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IIT JAM 2020 : Mathematics

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JAM 2020 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 this 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. MA 1 / 17 JAM 2020 MATHEMATICS - MA NOTATION 1. N = {1, 2, 3, } 2. R - the set of all real numbers 3. R \ {0} - the set of all non-zero real numbers 4. C - the set of all complex numbers 5. f g - composition of the functions f and g 6. f 0 and f 00 - first and second derivatives of the function f , respectively 7. f (n) - nth derivative of f + j y + k z 8. = i x H 9. - the line integral over an oriented closed curve C C 10. i, j, k - unit vectors along the Cartisean right handed rectangular co-ordinate system 11. n - unit outward normal vector 12. I - identity matrix of appropriate order 13. det(M ) - determinant of the matrix M 14. M 1 - inverse of the matrix M 15. M T - transpose of the matrix M 16. id - identity map 17. hai - cyclic subgroup generated by an element a of a group 18. Sn - permutation group on n symbols 19. S 1 = {z C : |z| = 1} 20. o(g) - order of the element g in a group MA 2 / 17 JAM 2020 MATHEMATICS - MA SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. 1 Q. 10 carry one mark each. ( 1)n , n N. Then the sequence {sn } is n (A) monotonically increasing and is convergent to 1 Q. 1 Let sn = 1 + (B) monotonically decreasing and is convergent to 1 (C) neither monotonically increasing nor monotonically decreasing but is convergent to 1 (D) divergent Q. 2 Let f (x) = 2x3 9x2 + 7. Which of the following is true? (A) f is one-one in the interval [ 1, 1] (B) f is one-one in the interval [2, 4] (C) f is NOT one-one in the interval [ 4, 0] (D) f is NOT one-one in the interval [0, 4] Q. 3 Which of the following is FALSE? x (A) lim x = 0 x e sin x =0 (C) lim+ x 0 1 + 2x 1 =0 x 0 xe1/x cos x (D) lim+ =0 x 0 1 + 2x (B) lim+ Q. 4 Let g : R R be a twice differentiable function. If f (x, y) = g(y) + xg 0 (y), then f 2f f +y = x x y y f 2f f (C) +x = x x y y (A) MA f 2f f +y = y x y x f 2f f (D) +x = y x y x (B) 3 / 17 JAM 2020 MATHEMATICS - MA Q. 5 If the equation of the tangent plane to the surface z = 16 x2 y 2 at the point P (1, 3, 6) is ax + by + cz + d = 0, then the value of |d| is (A) 16 (B) 26 (C) 36 (D) 46 Q. 6 If the directional derivative of the function z = y 2 e2x at (2, 1) along the unit vector b = i + j is zero, then | + | equals 1 1 (D) 2 2 (B) (C) 2 (A) 2 2 2 Q. 7 If u = x3 and v = y 2 transform the differential equation 3x5 dx y(y 2 x3 )dy = 0 to dv u = , then is du 2(u v) (A) 4 (C) 2 (B) 2 (D) 4 Q. 8 Let T : R2 R2 be the linear transformation given by T (x, y) = ( x, y). Then (A) T 2k = T for all k 1 (B) T 2k+1 = T for all k 1 (C) the range of T 2 is a proper subspace of the range of T (D) the range of T 2 is equal to the range of T Q. 9 The radius of convergence of the power series X n + 2 n2 n=1 n xn is (A) e2 MA 1 (B) e (C) 1 e (D) 1 e2 4 / 17 JAM 2020 MATHEMATICS - MA Q. 10 Consider the following group under matrix multiplication: 1 p q H = 0 1 r : p, q, r R . 0 0 1 Then the center of the group is isomorphic to (A) (R \ {0}, ) (B) (R, +) (C) (R2 , +) (D) (R, +) (R \ {0}, ) Q. 11 Q. 30 carry two marks each. an+1 . Which of the n an Q. 11 Let {an } be a sequence of positive real numbers. Suppose that l = lim following is true? (A) If l = 1, then lim an = 1 (B) If l = 1, then lim an = 0 (C) If l < 1, then lim an = 1 (D) If l < 1, then lim an = 0 n n n n r 1 + s2n , n 1. Which of the following is true? 1+ 1 1 , then {sn } is monotonically increasing and lim sn = n 1 1 , then {sn } is monotonically decreasing and lim sn = n 1 1 , then {sn } is monotonically increasing and lim sn = n 1 1 , then {sn } is monotonically decreasing and lim sn = n Q. 12 Define s1 = > 0 and sn+1 = (A) If s2n < (B) If s2n < (C) If s2n > (D) If s2n > MA 5 / 17 JAM 2020 MATHEMATICS - MA Q. 13 Suppose that S is the sum of a convergent series the series P P an . Define tn = an + an+1 + an+2 . Then n=1 tn n=1 (A) diverges (B) converges to 3S a1 a2 (C) converges to 3S a1 2a2 (D) converges to 3S 2a1 a2 Q. 14 Let a R. If f (x) = (x + a)2 , x 0 (x + a)3 , x > 0, then d2 f dx2 d2 f (B) 2 dx d2 f (C) 2 dx d2 f (D) 2 dx (A) does not exist at x = 0 for any value of a exists at x = 0 for exactly one value of a exists at x = 0 for exactly two values of a exists at x = 0 for infinitely many values of a Q. 15 Let f (x, y) = x2 sin x1 + y 2 sin y1 , x2 sin 1 , xy 6= 0 y 2 sin 0, y 6= 0, x = 0 x 1 , y x 6= 0, y = 0 x = y = 0. Which of the following is true at (0, 0)? (A) f is not continuous f f is continuous but is not continuous (B) x y (C) f is not differentiable f f (D) f is differentiable but both and are not continuous x y MA 6 / 17 JAM 2020 MATHEMATICS - MA Q. 16 Let S be the surface of the portion of the sphere with centre at the origin and radius 4, above If n the xy-plane. Let F = y i x j + yx3 k. is the unit outward normal to S, then ZZ ( F ) n dS S equals (A) 32 (B) 16 (C) 16 (D) 32 Q. 17 Let f (x, y, z) = x3 + y 3 + z 3 3xyz. A point at which the gradient of the function f is equal to zero is (A) ( 1, 1, 1) (B) ( 1, 1, 1) (C) ( 1, 1, 1) (D) (1, 1, 1) Q. 18 The area bounded by the curves x2 + y 2 = 2x and x2 + y 2 = 4x, and the straight lines y = x and y = 0 is 1 1 1 1 (A) 3 + + + + (B) 3 (C) 2 (D) 2 2 4 4 2 4 3 3 4 Q. 19 Let M be a real 6 6 matrix. Let 2 and 1 be two eigenvalues of M . If M 5 = aI + bM , where a, b R, then (A) a = 10, b = 11 (B) a = 11, b = 10 (C) a = 10, b = 11 (D) a = 10, b = 11 Q. 20 Let M be an n n (n 2) non-zero real matrix with M 2 = 0 and let R \ {0}. Then (A) is the only eigenvalue of (M + I) and (M I) (B) is the only eigenvalue of (M + I) and ( I M ) (C) is the only eigenvalue of (M + I) and (M I) (D) is the only eigenvalue of (M + I) and ( I M ) MA 7 / 17 JAM 2020 MATHEMATICS - MA Q. 21 Consider the differential equation L[y] = (y y 2 )dx + xdy = 0. The function f (x, y) is said to be an integrating factor of the equation if f (x, y)L[y] = 0 becomes exact. 1 If f (x, y) = 2 2 , then xy (A) f is an integrating factor and y = 1 kxy, k R is NOT its general solution (B) f is an integrating factor and y = 1 + kxy, k R is its general solution (C) f is an integrating factor and y = 1 + kxy, k R is NOT its general solution (D) f is NOT an integrating factor and y = 1 + kxy, k R is its general solution Q. 22 A solution of the differential equation 2x2 point (1, 1) is 1 (A) y = x (B) y = 1 x2 d2 y dy + 3x y = 0, x > 0 that passes through the 2 dx dx 1 (C) y = x (D) y = 1 x3/2 Q. 23 Let M be a 4 3 real matrix and let {e1 , e2 , e3 } be the standard basis of R3 . Which of the following is true? (A) If rank(M ) = 1, then {M e1 , M e2 } is a linearly independent set (B) If rank(M ) = 2, then {M e1 , M e2 } is a linearly independent set (C) If rank(M ) = 2, then {M e1 , M e3 } is a linearly independent set (D) If rank(M ) = 3, then {M e1 , M e3 } is a linearly independent set Q. 24 The value of the triple integral RRR (x2 y + 1) dxdydz, where V is the region given by x2 + y 2 V 1, 0 z 2 is (A) (B) 2 (C) 3 (D) 4 Q. 25 Let S be the part of the cone z 2 = x2 + y 2 between the planes z = 0 and z = 1. Then the value RR of the surface integral (x2 + y 2 ) dS is S (A) MA (B) 2 (C) 3 (D) 2 8 / 17 JAM 2020 MATHEMATICS - MA x, y, z R. Which of the following is FALSE? Q. 26 Let a = i + j + k and r = x i + y j + z k, (A) ( a r ) = a (B) ( a r ) = 0 (C) ( a r ) = a (D) (( a r ) r ) = 4( a r ) Q. 27 Let D = {(x, y) R2 : |x| + |y| 1} and f : D R be a non-constant continuous function. Which of the following is TRUE? (A) The range of f is unbounded (B) The range of f is a union of open intervals (C) The range of f is a closed interval (D) The range of f is a union of at least two disjoint closed intervals 1 1 = and Q. 28 Let f : [0, 1] R be a continuous function such that f 2 2 |f (x) f (y) (x y)| sin (|x y|2 ) for all x, y [0, 1]. Then R1 f (x) dx is 0 (A) 1 2 (B) 1 4 (C) 1 4 (D) 1 2 Q. 29 Let S 1 = {z C : |z| = 1} be the circle group under multiplication and i = set { R : hei2 i is infinite} is MA (A) empty (B) non-empty and finite (C) countably infinite (D) uncountable 1. Then the 9 / 17 JAM 2020 MATHEMATICS - MA Q. 30 Let F = { C : 2020 = 1}. Consider the groups ( ) z G= : F, z C 0 1 and ( ) 1 z H= :z C 0 1 under matrix multiplication. Then the number of cosets of H in G is (A) 1010 MA (B) 2019 (C) 2020 (D) infinite 10 / 17 JAM 2020 MATHEMATICS - MA SECTION B MULTIPLE SELECT QUESTIONS (MSQ) Q. 31 Q. 40 carry two marks each. Q. 31 Let a, b, c R such that a < b < c. Which of the following is/are true for any continuous function f : R R satisfying f (a) = b, f (b) = c and f (c) = a? (A) There exists (a, c) such that f ( ) = (B) There exists (a, b) such that f ( ) = (C) There exists (a, b) such that (f f )( ) = (D) There exists (a, c) such that (f f f )( ) = ( 1)n ( 1)n and t = , n = 0, 1, 2, ..., then n 2n + 3 4n 1 P P (A) sn is absolutely convergent (B) tn is absolutely convergent Q. 32 If sn = (C) n=0 P n=0 sn is conditionally convergent (D) n=0 P tn is conditionally convergent n=0 Q. 33 Let a, b R and a < b. Which of the following statement(s) is/are true? (A) There exists a continuous function f : [a, b] (a, b) such that f is one-one (B) There exists a continuous function f : [a, b] (a, b) such that f is onto (C) There exists a continuous function f : (a, b) [a, b] such that f is one-one (D) There exists a continuous function f : (a, b) [a, b] such that f is onto MA 11 / 17 JAM 2020 MATHEMATICS - MA Q. 34 Let V be a non-zero vector space over a field F . Let S V be a non-empty set. Consider the following properties of S: (I) For any vector space W over F , any map f : S W extends to a linear map from V to W. (II) For any vector space W over F and any two linear maps f, g : V W satisfying f (s) = g(s) for all s S, we have f (v) = g(v) for all v V . (III) S is linearly independent. (IV) The span of S is V. Which of the following statement(s) is /are true? (A) (I) implies (IV) (B) (I) implies (III) (C) (II) implies (III) (D) (II) implies (IV) dy d2 y + px + qy, where p, q are real constants. Let y1 (x) and y2 (x) be two 2 dx dx solutions of L[y] = 0, x > 0, that satisfy y1 (x0 ) = 1, y10 (x0 ) = 0, y2 (x0 ) = 0 and y20 (x0 ) = 1 for some x0 > 0. Then, Q. 35 Let L[y] = x2 (A) y1 (x) is not a constant multiple of y2 (x) (B) y1 (x) is a constant multiple of y2 (x) (C) 1, ln x are solutions of L[y] = 0 when p = 1, q = 0 (D) x, ln x are solutions of L[y] = 0 when p + q 6= 0 Q. 36 Consider the following system of linear equations x + y + 5z = 3, x + 2y + mz = 5 and x + 2y + 4z = k. The system is consistent if (A) m 6= 4 MA (B) k 6= 5 (C) m = 4 (D) k = 5 12 / 17 JAM 2020 MATHEMATICS - MA 1 1 1 1 2 (n 1) and b = lim + + + . + + + n n2 n n + 1 n2 n2 n+2 n+n Which of the following is/are true? a (A) a > b (B) a < b (C) ab = ln 2 (D) = ln 2 b Q. 37 Let a = lim Q. 38 Let S be that part of the surface of the paraboloid z = 16 x2 y 2 which is above the plane z = 0 and D be its projection on the xy-plane. Then the area of S equals (A) RR p 1 + 4(x2 + y 2 ) dxdy (B) D (C) R2 R4 RR p 1 + 2(x2 + y 2 ) dxdy D 1 + 4r2 drd (D) 0 0 R2 R4 1 + 4r2 rdrd 0 0 Q. 39 Let f be a real valued function of a real variable, such that |f (n) (0)| K for all n N, where K > 0. Which of the following is/are true? (n) n1 f (0) 0 as n (A) n! (n) n1 f (0) as n (B) n! (C) f (n) (x) exists for all x R and for all n N f (n) (0) P (D) The series is absolutely convergent n=1 (n 1)! Q. 40 Let G be a group with identity e. Let H be an abelian non-trivial proper subgroup of G with the property that H gHg 1 = {e} for all g / H. If K = g G : gh = hg for all h H , then (A) K is a proper subgroup of H (B) H is a proper subgroup of K (C) K = H (D) there exists no abelian subgroup L G such that K is a proper subgroup of L MA 13 / 17 JAM 2020 MATHEMATICS - MA SECTION C NUMERICAL ANSWER TYPE (NAT) Q. 41 Q. 50 carry one mark each. 1 Q. 41 Let xn = n n and yn = e1 xn , n N. Then the value of lim yn is n . Q. 42 Let F = x i + y j + z k and S be the sphere given by (x 2)2 + (y 2)2 + (z 2)2 = 4. If n is the unit outward normal to S, then ZZ 1 F n dS S . is Q. 43 Let f : R R be such that f, f 0 , f 00 are continuous functions with f > 0, f 0 > 0 and f 00 > 0. Then f (x) + f 0 (x) lim x 2 is . Q. 44 Let S = is 1 1 : n N and f : S R be defined by f (x) = . Then n x 1 1 <1 max : x < = f (x) f 3 3 . (rounded off to two decimal places) df Q. 45 Let f (x, y) = ex sin y, x = t3 + 1 and y = t4 + t. Then at t = 0 is dt to two decimal places) MA . (rounded off 14 / 17 JAM 2020 MATHEMATICS - MA Q. 46 Consider the differential equation dy + 10y = f (x), x > 0, dx where f (x) is a continuous function such that lim f (x) = 1. Then the value of x lim y(x) x is Q. 47 If . R1 R2 2 ex dxdy = k(e4 1), then k equals . 0 2y Q. 48 Let f (x, y) = 0 be a solution of the homogeneous differential equation (2x + 5y)dx (x + 3y)dy = 0. If f (x + , y 3) = 0 is a solution of the differential equation (2x + 5y 1)dx + (2 x 3y)dy = 0, then the value of is . Q. 49 Consider the real vector space P2020 = { n P ai xi : ai R and 0 n 2020}. Let W be the i=0 subspace given by W = ( n X ) ai xi P2020 : ai = 0 for all odd i . i=0 Then, the dimension of W is . Q. 50 Let : S3 S 1 be a non-trivial non-injective group homomorphism. Then, the number of elements in the kernel of is . MA 15 / 17 JAM 2020 MATHEMATICS - MA Q. 51 Q. 60 carry two marks each. Q. 51 The sum of the series 1 1 1 + + + is 2 2 1) 3(3 1) 4(4 1) 2(22 Q. 52 Consider the expansion of the function f (x) = 1 |x| < . Then the coefficient of x4 is 2 . 3 in powers of x, that is valid in (1 x)(1 + 2x) . Q. 53 The minimum value of the function f (x, y) = x2 + xy + y 2 3x 6y + 11 is Q. 54 Let f (x) = . x + x, x > 0 and g(x) = a0 + a1 (x 1) + a2 (x 1)2 be the sum of the first three terms of the Taylor series of f (x) around x = 1. If g(3) = 3, then . is Q. 55 Let C be the boundary of the square with vertices (0, 0), (1, 0), (1, 1) and (0, 1) oriented in the counter clockwise sense. Then, the value of the line integral I x2 y 2 dx + (x2 y 2 )dy C is . (rounded off to two decimal places) Q. 56 Let f : R R be a differentiable function with f 0 (x) = f (x) for all x. Suppose that f ( x) and f ( x) are two non-zero solutions of the differential equation 4 d2 y dy p + 3y = 0 dx2 dx satisfying f ( x)f ( x) = f (2x) and f ( x)f ( x) = f (x). Then, the value of p is MA . 16 / 17 JAM 2020 MATHEMATICS - MA Q. 57 If x2 + xy 2 = c, where c R, is the general solution of the exact differential equation M (x, y) dx + 2xy dy = 0, then M (1, 1) is 9 0 Q. 58 Let M = 0 0 . 2 7 1 7 2 1 3 . Then, the value of det (8I M ) is 0 11 6 0 5 0 . Q. 59 Let T : R7 R7 be a linear transformation with Nullity(T ) = 2. Then, the minimum possible value for Rank(T 2 ) is . Q. 60 Suppose that G is a group of order 57 which is NOT cyclic. If G contains a unique subgroup H of order 19, then for any g / H, o(g) is . END OF THE QUESTION PAPER MA 17 / 17

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