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IIT JAM 2018 : Mathematics (with Answer Keys)

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JAM 2018 Mathematics - MA Paper Specific Instructions 1. The examination is of 3 hours duration. There are a total of 60 questions carrying 100 marks. The entire paper is divided into three sections, A, B and C. All sections are compulsory. Questions in each section are of different types. 2. Section A contains a total of 30 Multiple Choice Questions (MCQ). Each MCQ type question has four choices out of which only one choice is the correct answer. Questions Q.1 Q.30 belong to this section 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. 3. Section B contains a total of 10 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 answers only and no wrong answers. Questions Q.31 Q.40 belong to this section and carry 2 marks each with a total of 20 marks. 4. Section C contains a total of 20 Numerical Answer Type (NAT) questions. For these NAT type questions, the answer is a real number which needs to be entered using the virtual keyboard on the monitor. No choices will be shown for these type of questions. Questions Q.41 Q.60 belong to this section and carry a total of 30 marks. Q.41 Q.50 carry 1 mark each and Questions Q.51 Q.60 carry 2 marks each. 5. 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, 1/3 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. 6. Only Virtual Scientific Calculator is allowed. Charts, graph sheets, tables, cellular phone or other electronic gadgets are NOT allowed in the examination hall. 7. The Scribble Pad will be provided for rough work. Useful information , , sup inf MA set of all natural numbers { , , , } set of all integers { , , , } set of all rational numbers set of all real numbers set of all complex numbers | , } -dimensional Euclidean space { , , , group of all permutations of distinct symbols group of congruence classes of integers modulo unit vectors having the directions of the positive , and axes of a three dimensional rectangular coordinate system + + real vector space of all matrices of order supremum infimum with entries in 1/12 JAM 2018 Mathematics - MA SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. 1 Q.10 carry one mark each. Q.1 Which one of the following is TRUE? (A) is cyclic if and only if is prime (B) Every proper subgroup of (C) Every proper subgroup of is cyclic is cyclic (D) If every proper subgroup of a group is cyclic, then the group is cyclic Q.2 = Let (A) Q.3 If { , , , where = , (B) = = and + , (C) . Then lim (D) is } is a linearly independent set of vectors in a vector space over , then which one of the following sets is also linearly independent? (A) { + (C) { + (B) { (D) { Q.4 Let [ , + , , + , then (C) , + + , (C) + } + + + = , , + + } } , + + } is a continuous and even function defined on the interval is equal to (B) (D) The tangent plane to the surface (A) MA , , be a positive real number. If (A) Q.5 = = + at (B) (D) , , + is given by + = = 2/12 JAM 2018 Q.6 Mathematics - MA (A) Q.7 + = at the point (B) , , is (C) + = (D) Let : be a scalar field, : be a vector field and let If represents the position vector (A) (B) (C) (D) Q.8 + In , the cosine of the acute angle between the surfaces | | = = = | | = + + + + , then which one of the following is FALSE? + , for / given by (C) Q.9 / / / + / = = / defined on . Let (A) Q.10 Let = + sequence { } (A) { } (B) { } ! + ! MA / = / = + / = / is / / over of polynomial functions of degree less than or equal to be defined by (C) for ! = . Then the rank of is (D) . Then which one of the following is TRUE for the converges in is a Cauchy sequence but does not converge in (C) the subsequence { (D) { } / (D) (B) + + / (B) / Consider the vector space be a constant vector. In , the family of trajectories orthogonal to the family of asteroids (A) and } is convergent in , only when is even natural number is not a Cauchy sequence 3/12 JAM 2018 Mathematics - MA Q. 11 Q. 30 carry two marks each. Q.11 + = Let , + . , , Then which one of the following is TRUE? (A) sup { (B) lim inf (C) sup { (D) lim inf } = | and inf { | } = } = and inf { | } = = lim sup | = = and lim sup = . Which of the following values of , , do NOT result in the convergence of the Q.12 Let , , series ? log (A) (C) Q.13 Let | | < , = , , < = and let (A) Both (C) (D) (C) = and , where = , = , is convergent but nor > , , > ! is (D) + { }. Then which one of the are convergent is convergent but (D) Neither MA (B) (B) following is TRUE? (B) . Then the sum of the series = + , (A) Q.14 Let , is not convergent is not convergent is convergent 4/12 JAM 2018 Mathematics - MA Q.15 Suppose that , are differentiable functions such that = strictly decreasing. Define (A) positive Q.16 sign of (A) lim = = (B) lim (C) , Let sin , , (C) dependent on = has infinitely many maxima and minima on the interval = = , , , , (A) For (B) For (C) For (D) For = (A) + = , + (C) = , is continuous and differentiable = , is continuous and differentiable = , is neither continuous nor differentiable = = Q.19 Consider the region (A) MA . If the region = : be a thrice differentiable function. If (D) dependent on , = , then which one of the following is TRUE? (B) + (D) in the and , ? at the point is continuous but not differentiable and . Then, for > , the , = , and let = but not differentiable at is . Then which one of the following is FALSE? Then which one of the following is TRUE for Q.18 Let , , = = is continuous at (D) Q.17 is (B) negative , let For and is strictly increasing and plane bounded by the line = , where = = and the curve + = , where is revolved about the -axis in , then the volume of the resulting solid is (B) (C) (D) 5/12 JAM 2018 Mathematics - MA Q.20 If , = + for , , then = , boundary of the triangular region bounded by the lines anti-clockwise direction, is (C) (B) (A) , and Q.21 Let = rank rank (A) nullity of dimension of = , dimension of = (B) (C) (D) < . when when Then < and < and < when < when = (B) and and log Q.25 Let be a group satisfying the property that (A) MA and : = range and be . Then a possible group (B) = , for is identically zero , = , where + hen (C) (B) = , : when when when + Q.24 A particular integral of the differential equation + (A) , is one-one, then + Q.23 An integrating factor of the differential equation (A) oriented in the is not identically zero and be the solution of the differential equation (A) , nullspace = , then = , dimension of (D) If dimension of , , = = nullity of (C) If dimension of = + (D) , then which one of the following is TRUE? (B) dimension of Q.22 Let 4 = nullspace , where is the = and be finite dimensional real vector spaces, : linear transformations. If range (C) is : (C) + + + (D) = log = is is (D) is a homomorphism implies (D) 6/12 JAM 2018 Mathematics - MA Q.26 Let I. be the quotient group / . Consider the following statements. Every cyclic subgroup of II. is finite. Every finite cyclic group is isomorphic to a subgroup of . Which one of the following holds? (A) I is TRUE but II is FALSE (B) II is TRUE but I is FALSE (C) both I and II are TRUE (D) neither I nor II is TRUE Q.27 Let denote the are (A) + identity matrix. If the roots of the characteristic polynomial of a , then = + (B) Q.28 Consider the group = { , following is a subgroup of ? (A) (B) (C) (D) { , | { , | divides { , { , = } | + | divides + (C) | , = } (D) matrix + } under component-wise addition. Then which of the } and divides } Q.29 Let : be a function and let be a bounded open interval in . Define , = sup { Which one of the following is FALSE? (A) (B) If , tends to as , then lim (C) If is discontinuous at a point (D) If is continuous at a point such that , | } . if is a bounded function in and interval MA , } inf { | < , then , , = , then for any given such that the length of the > there exists an interval 7/12 JAM 2018 Mathematics - MA Q.30 For , let > following is TRUE? = < (A) < (B) + (C) < (D) = log , < for < for < for < for < > > > < + = and . Then which one of the SECTION - B MULTIPLE SELECT QUESTIONS (MSQ) Q. 31 Q. 40 carry two marks each. Q.31 Let : \ { } be defined by one-one? (A) , (B) = + , (C) Q.32 The solution(s) of the differential equation = (A) = (C) . On which of the following interval(s) is sin , (D) = sin / (B) = = (D) , = is (are) satisfying sin cos Q.33 Suppose , , are permutations of the set { , , , } , where interchanges and but fixes and , g interchanges and but fixes and , interchanges and but fixes and . Which of the following permutations interchange(s) (A) Q.34 Let and (A) If (B) If (C) If (D) If MA (B) (C) and but fix(es) and ? (D) be two non-empty disjoint subsets of . Which of the following is (are) FALSE? and are compact, then and are not connected, then and are closed, then and are open, then is also compact is also not connected is closed is open 8/12 JAM 2018 Mathematics - MA Q.35 Let = \ { } denote the group of non-zero complex numbers under multiplication. Suppose = { | (A) Q.36 Suppose , , + + = }, (B) . Which of the following is (are) subgroup(s) of ? (C) (D) . Consider the following system of linear equations. = , + + = , + + = . If this system has at least one solution, then which of the following statements is (are) TRUE? = (A) If (C) If , (A) NOT possible? (B) < , (D) , , = = = = + = If (D) If = (B) = + + + + = along any simple closed curve = Q.39 Which of the following subsets of is (are) connected? (A) { (C) { MA | = then . Then which of the following is (are) for (C) There exists a scalar function : such that (D) = then , the smallest integer larger than or equal to the following is (are) TRUE? (A) , (B) = (C) Q.38 If = then , Q.37 Let = then + > } | | | < | |} (B) (D) { { , , , then which among = + | + < } + | | | > | |} 9/12 JAM 2018 Mathematics - MA be a subset of such that Q.40 Let is an interior point of . Which of the following is (are) TRUE? (A) contains an interval (B) There is a sequence in which does not converge to (C) There is an element , such that , such that | (D) There is a point is also an interior point of |= . SECTION C NUMERICAL ANSWER TYPE (NAT) Q. 41 Q. 50 carry one mark each. Q.41 The order of the element Q.42 Let of , , Q.44 Let Then = , , , = = for < = , at the point sin = + , is _________________ . Then the absolute value of the directional derivative , , Then + , at the point , , , , + , , is _____________ cos = ________________ , then is _______________ < . Then the value of for = , , , : [ , [ , be continuous on [ , and differentiable on = MA for : be given by Let Q.46 Let + in the direction of the line Q.43 Let Q.45 = in the group = ___________ is _____________ , > , > . , . If 10/12 JAM 2018 Mathematics - MA Q.47 Let about Q.48 Let order Q.49 Let = = . Then the radius of convergence of the power series + is __________ be the group of even permutations of in is __________________ be the real vector space of all zero. Let distinct symbols. Then the number of elements of be the real vector space of all matrices such that the sum of the entries in each row is matrices such that the sum of the entries in each is ______________ column is zero. Then the dimension of the space Q.50 The coefficient of in the power series expansion of about ____________ (correct up to three decimal places). = is Q. 51 Q. 60 carry two marks each. Q.51 Let = Then lim , = + + and is continuous on and + + + = , = / , where , and is ________________ (correct up to three decimal places). Q.53 Suppose , , are positive real numbers such that , then the value of MA = is _______________ (correct up to one decimal place). Q.52 Let : be such that Then lim + is ________________ + + = . If . = . is the maximum value of 11/12 JAM 2018 Mathematics - MA Q.54 If the volume of the solid in bounded by the surfaces Q.55 is , then If = / / = , = , = , = ________________ , then the value of = , + = , sin + = is ______________ Q.56 The value of the integral is ______________ (correct up to three decimal places). Q.57 Suppose is a matrix of rank . Let : transformation defined by = . Then the rank of be the linear is _______________ Q.58 The area of the parametrized surface = { + cos cos , + cos | sin , sin is ____________ (correct up to two decimal places). Q.59 If is the solution to the differential equation then the value of Q.60 If = = and = , + } , for > , satisfying is _____________ (correct up to two decimal places). sec is the solution of = , then tan + = , < = , < , satisfying is _______________ (correct up to two decimal places). END OF THE QUESTION PAPER MA 12/12 Paper Code : MA Q No Question Type (QT) Section Key/Range (KY) 1 MCQ A B 2 MCQ A D 3 MCQ A D 4 MCQ A A 5 MCQ A B 6 MCQ A C 7 MCQ A C 8 MCQ A B 9 MCQ A C 10 MCQ A B 11 MCQ A A 12 MCQ A C 13 MCQ A D 14 MCQ A B 15 MCQ A A 16 MCQ A D 17 MCQ A C 18 MCQ A A 19 MCQ A C 20 MCQ A B 21 MCQ A C 22 MCQ A A 23 MCQ A C Paper Code : MA Q No Question Type (QT) Section Key/Range (KY) 24 MCQ A B 25 MCQ A A 26 MCQ A C 27 MCQ A C 28 MCQ A D 29 MCQ A B 30 MCQ A Marks To All 31 MSQ B B 32 MSQ B A;B;C 33 MSQ B A;D 34 MSQ B B;C;D 35 MSQ B B;C;D 36 MSQ B A;B 37 MSQ B A;D 38 MSQ B A;B;C 39 MSQ B B;C;D 40 MSQ B A;B;C 41 NAT C 4 to 4 42 NAT C 6.5 to 7.5 43 NAT C 1 to 1 44 NAT C 1 to 1 45 NAT C 3 to 3 46 NAT C 9 to 9 Paper Code : MA Q No Question Type (QT) Section Key/Range (KY) 47 NAT C 2 to 2 48 NAT C 0 to 0 49 NAT C 4 to 4 50 NAT C -0.130 to -0.120 51 NAT C 0.4 to 0.6 52 NAT C 0.350 to 0.380 53 NAT C 1140 to 1160 54 NAT C 5.99 to 6.01 55 NAT C 2.9 to 3.1 56 NAT C 0.230 to 0.250 57 NAT C 6 to 6 58 NAT C 6.30 to 6.70 59 NAT C -2.80 to -2.70 60 NAT C 0.5 to 0.5

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