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GATE 2015 : Mathematics

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GATE 2015 MATHEMATICS MA List of Symbols, Notations and Data Binomial distribution with trials and success probability Uniform distribution on the interval Normal distribution with mean and variance and Probability of the event Poisson Poisson distribution with mean Expected value (mean) of the random variable then If Set of integers and Set of rational numbers Set of real numbers Set of complex numbers The cyclic group of order : Polynomial ring over the field Set of all real valued continuous functions on the interval Set of all real valued continuously differentiable functions on the interval Normed space of all square-summable real sequences Space of all square-Lebesgue integrable real valued functions on the interval The space The space with The orthogonal complement of with -dimensional Euclidean space Usual metric on The is given by identity matrix ( sup in an inner product space the identity matrix when order is NOT specified) The order of the element of a group MA 1/9 GATE 2015 MATHEMATICS MA Q. 1 Q. 25 carry one mark each. Q.1 Let be a linear map defined by Then the rank of Q.2 Let be a is equal to _________ matrix and suppose that then for some scalar Q.3 and are the eigenvalues of is equal to ___________ Let be a singular matrix and suppose that of linearly independent eigenvectors of and are eigenvalues of Then the number is equal to __________ Q.4 Let be a matrix such that and suppose that is equal to _______ some Q.5 Then Let be defined by Then the function If for sin is (A) uniformly continuous on but NOT on (B) uniformly continuous on but NOT on (C) uniformly continuous on both and (D) neither uniformly continuous on nor uniformly continuous on Q.6 where Consider the power series if if is even is odd The radius of convergence of the series is equal to __________ Q.7 Q.8 MA Let Let and Then Then is equal to ____________ is equal to ___________ 2/9 GATE 2015 Q.9 MATHEMATICS MA Let the random variable have the distribution function Then Q.10 if if if is equal to _________ if if if In an experiment, a fair die is rolled until two sixes are obtained in succession. The probability that the experiment will end in the fifth trial is equal to (B) Let random sample of size 6 from a a maximum likelihood estimate of (A) (A) (C) is equal to (B) Let the solution of the Dirichlet problem then MA if is equal to ___________ (A) Q.13 Let X be a random variable having the distribution function if Then Q.12 if Q.11 if is equal to (B) and be the observed values of a distribution, where is unknown. Then (C) be the open unit disc in (D) (D) with boundary If is in on (C) (D) 3/9 GATE 2015 MATHEMATICS MA Q.14 Let Q.15 Let be such that is a field. Then is equal to __________ and (A) dense in but NOT in (B) dense in but NOT in (C) dense in both and (D) neither dense in nor dense in Q.16 Let is equal to __________ Q.17 Let be defined by be the usual topology on . Let be the topology on Then the set (A) closed in but NOT in but NOT in (B) closed in and (C) closed in both nor closed in (D) neither closed in Q.18 Then is for all Then generated by sin is Let be a connected topological space such that there exists a non-constant continuous function Then where is equipped with the usual topology. Let (A) is countable but is uncountable is countable but is uncountable (B) and are countable (C) both and are uncountable (D) both Q.19 Let Let and denote the usual metric and the discrete metric on , respectively. be defined by Then (A) is continuous but is NOT continuous (B) is continuous but is NOT continuous are continuous (C) both and (D) neither nor is continuous Q.20 If the trapezoidal rule with single interval is exact for approximating the integral then the value of is equal to ________ Q.21 Suppose that the Newton-Raphson method is applied to the equation initial approximation sufficiently close to zero. Then, for the root convergence of the method is equal to _________ MA with an the order of 4/9 GATE 2015 MATHEMATICS MA Q.22 The minimum possible order of a homogeneous linear ordinary differential equation with real as a solution is equal to __________ constant coefficients having sin Q.23 The Lagrangian of a system in terms of polar coordinates is given by cos where is the mass, is the acceleration due to gravity and respect to time. Then the equations of motion are (A) (B) (C) (D) Q.24 If then Q.25 cos with sin cos sin cos sin cos sin satisfies the initial value problem is equal to __________ It is known that Bessel functions for all denotes the derivative of and for The value of satisfy the identity is equal to _________ Q. 26 Q. 55 carry two marks each. Q.26 Let and be two random variables having the joint probability density function if Then the conditional probability (A) Q.27 Let (B) be the sample space and let otherwise is equal to (C) be a probability function defined by if Then MA is equal to __________ (D) if x 5/9 GATE 2015 Q.28 MATHEMATICS MA and If Let Let Q.30 then Q.29 be independent and identically distributed random variables with is defined through the conditional expectation Poisson is equal to __________ where then distribution, where . For testing the null consider the critical region against the alternative hypothesis hypothesis , is unknown. If is the unbiased estimator of is equal to ___________ be a random sample from Let and where is some real constant. If the critical region has size 0.025 and power 0.7054, then the value of the sample size n is equal to ___________ Q.31 Let and be independently distributed central chi-squared random variables with degrees of freedom and respectively. If and then is equal to Q.32 Q.33 is equal to __________ cos Let Then Let Then MA be a solution of the initial value problem is equal to cos (C) (B) (A) Q.34 (D) be a sequence of independent and identically distributed random variables with If and for then Let lim (C) (B) (A) (D) be the solution of the initial value problem is equal to ________ 6/9 GATE 2015 Q.35 MATHEMATICS MA Span Let be a subspace of the Euclidean space square of the distance from the point Q.36 Let be a linear map such that the null space of and the rank of is then is equal to _______ Q.37 Let If the minimal polynomial of be an invertible Hermitian matrix and let (A) both (B) (C) (D) both Q.38 to the subspace and is singular but is non-singular but and (A) 1 (B) 2 The number of ring homomorphisms from Q.40 Let (A) (B) (C) (D) Q.41 is Then the number and to (D) 8 is equal to __________ and be two polynomials in and are both irreducible is reducible but is irreducible is irreducible but is reducible and are both reducible Consider the linear programming problem Maximize subject to Let Then are singular is non-singular is singular are non-singular Then the maximum value of the objective function is equal to ______ Q.42 is (C) 4 Q.39 Then, over is equal to ________ be such that Let with is equal to of elements in the center of the group Then the sin and Under the usual metric on (A) is closed but is NOT closed (B) is closed but is NOT closed (C) both and are closed (D) neither nor is closed MA 7/9 GATE 2015 Q.43 MATHEMATICS MA Let (A) is bounded (C) is a subspace Q.44 Then (B) is closed (D) has an interior point (A) (B) (C) (D) be the polynomial of degree at most that passes through the points in is equal to _________ and Then the coefficient of Q.45 Let Q.46 If, for some the integration formula holds for all polynomials Q.47 and then and let be given by Let be a closed subspace of If Span and is the orthogonal projection of on is Let then of degree at most be a continuous function on then the value of is equal to _____ satisfies whose Laplace transform exists. If cos is equal to _______ Q.48 Consider the initial value problem as then is equal to __________ If Q.49 Define by Then sin and (A) is continuous but is NOT continuous is continuous but is NOT continuous (B) (C) both and are continuous (D) neither nor is continuous Q.50 Consider the unit sphere at each point is equal to _______ MA on sin and the unit normal vector The value of the surface integral sin 8/9 GATE 2015 Q.51 MATHEMATICS MA Define Let on Then the minimum value of Q.52 Let for all Q.53 Let Q.55 The value of such that (B) (D) in the annulus be the Laurent series expansion of Then Q.54 is equal to ________ Then there exists a non-constant analytic function on (A) (C) is equal to _________ is equal to __________ Suppose that among all continuously differentiable functions with and , the function minimizes the functional Then (A) is equal to (B) (C) (D) END OF THE QUESTION PAPER MA 9/9 Graduate Aptitude Test in Engineering Notations : t.opticns shown in green color and with~ icon are correct. 2.0ptions shown in red color and with icon are incorrect. Question Paper Name: Number of Questions: Total Marks: MA: MATHEMATICS 1st Feb shift2 65 100.0 Wrong answer for MCQ will result in negative marks, (-1/3) for 1 mark Questions and (-2/3) for 2 marks Questions. General Aptitude Number of Questions: Section Marks: 10 15.0 Q.1 to Q.5 carry 1 mark each & Q.6 to Q.10 carry 2 marks each. Question Number : 1 Question Type : MCQ Choose the appropriate word/phrase. out of the four options given below, to complete the following sentence: Apparent lifelessness --------(A) harbours (B) leads to dormant life. (C) supports (D) affects Options : 1. ~A 2. x B 3. c 4. X D Question Number : 2 Question Type : MCQ Fill in the blank with the correct idiom/phrase. That boy from the town was a (A) dog out of herd (C) fish out of water Options : 1. X A 2. X B 3.~C 4. X D Question Number : 3 Question Type : MCQ in the sleepy village. (B) sheep from the heap (D) bird from the flock Choose the statement where underlined word is used correctly. (A) When the teacher eludes to different authors, he is being elusive. (B) (C) (D) When the thief keeps eluding the police, he is being elusive. Matters that are difficult to understand, identify or remember are allusive. Mirages can be allusive. but a better \Vay to express them is illusory. Options : i. A 2.~B c 3. 4. D Question Number : 4 Question Type : MCQ Tanya is older than Eric. Cliff is older than Tanya. Eric is older than Cliff. If the first rwo statemenrs are true, then the third statement is: (A) True (B) False (C) Uncertain (D) Data insufficient Options : 1. W A 2.~B c 3. 4. a D Question Number : 5 Question Type : MCQ Five teams have to compete in a league, with every team playing every other team exactly once, before going to the next round. How many matches will have to be held to complete the league round of matches? (A) 20 (B) 10 Options : 1. W A 2.~B c 3. 4. a D Question Number : 6 Question Type : MCQ (C) 8 (D) 5 Options : Question Number : 7 Question Type : MCQ Options : Question Number : 8 Question Type : NAT Correct Answer : 280 Question Number : 9 Question Type : MCQ Options : Question Number : 10 Question Type : MCQ Options : Number of Questions: Section Marks: 55 85.0 Q.11 to Q.35 carry 1 mark each & Q.36 to Q.65 carry 2 marks each. Question Number : 11 Question Type : NAT Correct Answer : 3 Question Number : 12 Question Type : NAT Correct Answer : 6 Question Number : 13 Question Type : NAT Correct Answer : 3 Question Number : 14 Question Type : NAT Correct Answer: 27 Question Number : 15 Question Type : MCQ Options : Question Number : 16 Question Type : NAT Correct Answer : 3 Question Number : 17 Question Type : NAT Correct Answer : -2 Question Number : 18 Question Type : NAT Correct Answer: 6 Question Number : 19 Question Type : NAT Correct Answer: 0.4 Question Number : 20 Question Type : NAT Correct Answer : 2.25 Question Number : 21 Question Type : MCQ Options : Question Number : 22 Question Type : MCQ Options : Question Number : 23 Question Type : MCQ Options : Question Number : 24 Question Type : NAT Correct Answer : 2 Question Number : 25 Question Type : MCQ Options : Question Number : 26 Question Type : NAT Correct Answer : 1 Question Number : 27 Question Type : MCQ Options : Question Number : 28 Question Type : MCQ Options : Question Number : 29 Question Type : MCQ Options : Question Number : 30 Question Type : NAT Correct Answer : 1.5 Question Number : 31 Question Type : NAT Correct Answer : 1 Question Number : 32 Question Type : NAT Correct Answer: 6 Question Number : 33 Question Type : MCQ Options : Question Number : 34 Question Type : NAT . Correct Answer: 6 Question Number : 35 Question Type : NAT Correct Answer : 1 Question Number : 36 Question Type : MCQ Options : Question Number : 37 Question Type : NAT Correct Answer : 0.25 Question Number : 38 Question Type : NAT Correct Answer : 2.5 Question Number : 39 Question Type : NAT Correct Answer : 9 Question Number : 40 Question Type : NAT Correct Answer : 25 Question Number : 41 Question Type : MCQ Options : Question Number : 42 Question Type : NAT Correct Answer: 1 Question Number : 43 Question Type : MCQ Options : Question Number : 44 Question Type : NAT Correct Answer: 4 Question Number : 45 Question Type : NAT Correct Answer : 2 Question Number : 46 Question Type : NAT Correct Answer : 1 Question Number : 47 Question Type : MCQ Options : Question Number : 48 Question Type : MCQ Options : Question Number : 49 Question Type : NAT Correct Answer: 1 Question Number : 50 Question Type : MCQ Options : Question Number : 51 Question Type : NAT Correct Answer: 24 Question Number : 52 Question Type : MCQ Options : Question Number : 53 Question Type : MCQ Options : Question Number : 54 Question Type : MCQ Options : Question Number : 55 Question Type : NAT Correct Answer : -2 Question Number : 56 Question Type : NAT Correct Answer : 4 Question Number : 57 Question Type : NAT Correct Answer : 28 Question Number : 58 Question Type : NAT Correct Answer : 2 Question Number : 59 Question Type : MCQ Options : Question Number : 60 Question Type : NAT Correct Answer : 4 Question Number : 61 Question Type : NAT Correct Answer: 150 Question Number : 62 Question Type : MCQ Options : Question Number : 63 Question Type : NAT Correct Answer: 5 Question Number : 64 Question Type : NAT Correct Answer : 2 Question Number : 65 Question Type : MCQ Options :

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