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IIT JAM 2015 : Mathematics

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JAM 2015 MATHEMATICS - MA JAM 2015: General Instructions during Examination 1. Total duration of the JAM 2015 examination is 180 minutes. 2. The clock will be set at the server. The countdown timer at 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 need not terminate the examination or submit your paper. 3. Any useful data required for your paper can be viewed by clicking on the Useful Data button that appears on the screen. 4. Use the scribble pad provided to you for any rough work. Submit the scribble pad at the end of the examination. 5. You are allowed to use only your own non-programmable calculator. 6. The Question Palette displayed on the right side of screen will show the status of each question using one of the following symbols: 7. 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, but marked for review', then the answer will be considered for evaluation unless the status is modified by the candidate. Navigating to a Question : 8. 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 the answer for a multiple choice type question and for the multiple select type question. Use the virtual numeric keypad to enter the answer for a numerical type question. c. Click on Save & Next to save your answer for the current question and then go to the next question. d. Click on Mark for Review & Next to save and to mark for review your answer for the current question, and then go to the next question. Caution: Note that your answer for the current question will not be saved, if you navigate to another question directly by clicking on a question number without saving the answer to the previous question. 9. You can view all the questions by clicking on the Question Paper button. This feature is provided, so that if you want you can just see the entire question paper at a glance. Answering a Question : 10. Procedure for answering a multiple choice question (MCQ): a. MA Choose the answer by selecting only one out of the 4 choices (A,B,C,D) given below the question and click on the bubble placed before the selected choice. 1/16 JAM 2015 MATHEMATICS - MA b. c. To change your chosen answer, click on the bubble of another choice. d. 11. To deselect your chosen answer, click on the bubble of the selected choice again or click on the Clear Response button. To save your answer, you MUST click on the Save & Next button. Procedure for answering a multiple select question (MSQ): a. b. To deselect one or more of your selected choice(s), click on the checkbox(es) of the choice(s) again. To deselect all the selected choices, click on the Clear Response button. c. To change a particular selected choice, deselect this choice that you want to change and click on the checkbox of another choice. d. 12. Choose the answer by selecting one or more than one out of the 4 choices (A,B,C,D) given below the question and click on the checkbox(es) placed before each of the selected choice (s). To save your answer, you MUST click on the Save & Next button. Procedure for answering a numerical answer type (NAT) question: a. To enter a number as your answer, use the virtual numerical keypad. b. A fraction (e.g. -0.3 or -.3) can be entered as an answer with or without '0' before the decimal point. As many as four decimal points, e.g. 12.5435 or 0.003 or -932.6711 or 12.82 can be entered. c. To clear your answer, click on the Clear Response button. d. To save your answer, you MUST click on the Save & Next button. 13. To mark a question for review, click on the Mark for Review & Next button. If an answer is selected (for MCQ and MSQ types) or entered (for NAT) for a question that is Marked for Review, that answer will be considered in the evaluation unless the status is modified by the candidate. 14. To change your answer to a question that has already been answered, first select that question and then follow the procedure for answering that type of question as described above. 15. Note that ONLY those questions for which answers are saved or marked for review after answering will be considered for evaluation. Choosing a Section : 16. Sections in this question paper are displayed on the top bar of the screen. All sections are compulsory. 17. Questions in a section can be viewed by clicking on the name of that section. The section you are currently viewing will be highlighted. 18. To select another section, simply click the name of the section on the top bar. You can shuffle between different sections any number of times. 19. When you select a section, you will only be able to see questions in this Section, and you can answer questions in the Section. 20. After clicking the Save & Next button for the last question in a section, you will automatically be taken to the first question of the next section in sequence. 21. You can move the mouse cursor over the name of a section to view the answering status for that section. MA 2/16 JAM 2015 MATHEMATICS - MA JAM 2015 Examination MA: Mathematics Duration: 180 minutes Maximum Marks: 100 Read the following instructions carefully. 1. To login, enter your Registration Number and Password provided to you. Kindly go through the various coloured symbols used in the test and understand their meaning before you start the examination. 2. Once you login and after the start of the examination, you can view all the questions in the question paper, by clicking on the Question Paper button in the screen. 3. This test paper has a total of 60 questions carrying 100 marks. The entire question paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are of different types. 4. Section A contains Multiple Choice Questions (MCQ). Each MCQ type question has four choices out of which only one choice is the correct answer. This section has 30 Questions and carry a total of 50 marks. Q.1 Q.10 carry 1 mark each and Questions Q.11 Q.30 carry 2 marks each. 5. Section B contains Multiple Select Questions (MSQ). Each MSQ type question is similar to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the four given choices. The candidate gets full credit if he/she selects all the correct choices only and no wrong choices. This section has 10 Questions and carry 2 marks each with a total of 20 marks. 6. Section C contains Numerical Answer Type (NAT) questions. For these NAT type questions, the answer is a real number which needs to be entered using the virtual numerical keypad on the monitor. No choices will be shown for these type of questions. This section has 20 Questions and carry a total of 30 marks. Q.1 Q.10 carry 1 mark each and Questions Q.11 Q.20 carry 2 marks each. 7. Depending upon the JAM test paper, there may be useful common data that may be required for answering the questions. If the paper has such useful data, the same can be viewed by clicking on the Useful Data button that appears at the top, right hand side of the screen. 8. The computer allotted to you at the examination centre runs specialized software that permits only one choice to be selected as answer for multiple choice questions using a mouse, one or more than one choices to be selected as answer for multiple select questions using a mouse and to enter a suitable number for the numerical answer type questions using the virtual numeric keypad and mouse. 9. Your answers shall be updated and saved on a server periodically and also at the end of the examination. The examination will stop automatically at the end of 180 minutes. 10. Multiple choice questions (Section-A) will have four choices against A, B, C, D, out of which only ONE choice is the correct answer. The candidate has to choose the correct answer by clicking on the bubble ( ) placed before the choice. 11. Multiple select questions (Section-B) will also have four choices against A, B, C, D, out of which ONE OR MORE THAN ONE choice(s) is /are the correct answer. The candidate has to choose the correct answer by clicking on the checkbox ( ) placed before the choices for each of the selected choice(s). 12. For numerical answer type questions (Section-C), each question will have a numerical answer and there will not be any choices. For these questions, the answer should be entered by using the mouse and the virtual numerical keypad that appears on the monitor. 13. In all questions, questions not attempted will result in zero mark. In Section A (MCQ), wrong answer will result in NEGATIVE marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. For all 2 marks questions, 2/3 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. MA 3/16 JAM 2015 MATHEMATICS - MA 14. Non-programmable calculators are allowed but sharing of calculators is not allowed. 15. Mobile phones, electronic gadgets other than calculators, charts, graph sheets, and mathematical tables are NOT allowed in the examination hall. 16. You can use the scribble pad provided to you at the examination centre for all your rough work. The scribble pad has to be returned at the end of the examination. Declaration by the candidate: I have read and understood all the above instructions. I have also read and understood clearly the instructions given on the admit card and shall follow the same. I also understand that in case I am found to violate any of these instructions, my candidature is liable to be cancelled. I also confirm that at the start of the examination all the computer hardware allotted to me are in proper working condition . MA 4/16 JAM 2015 MATHEMATICS - MA Notation - The set of natural numbers = - The set of integers - The set of rational numbers - The set of real numbers - The set of complex numbers - The group of permutations of - The group of integers modulo - The vector space of distinct symbols real matrices - Standard mutually orthogonal unit vectors - Imaginary number - Complex conjugate of - Complex conjugate of matrix - Transpose of matrix - Empty set sup - supremum inf - infimum - Derivative of MA 5/16 JAM 2015 MATHEMATICS - MA SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. 1 Q.10 carry one mark each. Q.1 Suppose (A) If (B) If (C) If (D) If Q.2 is a normal subgroup of a group . Which one of the following is true? is an infinite group then N is an infinite group is a nonabelian group then is a nonabelian group is a cyclic group then is an abelian group is an abelian group then is a cyclic group Let Then is (A) ln cos (C) Q.3 Let Q.6 (D) is a finite union of disjoint bounded intervals, then which one and be a convergent sequence of real numbers. If , then which one of the following is the limit of this sequence? (B) (C) The volume of the portion of the solid cylinder and bounded below by the -plane is (A) MA (C) If is not compact, then sup and inf Even if sup and inf need not be compact If sup and inf , then is compact Even if is compact, it is not necessary that sup and inf Let for (A) . Then an eigenvalue of the is Let be a nonempty subset of . If of the following is true? (A) (B) (C) (D) Q.5 (B) (D) ln sec (B) (A) sec . cos be a solution of the differential equation be distinct non-zero real numbers with matrix Q.4 sin (B) (C) (D) bounded above by the surface (D) 6/16 JAM 2015 Q.7 MATHEMATICS - MA be a differentiable function with Let which one of the following statements is true on is unbounded has at least one zero (A) (C) Q.8 Q.9 (B) (D) is increasing and bounded is periodic If an integral curve of the differential equation then is equal to (B) (A) , then . If for all ? passes through and , (D) (C) An integrating factor of the differential equation is (B) (A) Q.10 be a nonempty subset of Let (A) (B) (C) (D) (C) . Let (D) denote the set of interior points of . Then can be empty singleton a finite set containing more than one element countable but not finite Q. 11 Q. 30 carry two marks each. Q.11 Let be the group of permutations of three distinct symbols. The direct sum element of order (A) 4 Q.12 (D) 18 has an are (B) (D) Let be a nonabelian group. Let the element in (A) is 6 (C) is of the form MA (C) 9 The orthogonal trajectories of the family of curves (A) (C) Q.13 (B) 6 for have order and let have order . Then the order of (B) is 12 (D) need not be finite 7/16 JAM 2015 Q.14 MATHEMATICS - MA Let be the bounded surface of the cylinder cos Q.15 (B) sin cos (D) cos (C) and cos sin cos cos Suppose that the dependent variables and are functions of the independent variables and , where , defined by the equations Which one of the following is correct? (A) (C) . is equal to Then the value of the surface integral (A) cut by the planes and . (B) (D) Q.16 Let . Then and (A) an eigenvalue of is purely imaginary (B) an eigenvalue of is zero (C) all eigenvalues of are real (D) has a non-zero real eigenvalue Q.17 The limit lim is equal to (A) 0 Q.18 (B) sin sin (C) (D) be the vector space of polynomials in of degree at most 2 with real coefficients. Let be the vector space of real matrices. If a linear transformation is defined as Let then (A) (B) is one-one but not onto is onto but not one-one (C) Range ( (D) Null ( MA = span = span 8/16 JAM 2015 MATHEMATICS - MA Q.19 Let and a linear transformation such that then be ordered bases of , the matrix of with respect to . If and is , is , is equal to (A) (B) (C) (D) Q.20 Let . Which one of the following statements is FALSE? and in such that (A) There exist sequences (B) is an open set (C) If is an infinite subset of , then has a limit point (D) There exists an infinite subset of having no limit points Q.21 Let sequence be a strictly increasing continuous function. If is (A) increasing (C) convergent is a sequence in , then the (B) bounded (D) not necessarily bounded Q.22 Which one of the following statements is true for the series (A) (B) (C) (D) ? The series converges conditionally but not absolutely The series converges absolutely The sequence of partial sums of the series is bounded but not convergent The sequence of partial sums of the series is unbounded Q.23 The sequence cos (A) (B) (C) (D) tan is monotone and convergent monotone but not convergent convergent but not monotone neither monotone nor convergent Q.24 If is a solution of the differential equation lim is equal to (A) MA , then (B) (C) (D) 9/16 JAM 2015 MATHEMATICS - MA Q.25 For what real values of and y, does the integral (B) (D) (A) (C) Q.26 attain its maximum? The area of the planar region bounded by the curves (B) (A) and is (D) (C) Q.27 For , let sin . Then at be given by (A) local maximum if is even (B) local maximum if is even (D) local minimum if has a is odd (C) local minimum if , is odd Q.28 For Then at sin , define , is (A) differentiable for each pair with (B) differentiable for each pair with (C) not differentiable for each pair with (D) not differentiable for each pair with Q.29 is connected and Let and be nonempty subsets of , where Which one of the following statements is true for all such and ? (A) If is connected (B) If = , then is not connected (C) If , then (D) If MA = , then then is not connected. is connected is not connected 10/16 JAM 2015 MATHEMATICS - MA Q.30 Let Then the value of be given by (A) changes with tan but not with (C) changes with but not with (B) changes with and also with (D) neither changes with nor with SECTION - B MULTIPLE SELECT QUESTIONS (MSQ) Q. 1 Q. 10 carry two marks each. Q.1 be a function defined by Let In which of the following interval(s), takes the value 1? (A) (B) (C) . (D) Q.2 Which of the following statements is (are) true? (A) (B) (C) (D) MA is isomorphic to is isomorphic to is isomorphic to is isomorphic to 11/16 JAM 2015 MATHEMATICS - MA Q.3 Which of the following conditions implies (imply) the convergence of a sequence numbers? (A) Given there exists an such that for all , (B) Given there exists an such that for all , (C) Given there exists an such that for all , (D) Given there exists an such that for all with of real , Q.4 Let be a vector field given by the curve of intersection of the surfaces is (are) equal to ? (A) sin (C) sin is , for . If , then which of the following and matrices sin (B) sin (D) Q.5 Let be the set of with complex entries such that . Let . Then, under usual matrix addition and scalar be the set of matrices in with multiplication, which of the following is (are) true? (A) (B) (C) (D) is a vector space over is a vector space over is a vector space over is a vector space over Q.6 The initial value problem , has (A) at least two solutions if (B) no solution if (C) at least one solution if (D) a unique solution if MA 12/16 JAM 2015 Q.7 MATHEMATICS - MA Which of the following statements is (are) true on the interval (A) cos cos sin (C) Q.8 Let ? (B) tan ln (D) be defined by At , (A) is not continuous (B) is continuous, and both (C) is differentiable (D) and exist but and exist is not differentiable Q.9 be functions. Let and Let Which of the following statements is (are) true? (A) If for all (B) If for some (C) If for some (D) If for all Let be the ranges of and , respectively. be the function defined by , then sup inf , then inf , then inf , then sup sup sup inf Q.10 Then (A) is decreasing in (C) MA has two solutions in (B) (D) is increasing in has no solutions in 13/16 JAM 2015 MATHEMATICS - MA SECTION C NUMERICAL ANSWER TYPE (NAT) Q. 1 Q. 10 carry one mark each. Q.1 Let be the straight line segment from cos sin to , in the -plane. Then the value of is ______________ Q.2 bounded by the planes Let be the portion of the surface and . The surface area of , correct upto three decimal places, is ______________ Q.3 The number of distinct normal subgroups of is ______________ Q.4 Let be defined by Q.5 exists along the direction cos If the directional derivative of at sin , then the value of cot is ______________ Let sin , where be defined by The maximum rate of change of ______________ at sin , correct upto three decimal places, is Q.6 If the power series converges for and diverges for places, is ______________ , then the value of , correct upto three decimal Q.7 If MA ( mod ) and , then is equal to _________ 14/16 JAM 2015 MATHEMATICS - MA Q.8 If the set is linearly dependent in the vector space of all matrices with real entries, then is equal to ______________ Q.9 Let be defined by The number of points at which is continuous, is ______________ Q.10 Let be a continuously differentiable function such that and changes sign at exactly two of these points. Then for any in is ______________ number of solutions to has finitely many zeros in , the maximum Q. 11 Q. 20 carry two marks each. Q.11 Let be the planar region bounded by the lines and the curve , in the first quadrant. Let be the boundary of oriented counter-clockwise. Then the value of is ______________ Q.12 Suppose is a cyclic group and order of the smallest group containing Q.13 The limit are such that order( ) = 12 and order( ) = 21. Then the and is ______________ lim is equal to ______________ Q.14 Let be the vector space of Then the dimension of MA real matrices. Let is ______________ be a subspace of defined by 15/16 JAM 2015 MATHEMATICS - MA Q.15 be defined by Let sin Then the integral correct upto three decimal places, is ______________ Q.16 The coefficient of about the point in the Taylor series expansion of the function sin cos , correct upto three decimal places, is ______________ Q.17 If cos has the power series expansion decimal places, is equal to ______________ , then , correct upto three Q.18 Let be the length of the portion of the curve where satisfies The value of Q.19 and , , correct upto three decimal places, is ______________ The limit is equal to ______________ Q.20 between the lines Let rank and lim tan be two real matrices of size and , then rank is equal to ______________ , respectively. If rank and END OF THE QUESTION PAPER MA 16/16 JAM 2015 Answer Keys for the Test Paper: Mathematics (MA) Section A - MCQ Section B - MSQ Section C - NAT Multiple Choice Questions Multiple Select Questions Numerical Answer Type Questions Q. No. Key 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 C C B B A B A C B A B D D A A C D C D D B B A B A C D A D D Marks Q. No. 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 10 Key Marks Q. No. A;C;D A;D C;D A;B A;C;D A;C A;D B;C B;C;D A;B;C 2 2 2 2 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Range 1 6.28 3 -1 2.34 1.64 1 -1 2 3 8 84 0.25 2 0.125 1.41 0.10 5.09 3 2 to to to to to to to to to to to to to to to to to to to to 1 6.29 3 -1 2.35 1.65 1 -1 2 3 8 84 0.25 2 0.125 1.42 0.11 5.10 3 2 Marks 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2

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