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GATE 2013 : Mathematics

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2013 MATHEMATICS MA MA:MATHEMATICS Duration: Three Hours Maximum Marks:100 Please read the following instructions carefully: General Instructions: 1. Total duration of examination is 180 minutes (3 hours). 2. The clock will be set at the server. The countdown timer in the top right corner of screen will display the remaining time available for you to complete the examination. When the timer reaches zero, the examination will end by itself. You will not be required to end or submit your examination. 3. The Question Palette displayed on the right side of screen will show the status of each question using one of the following symbols: You have not visited the question yet. You have not answered the question. You have answered the question. You have NOT answered the question, but have marked the question for review. You have answered the question, but marked it for review. The Marked for Review status for a question simply indicates that you would like to look at that question again. If a question is answered and Marked for Review, your answer for that question will be considered in the evaluation. Navigating to a Question 4. To answer a question, do the following: a. Click on the question number in the Question Palette to go to that question directly. b. Select an answer for a multiple choice type question. Use the virtual numeric keypad to enter a number as answer for a numerical type question. c. Click on Save and Next to save your answer for the current question and then go to the next question. d. Click on Mark for Review and Next to save your answer for the current question, mark it for review, and then go to the next question. e. Caution: Note that your answer for the current question will not be saved, if you navigate to another question directly by clicking on its question number. 5. You can view all the questions by clicking on the Question Paper button. Note that the options for multiple choice type questions will not be shown. MA 1/14 2013 MATHEMATICS MA Answering a Question 6. Procedure for answering a multiple choice type question: a. To select your answer, click on the button of one of the options b. To deselect your chosen answer, click on the button of the chosen option again or click on the Clear Response button c. To change your chosen answer, click on the button of another option d. To save your answer, you MUST click on the Save and Next button e. To mark the question for review, click on the Mark for Review and Next button. If an answer is selected for a question that is Marked for Review, that answer will be considered in the evaluation. 7. Procedure for answering a numerical answer type question: a. To enter a number as your answer, use the virtual numerical keypad b. A fraction (eg.,-0.3 or -.3) can be entered as an answer with or without 0 before the decimal point c. To clear your answer, click on the Clear Response button d. To save your answer, you MUST click on the Save and Next button e. To mark the question for review, click on the Mark for Review and Next button. If an answer is entered for a question that is Marked for Review, that answer will be considered in the evaluation. 8. To change your answer to a question that has already been answered, first select that question for answering and then follow the procedure for answering that type of question. 9. Note that ONLY Questions for which answers are saved or marked for review after answering will be considered for evaluation. MA 2/14 2013 MATHEMATICS MA Paper specific instructions: 1. There are a total of 65 questions carrying 100 marks. Questions are of multiple choice type or numerical answer type. A multiple choice type question will have four choices for the answer with only one correct choice. For numerical answer type questions, the answer is a number and no choices will be given. A number as the answer should be entered using the virtual keyboard on the monitor. 2. Questions Q.1 Q.25 carry 1mark each. Questions Q.26 Q.55 carry 2marks each. The 2marks questions include two pairs of common data questions and two pairs of linked answer questions. The answer to the second question of the linked answer questions depends on the answer to the first question of the pair. If the first question in the linked pair is wrongly answered or is not attempted, then the answer to the second question in the pair will not be evaluated. 3. Questions Q.56 Q.65 belong to General Aptitude (GA) section and carry a total of 15 marks. Questions Q.56 Q.60 carry 1mark each, and questions Q.61 Q.65 carry 2marks each. 4. Questions not attempted will result in zero mark. Wrong answers for multiple choice type questions will result in NEGATIVE marks. For all 1 mark questions, mark will be deducted for each wrong answer. For all 2 marks questions, mark will be deducted for each wrong answer. However, in the case of the linked answer question pair, there will be negative marks only for wrong answer to the first question and no negative marks for wrong answer to the second question. There is no negative marking for questions of numerical answer type. 5. Calculator is allowed. Charts, graph sheets or tables are NOT allowed in the examination hall. 6. Do the rough work in the Scribble Pad provided. MA 3/14 2013 MATHEMATICS MA USEFUL DATA FOR MA: MATHEMATICS MA 4/14 2013 MATHEMATICS MA Q. 1 Q. 25 carry one mark each. Q.1 The possible set of eigen values of a 4 4 skew-symmetric orthogonal real matrix is (A) { } Q.2 (A) is 1 2 (B) Consider 2 ? P: Q: R:( S:( (D) {0, } (C) { 1} )2 in the Taylor series expansion of sin if ( )= 1 if = The coefficient of ( around Q.3 (B) { , 1} 2 1 2 (C) 1 6 (D) 1 6 with the usual topology. Which of the following statements are TRUE for all , = = ) = ) = . . . . (A) P and R only (B) P and S only (C) Q and R only (D) Q and S only Q.4 Let : be a continuous function with (1) = 5 and (3) = 11. If ( ) = (0) is equal to ______ then Q.5 Let be a 2 2 complex matrix such that trace( ) = 1 anddet( ) = ______ Q.6 Suppose that is a unique factorization domain and that , Which of the following statements is TRUE? 3 1 ( + ) 6. Then, trace( 4 3) is are distinct irreducible elements. (A) The ideal 1 + is a prime ideal (B) The ideal + is a prime ideal (C) The ideal 1 + is a prime ideal (D) The ideal is not necessarily a maximal ideal Q.7 Let be a compact Hausdorff topological space and let be a topological space. Let : bijective continuous mapping. Which of the following is TRUE? (A) (B) (C) (D) Q.8 be a is a closed mapbut not necessarily an open map is an open map but not necessarily a closed map is both an open map and a closed map need not be an open map or a closed map Consider the linear programming problem: Maximize + 3 2 subject to If 2 +3 16, +4 18, 0, 0. denotes the set of all solutions of the above problem, then (A) is empty (C) is a line segment MA (B) is a singleton (D) has positive area 5/14 2013 Q.9 MATHEMATICS MA Which of the following groups has a proper subgroup that is NOT cyclic? (A) 15 (B) 3 (C) ( , +) (D) ( , +) Q.10 77 The value of the integral 1 /2 0 is ______ Q.11 (A) ( ) = (B) ( ) = = 2 log . The density of if > 0 otherwise 0 2 2 if > 0 otherwise 0 1 2 (C) ( ) = 2 0 if >0 otherwise [0,2] if otherwise 1/2 (D) ( ) = 0 Q.12 has uniform distribution on [0,1] and Suppose the random variable is such that | ( )| Let be an entire function on then (1) (A) must be 2 (C) must be 100 log| |for each with| | 2. If ( ) = 2 , (B) must be 2 (D) cannot be determined from the given data Q.13 The number of group homomorphisms from Q.14 Let ( , ) be the solution to the wave equation 2 3 to 9 is ______ 2 ( , )= 2 2 ( , ), ( , 0) = cos(5 ), ( , 0) = 0. Then, the value of (1,1) is ______ Q.15 Let ( ) = (A) lim (C) lim MA 0 0 sin ( =1 2 ) . Then ( )=0 ( ) = 2 /6 (B) lim (D) lim 0 0 ( )=1 ( ) does not exist 6/14 2013 Q.16 MATHEMATICS MA {0,1,2, } and some Suppose is a random variable with ( = ) = (1 ) for (0,1). For the hypothesis testing problem 1 1 0: = 1: 2 2 consider the test Reject 0 if or if , where < are given positive integers. The type-I error of this test is (A) 1 + 2 (B) 1 2 (C) 1 + 2 (D) 1 2 2 +2 2 +2 1 1 Q.17 Let G be a group of order 231. The number of elements of order 11 in G is ______ Q.18 Let : {( , ) 2 2 2 ). The area of the image of the region be defined by ( , ) = ( + , : 0 < , < 1} under the mapping is (A) 1 Q.19 (B) 2 2 (D) 1 [ ] 2 2 +2 2 2 2 2 [ ] (C) [ ] (D) Let +2 [ ] (B) = 0. Define 0 +1 = cos for every } is increasing and convergent } is decreasing and convergent } is convergent and 2 < lim } is not convergent (A) { (B) { (C) { (D) { Q.21 (C) Which of the following is a field? (A) Q.20 1 Let 0. Then < 2 +1 for every be the contour | | = 2 oriented in the anti-clockwise direction. The value of the integral is 3/ (A) 3 Q.22 For each > 0, let Var(log ) (B) 5 (C) 7 be a random variable with exponential density (D) 9 on(0, ). Then, (A) is strictly increasing in (B) is strictly decreasing in (C) does not depend on (D) first increases and then decreases in MA 7/14 2013 Q.23 MATHEMATICS MA Let { } be the sequence of consecutive positive solutions of the equation tan = be the sequence of consecutive positive solutions of the equationtan = . Then (A) 1 =1 1 (C) Both Q.24 converges but =1 and 1 =1 diverges converge 1 (B) =1 1 (D) Both 1 diverges but =1 and =1 1 =1 Let be an analytic function on = { : | | 1}. Assume that | ( )| (0)? Then, which of the following is NOT a possible value of (A) 2 Q.25 1 =1 (B) 6 (C) 7 1/9 9 } and let { converges diverge 1 for each (D) 2 + . 2 The number of non-isomorphic abelian groups of order 24 is ______ Q. 26 to Q. 55 carry two marks each. Q.26 Let V be the real vector space of all polynomials in one variable with real coefficients and having degree at most 20. Define the subspaces 1 (1) = 0, (5) = 0, (7) = 0 , = 0, 1 = 2 1 (3) = 0, (4) = 0, (7) = 0 . = 0, 2 = 2 Then the dimension of Q.27 Let , [0,1] 1 2 is ______ be defined by if ( )= 0 = 1 for otherwise and [0,1] ( ) = 1 if 0 otherwise. Then (A) Both and are Riemann integrable (B) is Riemann integrable and is Lebesgue integrable (C) is Riemann integrable and is Lebesgue integrable (D) Neither nor is Riemann integrable Q.28 Consider the following linear programming problem: Maximize subject to +3 +6 5 + +6 +7 20, 6 +2 +2 +9 40, 0, 0, 0, 0. Then the optimal value is ______ Q.29 Suppose is a real-valued random variable. Which of the following values CANNOTbe attained by [ ] and [ 2 ], respectively? (A) 0 and 1 MA (B) 2 and 3 1 1 (C) 2and 3 (D) 2 and 5 8/14 2013 Q.30 MATHEMATICS MA Which of the following subsets of 2 is NOT compact? (A) {( , ) 2 1 1, = sin } (B) {( , ) 2 1 1, = (C) {( , ) 2 Q.31 Let > 0, 3 = sin 1 > 0, = 1 be the real vector space of 2 3 matrices with real entries. Let : 2 3 4 5 6 6 = 4 5 3 1 2 be defined by . is ______ The determinant of Let be a Hilbert space and let { 1} be an orthonormal basis of is a bounded linear operator. Which of the following CANNOT be true? (A) ( ) = 1 (B) ( ) = (C) ( ) = (D) ( ) = Q.33 2 ( , ) 1 Q.32 1} ) = 0} = 0, sin( 2 (D) ( , ) 8 1 for all +1 for all +1 1 . Suppose : 1 1 for all for all 2 and ( 1 ) = 0 The value of the limit lim 2 2 = +1 2 2 is (A) 0 Q.34 (B) some Let : \{3 } FALSE? (0,1) be defined by ( ) = (C) 1 +3 (D) . Which of the following statements about (A) is conformal on \{3 } (B) maps circles in \{3 } onto circles in (C) All the fixed points of are in the region { Im( ) > 0} (D) There is no straight line in \{3 } which is mapped onto a straight line in Q.35 The matrix 1 = 21 31 (A) [1 MA is by 1 2 0 = 1 3 1 can be decomposed uniquely into the product = , where 0 1 3 0 0 11 12 13 1 0 and = 0 = [1 2 2] is 22 23 . The solution of the system 0 0 33 32 1 1 1] (B) [1 1 0] (C) [0 1 1] (D) [1 0 1] 9/14 2013 Q.36 MATHEMATICS MA = Let 0, (A) 1 Q.37 (B) =1 < . Then the supremum of 1 (C) 0 The image of the region { Re( ) > 0, Im( ) > 0} (B){ (C){ | | > 1} (D) { 2 1 1 (B) 1 and 30 2 (C) 1 (D) (C) 0 and 25 (D) 1 and 25 (34 3)(34 32 )(34 33 ) (34 1)(34 2)(34 3) (34 1)(34 3)(34 32 ) 34 (34 1)(34 2) Let be a vector space of dimension 2. Let : be a linear transformation such that = 0 and 0 for some 1. Then which of the following is necessarily TRUE? (A) Rank ( ) Nullity ( (C) is diagonalizable ) (B) trace ( ) (D) = 0 Let be a convex region in the plane bounded by straight lines. Let have 7 vertices. Suppose ( , ) = + + has maximum value and minimum value on and < . Let ={ is a vertex of and < ( ) < }. If has elements, then which of the following statements is TRUE? (A) (C) cannot be 5 cannot be 3 (B) (D) can be 2 can be 4 Which of the following statements are TRUE? P: If Q: If R: If S: If 1( ), then is continuous. ) and lim| | ( ) exists, then the limit is zero. 1 ( ), then is bounded. 1( ) ( ) exists and equals zero. is uniformly continuous, then lim| | 1( (A) Q and S only MA 2 of order eight Let be the space of all4 3 matrices with entries in the finite field of three elements. Then the number of matrices of rank three in is +1 Q.44 2 4, Let X be an arbitrary random variable that takes values in{0, 1, , 10}. The minimum and maximum possible values of the variance of X are (A) (B) (C) (D) Q.43 (D) 8 (B) (A) 0 and 30 Q.42 Im( ) > 0, | | > 1} Let be a real symmetric positive-definite matrix. Consider the inner product on defined be the linear operator defined . Let be an real matrix and let : by , = by ( ) = for all for all . If is the adjoint of ,then ( ) = ,where is the matrix (A) Q.41 is Re( ) > 0, Im( ) > 0, | | > 1} (B) The dihedral group, 4 (C) The quaternion group, Q.40 2 Which of the following groups contains a unique normal subgroup of order four? (A) Q.39 (D) Re( ) > Im( ) > 0} under the mapping (A) { Q.38 is (B) P and R only (C) P and Q only (D) R and S only 10/14 2013 Q.45 MATHEMATICS MA 2 be a real valued harmonic function on . Let : Let be defined by 2 ( , )= ( + ) sin . 0 Which of the following statements is TRUE? (A) (B) (C) (D) Q.46 Let = { as ( ) = (A) (B) (C) (D) Q.47 is a harmonic polynomial is a polynomial but not harmonic is harmonic but not a polynomial is neither harmonic nor a polynomial 2 | | = 1} with the induced topology from . Then, which of the following is TRUE? is closed in [0,2] is open in [0,2] ( ) is closed in ( ) is open in and let : [0,2] be defined ( )is closed in ( ) is open in is closed in [0,2] is open in [0,2] 1 Assume that all the zeros of the polynomial + + 1 parts. If ( ) is any solution to the ordinary differential equation + + 1 0 have negative real 1 + 1 1 + + 1 + 0 = 0, ( ) is equal to then lim (A) 0 (B) 1 (C) 1 (D) Common Data Questions Common Data for Questions 48 and 49: Let 00 be the vector space of all complex sequences having finitely many non-zero terms. Equip 00 with for all = ( ) and = ( ) in 00 . Define : 00 by ( ) = the inner product , = =1 . Let be the kernel of . =1 Q.48 Which of the following is FALSE? (A) (B) is a continuous linear functional 6 (C) There does not exist any (D) {0} Q.49 such that ( ) = , for all 00 Which of the following is FALSE? (A) (B) (C) (D) MA 00 00 is closed is not a complete inner product space 00 = 00 11/14 2013 MATHEMATICS MA Common Data for Questions 50 and 51: Let 1 , 2 , , be an i.i.d. random sample from exponential distribution with mean they have density 1 / if > 0 ( )= 0 otherwise. Q.50 . In other words, Which of the following is NOT an unbiased estimate of ? (A) 1 1 (B) 1 ( 2 + 3 + + ) (C) (min{ 1 , 2 , , }) 1 (D) max{ 1 , 2 , , } Q.51 Consider the problem of estimating . The m.s.e (mean square error) of the estimate ( )= 1 + 2 + + +1 is 2 (A) (B) 1 +1 2 (C) 1 ( +1)2 2 2 (D) 2 ( +1)2 Linked Answer Questions Statement for Linked Answer Questions 52 and 53: Let Let 2 = {( , ) 0 = max{ 2 : < + 2 = 1} ([ 1,1] {0}) ({0} [ 1,1]). , there are distinct points 1 , , such that { 1, , Q.52 The value of Q.53 Let 1 , , 0 +1 be 0 + 1 distinct points and = . Let 1, , 0 +1 connected components of . The maximum possible value of is ______ 0 } is connected} is ______ be the number of Statement for Linked Answer Questions 54 and 55: Let ( 1 , 2 ) be the Wr nskian of two linearly independent solutions ( ) + ( ) = 0. Q.54 The product ( 1, 2) 1 2 1 1 2 (B) 1 2 of the equation + 2 1 (C) 1 2 2 1 (D) 2 1 1 If 1 (A) 4 MA 2 ( ) equals (A) Q.55 and = 2 and 2 = 2 2 , then the value of (0) is (B) 4 (C) 2 (D) 2 12/14 2013 MATHEMATICS MA General Aptitude (GA) Questions Q. 56 Q. 60 carry one mark each. Q.56 A number is as much greater than 75 as it is smaller than 117. The number is: (A) 91 Q.57 (B) 93 (D) 96 The professor ordered to the students to go out of the class. I II III IV Which of the above underlined parts of the sentence is grammatically incorrect? (A) I Q.58 (C) 89 (B) II (C) III (D) IV Which of the following options is the closest in meaning to the word given below: Primeval (A) Modern (C) Primitive Q.59 (B) Historic (D) Antique Friendship, no matter how _________it is, has its limitations. (A) cordial (B) intimate (C) secret (D) pleasant Q.60 Select the pair that best expresses a relationship similar to that expressed in the pair: Medicine: Health (A) Science: Experiment (C) Education: Knowledge (B) Wealth: Peace (D) Money: Happiness Q. 61 to Q. 65 carry two marks each. Q.61 X and Y are two positive real numbers such that 2 + 6 and + 2 8. For which of the following values of ( , ) the function ( , ) = 3 + 6 will give maximum value? (A) (4/3, 10/3) (B) (8/3, 20/3) (C) (8/3, 10/3) (D) (4/3, 20/3) Q.62 If |4 7| = 5 then the values of 2 | | (A) 2, 1/3 MA (B) 1/2, 3 | | is: (C) 3/2, 9 (D) 2/3, 9 13/14 2013 Q.63 MATHEMATICS MA Following table provides figures (in rupees) on annual expenditure of a firm for two years - 2010 and 2011. Category 2010 2011 Raw material 5200 6240 Power & fuel 7000 9450 Salary & wages 9000 12600 Plant & machinery 20000 25000 Advertising 15000 19500 Research & Development 22000 26400 In 2011, which of the following two categories have registered increase by same percentage? (A) Raw material and Salary & wages (B) Salary & wages and Advertising (C) Power & fuel and Advertising (D) Raw material and Research & Development Q.64 A firm is selling its product at Rs. 60 per unit. The total cost of production is Rs. 100 and firm is earning total profit of Rs. 500. Later, the total cost increased by 30%. By what percentage the price should be increased to maintained the same profit level. (A) 5 Q.65 (B) 10 (C) 15 (D) 30 Abhishek is elder to Savar. Savar is younger to Anshul. Which of the given conclusions is logically valid and is inferred from the above statements? (A) Abhishek is elder to Anshul (B) Anshul is elder to Abhishek (C) Abhishek and Anshul are of the same age (D) No conclusion follows END OF THE QUESTION PAPER MA 14/14 GATE 2013 : Answer keys for MA - Mathematics Paper Q.No Key(s)/Value(s) Paper Q.No Key(s)/Value(s) MA 1 A MA 36 A MA 2 C MA 37 C MA 3 B MA 38 Marks to All MA 4 6 MA 39 A MA 5 78 MA 40 C MA 6 D MA 41 C MA 7 D MA 42 A MA 8 C MA 43 D MA 9 D MA 44 A MA 10 2 MA 45 A MA 11 C MA 46 A MA 12 B MA 47 A MA 13 3 MA 48 D MA 14 1 MA 49 D MA 15 A MA 50 D MA 16 C MA 51 B MA 17 10 MA 52 4 MA 18 D MA 53 8 MA 19 C MA 54 A MA 20 C MA 55 B MA 21 D MA 56 D MA 22 C MA 57 B MA 23 B MA 58 C MA 24 B MA 59 B MA 25 3 MA 60 C MA 26 15 MA 61 A MA 27 B MA 62 B MA 28 60 MA 63 D MA 29 B MA 64 A MA 30 C MA 65 D MA 31 -1 MA 32 A MA 33 D MA 34 C MA 35 A

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