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IIT JAM 2021 : Mathematics (with Answer Keys)

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JAM 2021 MATHEMATICS - MA N = {1, 2, . . .}. Z = {. . . , 2, 1, 0, 1, 2, . . .}. Q = the set of rational numbers. R = the set of real numbers. Rn = the n-dimensional real space with the Euclidean topology. C = the set of complex numbers. Cn = the n-dimensional complex space with the Euclidean topology. Mn (R), Mn (C) = the vector space of n n real or complex matrices, respectively. f , f = the first and second derivatives of the function f , respectively. f (n) = the nth. derivative of the function f . R C stands for the line integral over the curve C. In = the n n identity matrix. A 1 = the inverse of an invertible matrix A. Sn = the permutation group on n symbols. i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1). ln x = the natural logarithm of x (to the base e). |X| = the number of elements in a finite set X. Zn = the additive group of integers modulo n. arctan(x) denotes the unique ( /2, /2) such that tan = x. All vector spaces are over the real or complex field, unless otherwise stated. MA 1/1 JAM 2021 MATHEMATICS - MA SECTION A MULTIPLE CHOICE QUESTIONS (MCQ) Q. 1 Q. 10 carry one mark each. Q. 1 Let 0 < < 1 be a real number. The number of differentiable functions y : [0, 1] [0, ), having continuous derivative on [0, 1] and satisfying y (t) = (y(t)) , t [0, 1], y(0) = 0, is (A) exactly one. (B) exactly two. (C) finite but more than two. (D) infinite. Q. 2 Let P : R R be a continuous function such that P (x) > 0 for all x R. Let y be a twice differentiable function on R satisfying y (x) + P (x)y (x) y(x) = 0 for all x R. Suppose that there exist two real numbers a, b (a < b) such that y(a) = y(b) = 0. Then (A) y(x) = 0 for all x [a, b]. (B) y(x) > 0 for all x (a, b). (C) y(x) < 0 for all x (a, b). (D) y(x) changes sign on (a, b). Q. 3 Let f : R R be a continuous function satisfying f (x) = f (x + 1) for all x R. Then (A) f is not necessarily bounded above. (B) there exists a unique x0 R such that f (x0 + ) = f (x0 ). (C) there is no x0 R such that f (x0 + ) = f (x0 ). (D) there exist infinitely many x0 R such that f (x0 + ) = f (x0 ). MA 1 / 17 JAM 2021 MATHEMATICS - MA Q. 4 Let f : R R be a continuous function such that for all x R, Z 1 f (xt) dt = 0. ( ) 0 Then (A) f must be identically 0 on the whole of R. (B) there is an f satisfying ( ) that is identically 0 on (0, 1) but not identically 0 on the whole of R. (C) there is an f satisfying ( ) that takes both positive and negative values. (D) there is an f satisfying ( ) that is 0 at infinitely many points, but is not identically zero. Q. 5 Let p and t be positive real numbers. Let Dt be the closed disc of radius t centered at (0, 0), i.e., Dt = {(x, y) R2 : x2 + y 2 t2 }. Define ZZ dxdy I(p, t) = . 2 2 2 p Dt (p + x + y ) Then limt I(p, t) is finite (A) only if p > 1. (B) only if p = 1. (C) only if p < 1. (D) for no value of p. Q. 6 How many elements of the group Z50 have order 10? (A) 10 MA (B) 4 (C) 5 (D) 8 2 / 17 JAM 2021 MATHEMATICS - MA Q. 7 For every n N, let fn : R R be a function. From the given choices, pick the statement that is the negation of For every x R and for every real number > 0, there exists an integer N > 0 such that Pp i=1 |fN +i (x)| < for every integer p > 0. (A) For every x R and for every real number > 0, there does not exist any integer N > 0 P such that pi=1 |fN +i (x)| < for every integer p > 0. (B) For every x R and for every real number > 0, there exists an integer N > 0 such that Pp i=1 |fN +i (x)| for some integer p > 0. (C) There exists x R and there exists a real number > 0 such that for every integer N > 0, P there exists an integer p > 0 for which the inequality pi=1 |fN +i (x)| holds. (D) There exists x R and there exists a real number > 0 such that for every integer N > 0 P and for every integer p > 0 the inequality pi=1 |fN +i (x)| holds. Q. 8 Which one of the following subsets of R has a non-empty interior? (A) The set of all irrational numbers in R. (B) The set {a R : sin(a) = 1}. (C) The set {b R : x2 + bx + 1 = 0 has distinct roots}. (D) The set of all rational numbers in R. Q. 9 For an integer k 0, let Pk denote the vector space of all real polynomials in one variable of degree less than or equal to k. Define a linear transformation T : P2 P3 by T f (x) = f (x) + xf (x). Which one of the following polynomials is not in the range of T ? (A) x + x2 MA (B) x2 + x3 + 2 (C) x + x3 + 2 (D) x + 1 3 / 17 JAM 2021 MATHEMATICS - MA Q. 10 Let n > 1 be an integer. Consider the following two statements for an arbitrary n n matrix A with complex entries. I. If Ak = In for some integer k 1, then all the eigenvalues of A are k th roots of unity. II. If, for some integer k 1, all the eigenvalues of A are k th roots of unity, then Ak = In . Then MA (A) both I and II are TRUE. (B) I is TRUE but II is FALSE. (C) I is FALSE but II is TRUE. (D) neither I nor II is TRUE. 4 / 17 JAM 2021 MATHEMATICS - MA Q. 11 Q. 30 carry two marks each. Q. 11 Let Mn (R) be the real vector space of all n n matrices with real entries, n 2. Let A Mn (R). Consider the subspace W of Mn (R) spanned by {In , A, A2 , . . .}. Then the dimension of W over R is necessarily (B) n2 . (A) . (C) n. (D) at most n. Q. 12 Let y be the solution of 1 y(x) = 0, x ( 1, ), 1+x y(0) = 1, y (0) = 0. (1 + x)y (x) + y (x) Then (A) y is bounded on (0, ). (B) y is bounded on ( 1, 0]. (C) y(x) 2 on ( 1, ). (D) y attains its minimum at x = 0. If n Q. 13 Consider the surface S = {(x, y, xy) R3 : x2 + y 2 1}. Let F~ = y i + x j + k. is the continuous unit normal field to the surface S with positive z-component, then ZZ F~ n dS S equals (A) 4 . (B) 2 . (C) . (D) 2 . Q. 14 Consider the following statements. I. The group (Q, +) has no proper subgroup of finite index. II. The group (C \ {0}, ) has no proper subgroup of finite index. Which one of the following statements is true? MA (A) Both I and II are TRUE. (B) I is TRUE but II is FALSE. (C) II is TRUE but I is FALSE. (D) Neither I nor II is TRUE. 5 / 17 JAM 2021 MATHEMATICS - MA Q. 15 Let f : N N be a bijective map such that X f (n) n=1 n2 < + . The number of such bijective maps is (A) exactly one. (B) zero. (C) finite but more than one. (D) infinite. Q. 16 Define Then (A) S = 1/2. 1 1 1 S = lim 1 2 1 2 1 2 . n 2 3 n (B) S = 1/4. (C) S = 1. (D) S = 3/4. Q. 17 Let f : R R be an infinitely differentiable function such that for all a, b R with a < b, Then f (b) f (a) a+b . =f b a 2 (A) f must be a polynomial of degree less than or equal to 2. (B) f must be a polynomial of degree greater than 2. (C) f is not a polynomial. (D) f must be a linear polynomial. MA 6 / 17 JAM 2021 MATHEMATICS - MA Q. 18 Consider the function ( f (x) = 1 1 1 p if x (R \ Q) {0}, if x = np , n Z \ {0}, p N and gcd(n, p) = 1. Then (A) all x Q \ {0} are strict local minima for f . (B) f is continuous at all x Q. (C) f is not continuous at all x R \ Q. (D) f is not continuous at x = 0. Q. 19 Consider the family of curves x2 y 2 = ky with parameter k R. The equation of the orthogonal trajectory to this family passing through (1, 1) is given by (A) x3 + 3xy 2 = 4. (B) x2 + 2xy = 3. (C) y 2 + 2x2 y = 3. (D) x3 + 2xy 2 = 3. Q. 20 Which one of the following statements is true? (A) Exactly half of the elements in any even order subgroup of S5 must be even permutations. (B) Any abelian subgroup of S5 is trivial. (C) There exists a cyclic subgroup of S5 of order 6. (D) There exists a normal subgroup of S5 of index 7. Q. 21 Let f : [0, 1] [0, ) be a continuous function such that Z t 2 f (s) ds, for all t [0, 1]. f (t) < 1 + 2 0 Then MA (A) f (t) < 1 + t for all t [0, 1]. (B) f (t) > 1 + t for all t [0, 1]. (C) f (t) = 1 + t for all t [0, 1]. (D) f (t) < 1 + t 2 for all t [0, 1]. 7 / 17 JAM 2021 MATHEMATICS - MA ! X11 X12 Q. 22 Let A be an n n invertible matrix and C be an n n nilpotent matrix. If X = X21 X22 is a 2n !2n matrix (each Xij being n n) that commutes with the 2n 2n matrix B = A 0 , then 0 C (A) X11 and X22 are necessarily zero matrices. (B) X12 and X21 are necessarily zero matrices. (C) X11 and X21 are necessarily zero matrices. (D) X12 and X22 are necessarily zero matrices. Q. 23 Let D R2 be defined by D = R2 \ {(x, 0) : x R}. Consider the function f : D R defined by 1 f (x, y) = x sin . y Then (A) f is a discontinuous function on D. (B) f is a continuous function on D and cannot be extended continuously to any point outside D. (C) f is a continuous function on D and can be extended continuously to D {(0, 0)}. (D) f is a continuous function on D and can be extended continuously to the whole of R2 . Q. 24 Which one of the following statements is true? (A) (Z, +) is isomorphic to (R, +). (B) (Z, +) is isomorphic to (Q, +). (C) (Q/Z, +) is isomorphic to (Q/2Z, +). (D) (Q/Z, +) is isomorphic to (Q, +). MA 8 / 17 JAM 2021 MATHEMATICS - MA Q. 25 Let y be a twice differentiable function on R satisfying y (x) = 2 + e |x| , x R, y(0) = 1, y (0) = 0. Then (A) y = 0 has exactly one root. (B) y = 0 has exactly two roots. (C) y = 0 has more than two roots. (D) there exists an x0 R such that y(x0 ) y(x) for all x R. Q. 26 Let f : [0, 1] [0, 1] be a non-constant continuous function such that f f = f . Define Ef = {x [0, 1] : f (x) = x}. Then (A) Ef is neither open nor closed. (B) Ef is an interval. (C) Ef is empty. (D) Ef need not be an interval. Q. 27 Let g be an element of S7 such that g commutes with the element (2, 6, 4, 3). The number of such g is (A) 6. (B) 4. (C) 24. (D) 48. Q. 28 Let G be a finite abelian group of odd order. Consider the following two statements: I. The map f : G G defined by f (g) = g 2 is a group isomorphism. Q II. The product g G g = e. MA (A) Both I and II are TRUE. (B) I is TRUE but II is FALSE. (C) II is TRUE but I is FALSE. (D) Neither I nor II is TRUE. 9 / 17 JAM 2021 MATHEMATICS - MA Q. 29 Let n 2 be an integer. Let A : Cn Cn be the linear transformation defined by A(z1 , z2 , . . . , zn ) = (zn , z1 , z2 , . . . , zn 1 ). Which one of the following statements is true for every n 2? (A) A is nilpotent. (B) All eigenvalues of A are of modulus 1. (C) Every eigenvalue of A is either 0 or 1. (D) A is singular. Q. 30 Consider the two series I. X n=1 1 n1+(1/n) and II. X n=1 1 n2 n1/n . Which one of the following holds? MA (A) Both I and II converge. (B) Both I and II diverge. (C) I converges and II diverges. (D) I diverges and II converges. 10 / 17 JAM 2021 MATHEMATICS - MA SECTION B MULTIPLE SELECT QUESTIONS (MSQ) Q. 31 Q. 40 carry two marks each. Q. 31 Let f : R R be a function with the property that for every y R, the value of the expression sup [xy f (x)] x R is finite. Define g(y) = supx R [xy f (x)] for y R. Then f (x) = + . |x| |x| (A) g is even if f is even. (B) f must satisfy lim (C) g is odd if f is even. (D) f must satisfy lim f (x) = . |x| |x| Q. 32 Consider the equation x2021 + x2020 + + x 1 = 0. Then (A) all real roots are positive. (B) exactly one real root is positive. (C) exactly one real root is negative. (D) no real root is positive. Q. 33 Let D = R2 \ {(0, 0)}. Consider the two functions u, v : D R defined by u(x, y) = x2 y 2 and v(x, y) = xy. Consider the gradients u and v of the functions u and v, respectively. Then (A) u and v are parallel at each point (x, y) of D. (B) u and v are perpendicular at each point (x, y) of D. (C) u and v do not exist at some points (x, y) of D. (D) u and v at each point (x, y) of D span R2 . MA 11 / 17 JAM 2021 MATHEMATICS - MA Q. 34 Consider the two functions f (x, y) = x + y and g(x, y) = xy 16 defined on R2 . Then (A) the function f has no global extreme value subject to the condition g = 0. (B) the function f attains global extreme values at (4, 4) and ( 4, 4) subject to the condition g = 0. (C) the function g has no global extreme value subject to the condition f = 0. (D) the function g has a global extreme value at (0, 0) subject to the condition f = 0. Q. 35 Let f : (a, b) R be a differentiable function on (a, b). Which of the following statements is/are true? (A) f > 0 in (a, b) implies that f is increasing in (a, b). (B) f is increasing in (a, b) implies that f > 0 in (a, b). (C) If f (x0 ) > 0 for some x0 (a, b), then there exists a > 0 such that f (x) > f (x0 ) for all x (x0 , x0 + ). (D) If f (x0 ) > 0 for some x0 (a, b), then f is increasing in a neighbourhood of x0 . Q. 36 Let G be a finite group of order 28. Assume that G contains a subgroup of order 7. Which of the following statements is/are true? (A) G contains a unique subgroup of order 7. (B) G contains a normal subgroup of order 7. (C) G contains no normal subgroup of order 7. (D) G contains at least two subgroups of order 7. Q. 37 Which of the following subsets of R is/are connected? MA (A) The set {x R : x is irrational}. (B) The set {x R : x3 1 0}. (C) The set {x R : x3 + x + 1 0}. (D) The set {x R : x3 2x + 1 0}. 12 / 17 JAM 2021 MATHEMATICS - MA Q. 38 Consider the four functions from R to R: f1 (x) = x4 + 3x3 + 7x + 1, f2 (x) = x3 + 3x2 + 4x, f3 (x) = arctan(x) and f4 (x) = ( x 0 if x / Z, if x Z. Which of the following subsets of R are open? (A) The range of f1 . (B) The range of f2 . (C) The range of f3 . (D) The range of f4 . Q. 39 Let V be a finite dimensional vector space and T : V V be a linear transformation. Let R(T ) denote the range of T and N (T ) denote the null space {v V : T v = 0} of T . If rank(T ) = rank(T 2 ), then which of the following is/are necessarily true? (A) N (T ) = N (T 2 ). (B) R(T ) = R(T 2 ). (C) N (T ) R(T ) = {0}. (D) N (T ) = {0}. Q. 40 Let m > 1 and n > 1 be integers. Let A be an m n matrix such that for some m 1 matrix b1 , the equation Ax = b1 has infinitely many solutions. Let b2 denote an m 1 matrix different from b1 . Then Ax = b2 has MA (A) infinitely many solutions for some b2 . (B) a unique solution for some b2 . (C) no solution for some b2 . (D) finitely many solutions for some b2 . 13 / 17 JAM 2021 MATHEMATICS - MA SECTION C NUMERICAL ANSWER TYPE (NAT) Q. 41 Q. 50 carry one mark each. Q. 41 The number of cycles of length 4 in S6 is . Q. 42 The value of lim n is . n n 3 +5 +7 n n1 Q. 43 Let B = {(x, y, z) R3 : x2 + y 2 + z 2 1} and define u(x, y, z) = sin (1 x2 y 2 z 2 )2 for (x, y, z) B. Then the value of ZZZ 2 u 2u 2u dxdydz + + x2 y 2 z 2 B is . Q. 44 Consider the subset S = {(x, y) : x2 + y 2 > 0} of R2 . Let y x P (x, y) = 2 and Q(x, y) = 2 2 x +y x + y2 for (x, y) S. If C denotes the unit circle traversed in the counter-clockwise direction, then the value of Z 1 (P dx + Qdy) C is . Q. 45 Consider the set A = {a R : x2 = a(a+1)(a+2) has a real root }. The number of connected components of A is . Q. 46 Let V be the real vector space of all continuous functions f : [0, 2] R such that the restriction of f to the interval [0, 1] is a polynomial of degree less than or equal to 2, the restriction of f to the interval [1, 2] is a polynomial of degree less than or equal to 3 and f (0) = 0. Then the dimension of V is equal to . MA 14 / 17 JAM 2021 MATHEMATICS - MA Q. 47 The number of group homomorphisms from the group Z4 to the group S3 is Q. 48 Let y : 9 ,3 10 R be a differentiable function satisfying 9 dy ,3 , (x 2y) + (2x + y) = 0, x dx 10 Then y(2) equals . and y(1) = 1. . Q. 49 Let F~ = (y + 1)ey cos(x) i + (y + 2)ey sin(x) j be a vector field in R2 and C be a continuously differentiable path with the starting point (0, 1) and the end point ( 2 , 0). Then Z F~ d~r C . equals Q. 50 The value of is MA . cos cos n+1 lim cos 2 n 4 8 2 15 / 17 JAM 2021 MATHEMATICS - MA Q. 51 Q. 60 carry two marks each. Q. 51 The number of elements of order two in the group S4 is equal to . Q. 52 The least possible value of k, accurate up to two decimal places, for which the following problem y (t) + 2y (t) + ky(t) = 0, t R, y(0) = 0, y(1) = 0, y(1/2) = 1, . has a solution is Q. 53 Consider those continuous functions f : R R that have the property that given any x R, f (x) Q if and only if f (x + 1) R \ Q. . The number of such functions is Q. 54 The largest positive number a such that Z Z 5 f (x)dx + 0 3 f 1 (x)dx a 0 for every strictly increasing surjective continuous function f : [0, ) [0, ) is . Q. 55 Define the sequence sn = m n 2 1 X 2j 2 n 2 j=0 n 1 1 X 2j 2 2n j=0 if n > 0 is even, if n > 0 is odd. 1 X Define m = sn . The number of limit points of the sequence { m } is m n=1 MA . 16 / 17 JAM 2021 MATHEMATICS - MA Q. 56 The determinant of the matrix is 2021 2021 2021 2021 . 2020 2021 2021 2021 2020 2020 2021 2021 2020 2020 2020 2021 Q. 57 The value of lim n is . Z 1 2 ex sin(nx) dx 0 Q. 58 Let S be the surface defined by {(x, y, z) R3 : z = 1 x2 y 2 , z 0}. Let F~ = y i + (x 1) j + z 2 k and n be the continuous unit normal field to the surface S with positive z-component. Then the value of ZZ 1 F~ n dS S is . 2 1 3 Q. 59 Let A = 2 1 3 . Then the largest eigenvalue of A is 3 2 1 . 1 0 0 0 0 1 0 0 Q. 60 Let A = . Consider the linear map TA from the real vector space M4 (R) 0 0 1 0 0 0 0 1 to itself defined by TA (X) = AX XA, for all X M4 (R). The dimension of the range of . TA is END OF THE QUESTION PAPER MA 17 / 17 Answer Key of JAM-2021 Mathematics (MA) Paper Note: Question numbers pertain to the question paper published on the JAM 2021 website Q. No. Answer Q. No. Answer 1 D 31 A, B 2 A 32 A, B 3 D 33 B, D 4 A 34 A, D 5 A 35 A, C 6 B 36 A, B 7 C 37 B, C 8 C 38 B, C, D 9 D 39 A, B, C 10 B 40 MTA 11 D 41 90 12 D 42 7 13 B 43 0 14 A 44 -2 15 B 45 2 16 A 46 5 17 A 47 4 18 A 48 3 19 A 49 1 20 C 50 1 21 A 51 9 22 B 52 10.8 to 10.9 23 C 53 0 24 C 54 15 25 B 55 0 26 B 56 2021 27 C 57 0 28 A 58 2 29 B 59 4 30 B 60 8

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