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GATE 2018 : Mathematics (with Final Answer Keys)

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GATE 2018 General Aptitude (GA) Set-2 Q. 1 Q. 5 carry one mark each. Q.1 The dress _________ her so well that they all immediately _________ her on her appearance. The words that best fill the blanks in the above sentence are (A) complemented, complemented (C) complimented, complimented Q.2 (B) complimented, complemented (D) complemented, complimented The judge s standing in the legal community, though shaken by false allegations of wrongdoing, remained _________. The word that best fills the blank in the above sentence is (A) undiminished Q.3 (C) illegal (D) uncertain Find the missing group of letters in the following series: BC, FGH, LMNO, _____ (A) UVWXY Q.4 (B) damaged (B) TUVWX (C) STUVW (D) RSTUV The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the following statements is true? (A) The circle has the largest area. (B) The square has the largest area. (C) The equilateral triangle has the largest area. (D) All the three shapes have the same area. Q.5 The value of the expression (A) -1 (B) 0 1 1+log + 1 1+log (C) 1 + 1 1+log is _________. (D) 3 Q. 6 Q. 10 carry two marks each. Q.6 Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films? (A) 0 GA (B) 2 (C) 4 (D) 8 1/3 GATE 2018 General Aptitude (GA) Set-2 Q.7 A wire would enclose an area of 1936 m2, if it is bent into a square. The wire is cut into two pieces. The longer piece is thrice as long as the shorter piece. The long and the short pieces are bent into a square and a circle, respectively. Which of the following choices is closest to the sum of the areas enclosed by the two pieces in square meters? (A) 1096 Q.8 (C) 1243 (D) 2486 A contract is to be completed in 52 days and 125 identical robots were employed, each operational for 7 hours a day. After 39 days, five-seventh of the work was completed. How many additional robots would be required to complete the work on time, if each robot is now operational for 8 hours a day? (A) 50 Q.9 (B) 1111 (B) 89 (C) 146 (D) 175 A house has a number which needs to be identified. The following three statements are given that can help in identifying the house number. i. If the house number is a multiple of 3, then it is a number from 50 to 59. ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69. iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79. What is the house number? (A) 54 GA (B) 65 (C) 66 (D) 76 2/3 GATE 2018 General Aptitude (GA) Set-2 Q.10 An unbiased coin is tossed six times in a row and four different such trials are conducted. One trial implies six tosses of the coin. If H stands for head and T stands for tail, the following are the observations from the four trials: (1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __. Which statement describing the last two coin tosses of the fourth trial has the highest probability of being correct? (A) Two T will occur. (B) One H and one T will occur. (C) Two H will occur. (D) One H will be followed by one T. END OF THE QUESTION PAPER GA 3/3 GATE 2018 MATHEMATICS Q.1 Q.25 carry one mark each Q.1 The principal value of ( 1)( 2i/ ) is (A) e2 (B) e2i (C) e 2i (D) e 2 Q.2 Let f : C C be an entire function with f (0) = 1, f (1) = 2 and f (0) = 0. If there exists M > 0 such that |f (z)| M for all z C, then f (2) = (A) 2 (B) 5 (C) 2 + 5i (D) 5 + 2i Q.3 In the Laurent series expansion of f (z) = 1 is z 1 (A) 2 (B) 1 (C) 0 1 valid for |z 1| > 1, the coefficient of z(z 1) (D) 1 Q.4 Let X and Y be metric spaces, and let f : X Y be a continuous map. For any subset S of X, which one of the following statements is true? (A) If S is open, then f (S) is open (B) If S is connected, then f (S) is connected (C) If S is closed, then f (S) is closed (D) If S is bounded, then f (S) is bounded Q.5 The general solution of the differential equation p xy = y + x2 + y 2 for x > 0 is given by (with an arbitrary positive constant k) p (A) ky 2 = x + x2 + y 2 p (B) kx2 = x + x2 + y 2 p (C) kx2 = y + x2 + y 2 p (D) ky 2 = y + x2 + y 2 MA 1/10 GATE 2018 MATHEMATICS Q.6 Let pn (x) be the polynomial solution of the differential equation d [(1 x2 )y ] + n(n + 1)y = 0 dx with pn (1) = 1 for n = 1, 2, 3, . . . . If d [pn+2 (x) pn (x)] = n pn+1 (x), dx then n is (A) 2n (B) 2n + 1 (C) 2n + 2 (D) 2n + 3 Q.7 In the permutation group S6 , the number of elements of order 8 is (A) 0 (B) 1 (C) 2 (D) 4 Q.8 Let R be a commutative ring with 1 (unity) which is not a field. Let I R be a proper ideal such that every element of R not in I is invertible in R. Then the number of maximal ideals of R is (A) 1 (B) 2 (C) 3 (D) infinite Q.9 Let f : R R be a twice continuously differentiable function. The order of convergence of the secant method for finding root of the equation f (x) = 0 is 1+ 5 1+ 5 2 3 (A) (C) (B) (D) 2 3 1+ 5 1+ 5 Q.10 The Cauchy problem uux + yuy = x with u(x, 1) = 2x, when solved using its characteristic equations with an independent variable t, is found to admit of a solution in the form 3 1 x = set se t , 2 2 y = et , u = f (s, t). Then f (s, t) = (A) 3 t 1 t se + se 2 2 (B) 1 t 3 t se + se 2 2 (C) 1 t 3 t se se 2 2 (D) 3 t 1 t se se 2 2 Q.11 An urn contains four balls, each ball having equal probability of being white or black. Three black balls are added to the urn. The probability that five balls in the urn are black is (A) 2/7 MA (B) 3/8 (C) 1/2 (D) 5/7 2/10 GATE 2018 MATHEMATICS Q.12 For a linear programming problem, which one of the following statements is FALSE? (A) If a constraint is an equality, then the corresponding dual variable is unrestricted in sign (B) Both primal and its dual can be infeasible (C) If primal is unbounded, then its dual is infeasible (D) Even if both primal and dual are feasible, the optimal values of the primal and the dual can differ a 2f 0 Q.13 Let A = 2f b 3f , where a, b, c, f are real numbers and f 6= 0. The geometric multi0 3f c . plicity of the largest eigenvalue of A equals Q.14 Consider the subspaces W1 = {(x1 , x2 , x3 ) R3 : x1 = x2 + 2x3 } W2 = {(x1 , x2 , x3 ) R3 : x1 = 3x2 + 2x3 } of R3 . Then the dimension of W1 + W2 equals . Q.15 Let V be the real vector space of all polynomials of degree less than or equal to 2 with real coefficients. Let T : V V be the linear transformation given by T (p) = 2p + p for p V, where p is the derivative of p. Then the number of nonzero entries in the Jordan canonical . form of a matrix of T equals Q.16 Let I = [2, 3), J be the set of all rational numbers in the interval [4, 6], K be the Cantor (ternary) set, and let L = {7 + x : x K}. Then the Lebesgue measure of the set I J L . equals Q.17 Let u(x, y, z) = x2 2y + 4z 2 for (x, y, z) R3 . Then the directional derivative of u in the 3 4 . direction i k at the point (5, 1, 0) is 5 5 Q.18 If the Laplace transform of y(t) is given by Y (s) = L(y(t)) = then y(0) + y (0) = MA . 5 2 1 + , 2(s 1) s 2 2(s 3) 3/10 GATE 2018 MATHEMATICS Q.19 The number of regular singular points of the differential equation [(x 1)2 sin x]y + [cos x sin(x 1)]y + (x 1)y = 0 is equal to in the interval 0, . 2 Q.20 Let F be a field with 7 6 elements and let K be a subfield of F with 49 elements. Then the . dimension of F as a vector space over K is Q.21 Let C([0, 1]) be the real vector space of all continuous real valued functions on [0, 1], and let T be the linear operator on C([0, 1]) given by Z 1 sin(x + y)f (y) dy, x [0, 1]. (T f )(x) = 0 Then the dimension of the range space of T equals . Q.22 Let a ( 1, 1) be such that the quadrature rule Z 1 f (x) dx f ( a) + f (a) 1 is exact for all polynomials of degree less than or equal to 3. Then 3a2 = . Q.23 Let X and Y have joint probability density function given by 2, 0 x 1 y, 0 y 1 fX,Y (x, y) = 0, otherwise. If fY denotes the marginal probability density function of Y , then fY (1/2) = . Q.24 Let the cumulative distribution function of the random variable X be given by 0, x < 0, x, 0 x < 1/2, FX (x) = (1 + x)/2, 1/2 x < 1, 1, x 1. Then P(X = 1/2) = . Q.25 Let {Xj } be a sequence of independent Bernoulli random variables with P(Xj = 1) = 1/4 1 Pn and let Yn = X 2 . Then Yn converges, in probability, to . n j=1 j MA 4/10 GATE 2018 MATHEMATICS Q.26 Q.55 carry two marks each Q.26 Let be the circle given by z = 4ei , where varies from 0 to 2 . Then I ez dz = 2 z 2z (A) 2 i(e2 1) (B) i(1 e2 ) (C) i(e2 1) (D) 2 i(1 e2 ) Q.27 The image of the half plane Re(z) + Im(z) > 0 under the map w = (A) Re(w) > 0 (B) Im(w) > 0 (C) |w| > 1 z 1 is given by z+i (D) |w| < 1 Q.28 Let D R2 denote the closed disc with center at the origin and radius 2. Then ZZ 2 2 e (x +y ) dxdy = D (A) (1 e 4 ) (B) (1 e 4 ) 2 (C) (1 e 2 ) (D) (1 e 2 ) 2 Q.29 Consider the polynomial p(X) = X 4 + 4 in the ring Q[X] of polynomials in the variable X with coefficients in the field Q of rational numbers. Then (A) the set of zeros of p(X) in C forms a group under multiplication (B) p(X) is reducible in the ring Q[X] (C) the splitting field of p(X) has degree 3 over Q (D) the splitting field of p(X) has degree 4 over Q Q.30 Which one of the following statements is true? (A) Every group of order 12 has a non-trivial proper normal subgroup (B) Some group of order 12 does not have a non-trivial proper normal subgroup (C) Every group of order 12 has a subgroup of order 6 (D) Every group of order 12 has an element of order 12 MA 5/10 GATE 2018 MATHEMATICS Q.31 For an odd prime p, consider the ring Z[ p] = {a + b p : a, b Z} C. Then the element 2 in Z[ p] is (A) a unit (B) a square (C) a prime (D) irreducible Q.32 Consider the following two statements: 0 5 has infinitely many LU factorizations, where L is lower triangular P: The matrix 0 7 with each diagonal entry 1 and U is upper triangular. 0 0 has no LU factorization, where L is lower triangular with each diagQ: The matrix 2 5 onal entry 1 and U is upper triangular. Then which one of the following options is correct? (A) P is TRUE and Q is FALSE (B) Both P and Q are TRUE (C) P is FALSE and Q is TRUE (D) Both P and Q are FALSE Q.33 If the characteristic curves of the partial differential equation xuxx + 2x2 uxy = ux 1 are (x, y) = c1 and (x, y) = c2 , where c1 and c2 are constants, then (A) (x, y) = x2 y, (x, y) = y (B) (x, y) = x2 + y, (x, y) = y (C) (x, y) = x2 + y, (x, y) = x2 (D) (x, y) = x2 y, (x, y) = x2 MA 6/10 GATE 2018 MATHEMATICS Q.34 Let f : X Y be a continuous map from a Hausdorff topological space X to a metric space Y . Consider the following two statements: P: f is a closed map and the inverse image f 1 (y) = {x X : f (x) = y} is compact for each y Y. Q: For every compact subset K Y, the inverse image f 1 (K) is a compact subset of X. Which one of the following is true? (A) Q implies P but P does NOT imply Q (B) P implies Q but Q does NOT imply P (C) P and Q are equivalent (D) neither P implies Q nor Q implies P Q.35 Let X denote R2 endowed with the usual topology. Let Y denote R endowed with the co-finite topology. If Z is the product topological space Y Y, then (A) the topology of X is the same as the topology of Z (B) the topology of X is strictly coarser (weaker) than that of Z (C) the topology of Z is strictly coarser (weaker) than that of X (D) the topology of X cannot be compared with that of Z Q.36 Consider Rn with the usual topology for n = 1, 2, 3. Each of the following options gives topological spaces X and Y with respective induced topologies. In which option is X homeomorphic to Y ? (A) X = {(x, y, z) R3 : x2 + y 2 = 1}, Y = {(x, y, z) R3 : z = 0, x2 + y 2 6= 0} (B) X = {(x, y) R2 : y = sin(1/x), 0 < x 1} {(x, y) R2 : x = 0, 1 y 1}, Y = [0, 1] R (C) X = {(x, y) R2 : y = x sin(1/x), 0 < x 1}, (D) X = {(x, y, z) R3 : x2 + y 2 = 1}, Y = [0, 1] R Y = {(x, y, z) R3 : x2 + y 2 = z 2 6= 0} 1 Pn Xi . Then Q.37 Let {Xi } be a sequence of independent Poisson( ) variables and let Wn = n i=1 the limiting distribution of n(Wn ) is the normal distribution with zero mean and variance given by (A) 1 (B) (C) (D) 2 MA 7/10 GATE 2018 MATHEMATICS Q.38 Let X1 , X2 , . . . , Xn be independent and identically distributed random variables with probability density function given by (x 1) e , x 1, fX (x; ) = 0 otherwise. 1 Pn Xi . Then the maximum likelihood estimator of is n i=1 (B) 1/X 1 (C) 1/ X 1 (D) X Also, let X = (A) 1/X Q.39 Consider the Linear Programming Problem (LPP): Maximize x1 + x2 Subject to 2x1 + x2 6, x1 + x2 1, x1 + x2 4, x1 0, x2 0, where is a constant. If (3, 0) is the only optimal solution, then (A) < 2 (B) 2 < < 1 (C) 1 < < 2 (D) > 2 Q.40 Let M2 (R) be the vector space of all 2 2 real matrices over the field R. Define the linear transformation S : M2 (R) M2 (R) by S(X) = 2X + X T , where X T denotes the transpose of the matrix X. Then the trace of S equals . Q.41 Consider R3 with the usual inner product. If d is the distance from (1, 1, 1) to the subspace . span{(1, 1, 0), (0, 1, 1)} of R3 , then 3d2 = 1 [1, 1, 1, 1, 1, 1, 1, 1, 1], where I9 is the 9 9 3 identity matrix and uT is the transpose of u. If and are two distinct eigenvalues of A, then | | = . Q.42 Consider the matrix A = I9 2uT u with u = 2 Q.43 Let f (z) = z 3 ez for z C and let be the circle z = ei , where varies from 0 to 4 . Then I f (z) 1 dz = . 2 i f (z) MA 8/10 GATE 2018 MATHEMATICS Q.44 Let S be the surface of the solid V = {(x, y, z) : 0 x 1, 0 y 2, 0 z 3}. Let n denote the unit outward normal to S and let F~ (x, y, z) = x i + y j + z k, ZZ F~ n dS equals Then the surface integral (x, y, z) V. . S Q.45 Let A be a 3 3 matrix with real entries. If three solutions of the linear system of differential equations x(t) = Ax(t) are given by et e2t et + e2t , et + e2t e2t e t e2t e t e2t + e t and e t + 2et e t 2et , e t + 2et then the sum of the diagonal entries of A is equal to . 2 Q.46 If y1 (x) = e x is a solution of the differential equation xy + y + x3 y = 0 for some real numbers and , then = . Q.47 Let L2 ([0, 1]) be the Hilbert space of all real valued square integrable functions on [0, 1] with the usual inner product. Let be the linear functional on L2 ([0, 1]) defined by (f ) = Z 3/4 3 2f d , 1/4 where denotes the Lebesgue measure on [0, 1]. Then k k = . Q.48 Let U be an orthonormal set in a Hilbert space H and let x H be such that kxk = 2. Consider the set 1 E = u U : |hx, ui| . 4 Then the maximum possible number of elements in E is MA . 9/10 GATE 2018 MATHEMATICS Q.49 If p(x) = 2 (x + 1) + x(x + 1) x(x + 1)(x ) interpolates the points (x, y) in the table x y then + = Q.50 If sin( x) = a0 + 1 2 0 1 1 2 2 7 . P n=1 an cos(n x) for 0 < x < 1, then (a0 + a1 ) = 2nxn 1 , x [0, 1]. Then lim Q.51 For n = 1, 2, . . . , let fn (x) = n 1+x Z . 1 fn (x)dx = . 0 Q.52 Let X1 , X2 , X3 , X4 be independent exponential random variables with mean 1, 1/2, 1/3, 1/4, respectively. Then Y = min(X1 , X2 , X3 , X4 ) has exponential distribution with mean equal to . Q.53 Let X be the number of heads in 4 tosses of a fair coin by Person 1 and let Y be the number of heads in 4 tosses of a fair coin by Person 2. Assume that all the tosses are independent. Then the value of P(X = Y ) correct up to three decimal places is . Q.54 Let X1 and X2 be independent geometric random variables with the same probability mass function given by P(X = k) = p(1 p)k 1 , k = 1, 2, . . .. Then the value of P(X1 = 2|X1 + X2 = 4) correct up to three decimal places is . Q.55 A certain commodity is produced by the manufacturing plants P1 and P2 whose capacities are 6 and 5 units, respectively. The commodity is shipped to markets M1 , M2 , M3 and M4 whose requirements are 1, 2, 3 and 5 units, respectively. The transportation cost per unit from plant Pi to market Mj is as follows: P1 P2 M1 1 2 1 Then the optimal cost of transportation is M2 3 5 2 M3 5 6 3 M4 8 7 5 6 5 . END OF THE QUESTION PAPER MA 10/10 Q.No. Type Section Key/Range Marks 1 MCQ GA D 1 2 MCQ GA A 1 3 MCQ GA B 1 4 MCQ GA A 1 5 MCQ GA C 1 6 MCQ GA C 2 7 MCQ GA C 2 8 MCQ GA Marks to All 2 9 MCQ GA D 2 10 MCQ GA B 2 1 MCQ MA A 1 2 MCQ MA B 1 3 MCQ MA C 1 4 MCQ MA B 1 5 MCQ MA C 1 6 MCQ MA D 1 7 MCQ MA A 1 8 MCQ MA A 1 9 MCQ MA A 1 10 MCQ MA A 1 11 MCQ MA B 1 12 MCQ MA D 1 13 NAT MA 1 to 1 1 Q.No. Type Section Key/Range Marks 14 NAT MA 3 to 3 1 15 NAT MA 5 to 5 1 16 NAT MA 1 to 1 1 17 NAT MA 6 to 6 1 18 NAT MA 1 to 1 1 19 NAT MA 2 to 2 1 20 NAT MA 3 to 3 1 21 NAT MA 2 to 2 1 22 NAT MA 1 to 1 1 23 NAT MA 1 to 1 1 24 NAT MA 0.25 to 0.25 1 25 NAT MA 0.25 to 0.25 1 26 MCQ MA C 2 27 MCQ MA D 2 28 MCQ MA A 2 29 MCQ MA B 2 30 MCQ MA A 2 31 MCQ MA D 2 32 MCQ MA B 2 33 MCQ MA A 2 34 MCQ MA C 2 35 MCQ MA C 2 36 MCQ MA A 2 Q.No. Type Section Key/Range Marks 37 MCQ MA C 2 38 MCQ MA C 2 39 MCQ MA D 2 40 NAT MA 10 to 10 2 41 NAT MA 1 to 1 2 42 NAT MA 2 to 2 2 43 NAT MA 6 to 6 2 44 NAT MA 18 to 18 2 45 NAT MA 2 to 2 2 46 NAT MA 4 to 4 2 47 NAT MA 3 to 3 2 48 NAT MA 64 to 64 2 49 NAT MA 3 to 3 2 50 NAT MA 2 to 2 2 51 NAT MA 1 to 1 2 52 NAT MA 0.1 to 0.1 2 53 NAT MA 0.272 to 0.274 2 54 NAT MA 0.332 to 0.334 2 55 NAT MA 57 to 57 2

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