) 3 3 2 MA-12 / 36 A MA-13 / 36 A 4 Q.19 (a) Evaluate 2 x =0 3 e y dy dx. y = 4 x (6) 3 bounded above by (b) Using multiple integral, find the volume of the solid region in 2 2 the hemisphere z = 1 + 1 x y and bounded below by the cone z = x 2 + y 2 . (9) MA-14 / 36 A MA-15 / 36 A Q.20 Find the area of the portion of the surface z = x 2 y 2 in cylinder x 2 + y 2 1 . MA-16 / 36 3 which lies inside the solid (15) A MA-17 / 36 A d2y y = 0 such that y ( 0 ) = 2 dx 2 such that the infimum of the set Q.21 Let y ( x ) be the solution of the differential equation and y (0) = 2 . Find all values of [ 0,1) { y ( x) x } is greater than or equal to 1. MA-18 / 36 (15) A MA-19 / 36 A Q.22 (a) Assume that y1 ( x ) = x and y2 ( x ) = x3 are two linearly independent solutions of the d2y dy 3x + 3 y = 0. Using the method of 2 dx dx a particular solution of the differential homogeneous differential equation x 2 variation of parameters, find d2y dy equation x 2 2 3x + 3 y = x5 . dx dx dy 5 y 5 x 4 + = subject to the condition y (1) = 1. (b) Solve the differential equation dx 6 x y 5 MA-20 / 36 (6) (9) A MA-21 / 36 A be the position vector field in Q.23 (a) Let r = xi + yj + zk { } 3 and let f : be a differentiable function. Show that f ( r ) r = 0 for r 0 . (6) (b) Let W be the region inside the solid cylinder x 2 + y 2 4 between the plane z = 0 and the paraboloid z = x 2 + y 2 . Let S be the boundary of W . Using Gauss s dS , where F n + ( 3 xy ) F = ( x + y 4) i j + ( 2 xz + z ) k divergence theorem, evaluate S 2 2 is the outward unit normal vector to S. and n MA-22 / 36 2 (9) A MA-23 / 36 A Q.24 (a) Let G be a finite group whose order is not divisible by 3. Show that for every g G , there exists an h G such that g = h3 . (6) (b) Let A be the group of all rational numbers under addition, B be the group of all non-zero rational numbers under multiplication and C the group of all positive rational numbers under multiplication. Show that no two of the groups A, B and C are isomorphic. (9) MA-24 / 36 A MA-25 / 36 A Q.25 (a) Let I be an ideal of a commutative ring R. Define A = {r R | r n I for some n }. Show that A is an ideal of R. (6) (b) Let F be a field. For each p ( x ) F [ x ] (the polynomial ring in x over F ) define : F [ x ] F F by ( p ( x ) ) = ( p ( 0 ) , p (1) ) . (i) Prove that is a ring homomorphism. (ii) Prove that the quotient ring F [ x ] ( x 2 x ) is isomorphic to the ring F F . MA-26 / 36 (9) A MA-27 / 36 A Q.26 (a) Let P, D and A be real square matrices of the same order such that P is invertible, D is diagonal and D = PAP 1. If An = 0 for some n , then show that A = 0. (6) (b) Let T : V W be a linear transformation of vector spaces. Prove the following: {v1 , v2 , , vk } spans V , and T is onto, then {Tv1 , Tv2 , , Tvk } spans W . If {v1 , v2 , , vk } is linearly independent in V , and T is one-one, then {Tv1 , Tv2 , , Tvk } is linearly independent in W . If {v1 , v2 , , vk } is a basis of V , and T is bijective, then {Tv1 , Tv2 , , Tvk } is (i) If (ii) (iii) a basis of W . (9) MA-28 / 36 A MA-29 / 36 A Q.27 (a) Let {v1 , v2 , v3 } be a basis of a vector space V over . Let T : V V be the linear transformation determined by Tv1 = v1 , Tv2 = v2 v3 and Tv3 = v2 + 2v3 . Find the matrix of the transformation T with {v1 + v2 , v1 v2 , v3 } as a basis of both the domain and the co-domain of T . (6) (b) Let W be a three dimensional vector space over and let S : W W be a linear transformation. Further, assume that every non-zero vector of W is an eigenvector of S. Prove that there exists an such that S = I , where I : W W is the identity transformation. (9) MA-30 / 36 A MA-31 / 36 A Q.28 (a) Show that the function uniformly continuous. (b) For each n , let f n : { fn} f: , defined by f ( x ) = x 2 for x , is not (6) converges uniformly on uniformly continuous. be a uniformly continuous function. If the sequence to a function f : , then show that f is (9) MA-32 / 36 A MA-33 / 36 A Q.29 (a) Let A be a nonempty bounded subset of is a closed subset of . Show that { x | x a for all a A} . (b) Let {xn } be a sequence in (6) such that | xn +1 xn | < sequence {xn } is convergent. 1 for all n . Show that the n2 (9) MA-34 / 36 A MA-35 / 36 A Space for rough work MA-36 / 36 A 2012 - MA Objective Part (Question Number 1 15) Total Marks Signature Subjective Part Question Number 16 Question Number Marks 23 17 24 18 25 19 26 20 27 21 28 22 29 Total Marks in Subjective Part Total (Objective Part) : Total (Subjective Part) : Grand Total : Total Marks (in words) : Signature of Examiner(s) : Signature of Head Examiner(s) : Signature of Scrutinizer : Signature of Chief Scrutinizer : Signature of Coordinating Head Examiner : MA-iii / 36 Marks A MA-iv / 36

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