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GATE 2016 : Mathematics

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GATE 2016 General Aptitude - GA Set-4 Q. 1 Q. 5 carry one mark each. Q.1 An apple costs Rs. 10. An onion costs Rs. 8. Select the most suitable sentence with respect to grammar and usage. (A) The price of an apple is greater than an onion. (B) The price of an apple is more than onion. (C) The price of an apple is greater than that of an onion. (D) Apples are more costlier than onions. Q.2 The Buddha said, Holding on to anger is like grasping a hot coal with the intent of throwing it at someone else; you are the one who gets burnt. Select the word below which is closest in meaning to the word underlined above. (A) burning Q.3 (B) igniting (C) clutching (D) flinging M has a son Q and a daughter R. He has no other children. E is the mother of P and daughter-inlaw of M. How is P related to M? (A) P is the son-in-law of M. (C) P is the daughter-in law of M. Q.4 (B) P is the grandchild of M. (D) P is the grandfather of M. The number that least fits this set: (324, 441, 97 and 64) is ________. (A) 324 Q.5 (B) 441 (C) 97 (D) 64 It takes 10 s and 15 s, respectively, for two trains travelling at different constant speeds to completely pass a telegraph post. The length of the first train is 120 m and that of the second train is 150 m. The magnitude of the difference in the speeds of the two trains (in m/s) is ____________. (A) 2.0 (B) 10.0 (C) 12.0 (D) 22.0 1/3 GATE 2016 General Ap ptitude - GA Set-4 4 Q. 6 Q. 10 carry tw wo marks each. Q.6 The e velocity V of a vehicl le along a st traight line is i measured in m/s and plotted as shown s with resp pect to time e in seconds. At the end d of the 7 seconds, s how w much will l the odome eter reading incr rease by (in m)? (A) ) 0 Q.7 (B) 3 (C) 4 (D) 5 The e overwhelm ming number r of people in nfected with h rabies in In ndia has been n flagged by y the World Hea alth Organiz zation as a so ource of con ncern. It is es stimated that t inoculating g 70% of pet ts and stray dog gs against rab bies can lead d to a signific cant reductio on in the num mber of peopl le infected with w rabies. Wh hich of the fo ollowing can be logically y inferred from m the above sentences? (A) ) The numb ber of people in India infe ected with ra abies is high. ) The number of people in other part ts of the world who are in nfected with rabies is low w. (B) (C) ) Rabies can n be eradicated in India by b vaccinatin ng 70% of str ray dogs. ) Stray dogs s are the main source of rabies r worldw wide. (D) Q.8 A flat f is shared by four first t year underg graduate stud dents. They agreed a to allo ow the oldest t of them to enjoy some ext tra space in the flat. Ma anu is two months m older r than Sravan n, who is th hree months you unger than Tr rideep. Pava an is one mon nth older than n Sravan. Who W should oc ccupy the ex xtra space in the flat? (A) ) Manu Q.9 (B) Sravan (C) Trideep (D) Pavan Fin nd the area bo ounded by th he lines 3x+2 2y=14, 2x-3y y=5 in the fir rst quadrant. (A) ) 14.95 (B) 15.25 (C) 15.70 (D) 20.35 2/3 3 GATE 2016 Q.10 General Aptitude - GA Set-4 A straight line is fit to a data set (ln x, y). This line intercepts the abscissa at ln x = 0.1 and has a slope of 0.02. What is the value of y at x = 5 from the fit? (A) 0.030 (B) 0.014 (C) 0.014 (D) 0.030 END OF THE QUESTION PAPER 3/3 GATE 2016 Mathematics - MA List of Symbols Notations and Data i i d independent and identically distributed Normal distribution with mean and variance Expected value mean of the random variable the greatest integer less than or equal to Set of integers Set of integers modulo n Set of real numbers Set of complex numbers n dimensional Euclidean space Usual metric d on is given by Normed linear space of all square summable real sequences Set of all real valued continuous functions on the interval Conjugate transpose of the matrix M Transpose of the matrix M Id )dentity matrix of appropriate order Range space of M Null space of M MA Orthogonal complement of the subspace W 1/16 GATE 2016 Mathematics - MA Q. 1 Q. 25 carry one mark each. Q Let be a basis of P Consider the following statements P and Q is a basis of Q is a basis of Which of the above statements hold TRUE A both P and Q C only Q Q D Neither P nor Q P )f then M is singular . Q Let S be a diagonalizable matrix )f T is a matrix such that S + 5 T = Id then T is diagonalizable Which of the above statements hold TRUE A both P and Q C only Q Q B only P D Neither P nor Q Consider the following statements P and Q P )f M is an complex matrix then Q There exists a unitary matrix with an eigenvalue such that Which of the above statements hold TRUE A both P and Q B only P Consider the following statements P and Q C only Q B only P D Neither P nor Q MA 2/16 GATE 2016 Q Mathematics - MA Consider a real vector space V of dimension n and a non zero linear transformation )f dimension and for some which of the following statements is TRUE A determinant B There exists a non trivial subspace C T is invertible of V such that then for all D is the only eigenvalue of T Q Let and be a strictly increasing function such that is connected Which of the following statements is TRUE A has exactly one discontinuity B has exactly two discontinuities C has infinitely many discontinuities D is continuous Q Let and lim is equal to Q Let Maximum Q Then such that MA C is equal to has a unique solution if A Then the Cauchy problem on B D 3/16 GATE 2016 Q Mathematics - MA Let be the d Alembert s solution of the initial value problem for the wave equation where c is a positive real number and equal to Q Q Let where continuous functions )f Q is otherwise Let V be the set of all solutions of the equation Let the probability density function of a random variable X be Then the value of c is equal to Q are smooth odd functions Then Let satisfying are positive real numbers Then dimension V is equal to where and and are are two linearly independent solutions of the above equation then is equal to be the Legendre polynomial of degree and Q if if is a non negative integer Consider the following statements P and Q P where k is an odd integer Which of the above statements hold TRUE A both P and Q MA C only Q B only P D Neither P nor Q 4/16 GATE 2016 Q Mathematics - MA Consider the following statements P and Q P Q solutions near solutions near has two linearly independent Frobenius series has two linearly independent Frobenius series Which of the above statements hold TRUE A both P and Q B only P C only Q Q Let the polynomial interpolates at over the interval Q Let lim D Neither P nor Q be approximated by a polynomial of degree and Then the maximum absolute interpolation error is equal to be a sequence of distinct points in Consider the following statements P and Q P There exists a unique analytic function f on all n Q There exists an analytic function f on and which such that such that if n is odd with for if n is even Which of the above statements hold TRUE A both P and Q C only Q Q Let is B only P D Neither P nor Q be a topological space with the cofinite topology Every infinite subset of A Compact but NOT connected B Both compact and connected C NOT compact but connected MA D Neither compact nor connected 5/16 GATE 2016 Mathematics - MA Q Let Q Consider Then dimension defined by Q and is equal to where Then maximum Let be and the norm preserving linear extension of to is equal to is called a shrinking map if and a contraction if there exists an for all for all such that Which of the following statements is TRUE for the function A is both a shrinking map and a contraction B is a shrinking map but NOT a contraction C is NOT a shrinking map but a contraction D is Neither a shrinking map nor a contraction Q Let be the set of all real matrices with the usual norm topology Consider the following statements P and Q P The set of all symmetric positive definite matrices in is connected Q The set of all invertible matrices in is compact Which of the above statements hold TRUE A both P and Q C only Q B only P D Neither P nor Q MA 6/16 GATE 2016 Q Mathematics - MA Let function for be a random sample from the following probability density otherwise (ere and are unknown parameters Which of the following statements is TRUE A Maximum likelihood estimator of only exists B Maximum likelihood estimator of only exists C Maximum likelihood estimators of both and D Maximum likelihood estimator of Neither Q nor exist exists Suppose X and Y are two random variables such that variable for all is a normal random Consider the following statements P Q R and S P X is a standard normal random variable Q The conditional distribution of X given Y is normal R The conditional distribution of X given S has mean is normal Which of the above statements ALWAYS hold TRUE A both P and Q B both Q and R C both Q and S Q D both P and S Consider the following statements P and Q P )f is a normal subgroup of order of the symmetric group Q )f abelian is the quaternion group then Which of the above statements hold TRUE C only Q Q MA is abelian D Neither P nor Q Let be a field of order the equation is B only P A both P and Q then Then the number of non zero solutions is equal to of 7/16 GATE 2016 Mathematics - MA Q. 26 Q. 55 carry two marks each. Q Let be oriented in the counter clockwise direction Let Then the value of is equal to Q Q Let harmonic conjugate )f clockwise direction in then Then C is equal to MA and B its in the counter D Let X be a random variable with the following cumulative distribution function is equal to A Q be a harmonic function and Let y be the solution of Then Let be the triangular path connecting the points is equal to Q Then is equal to 8/16 GATE 2016 Q Mathematics - MA Let be the curve which passes through A orthogonally Then also passes through the point C Q Let B be a continuous function on is equal to Let A ln C ln Q Q MA Let equal to B ln distance let in imum is equal to and Then is equal to D ln Then Then and For any )f then Q be the Fourier series of the periodic function defined by Let D is equal to Q and intersects each curve of the family Then is 9/16 GATE 2016 Q Mathematics - MA Let be a real matrix with eigenvalues and )f the eigenvectors corresponding to and are is equal to Q Let respectively then the value of )f and is equal to Q and Let the integral where then Consider the following statements P and Q P )f is the value of the integral obtained by the composite trapezoidal rule with two equal sub intervals then is exact Q )f is the value of the integral obtained by the composite trapezoidal rule with three equal sub intervals then is exact Which of the above statements hold TRUE A both P and Q B only P C only Q Q The difference between the least two eigenvalues of the boundary value problem is equal to Q MA D Neither P nor Q The number of roots of the equation to cos in the interval is equal 10/16 GATE 2016 Q Mathematics - MA consider the following For the fixed point iteration statements P and Q Q )f then the fixed point iteration converges to for all P )f then the fixed point iteration converges to for all Which of the above statements hold TRUE A both P and Q C only Q Q Let Then A B C B only P D Q D Neither P nor Q be defined by but bounded is unbounded Minimize subject to Then the minimum value of is equal to Q Maximize subject to MA Then the maximum value of is equal to 11/16 GATE 2016 Q Mathematics - MA Let be a sequence of i i d random variables with mean )f N is a geometric random variable with the probability mass function s then and it is independent of the Q Let be an exponential random variable with mean and variable with mean and variance )f of random variables Then the value lim ln ln Let X be a standard normal random variable Then A Let where a gamma random be a sequence of i i d uniform Let C Q is equal to are independently distributed then is equal to Q and is equal to Q B D is equal to be a random sample from the probability density function otherwise are parameters Consider the following testing problem versus Which of the following statements is TRUE A Uniformly Most Powerful test does NOT exist B Uniformly Most Powerful test is of the form C Uniformly Most Powerful test is of the form MA D Uniformly Most Powerful test is of the form some for some for some for 12/16 GATE 2016 Q Mathematics - MA Let lim is equal to Q Let be a sequence of i i d Maximum A then the UMVUE of is B D Let The remainder when Q Let be a group whose presentation is MA is divided by is equal to Then is isomorphic to C )f Then the maximum likelihood estimator of Q A for some be a random sample from a Poisson random variable with mean where is equal to be a random sample from uniform C Q random variables Then B D 13/16 GATE 2016 Mathematics - MA END OF THE QUESTION PAPER MA 14/16 GATE 2016 MA Mathematics - MA 15/16 GATE 2016 Mathematics - MA MA 16/16 Q. No 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Type Section Key Marks MCQ GA C 1 MCQ GA C 1 MCQ GA B 1 MCQ GA C;D 1 MCQ GA A 1 MCQ GA D 2 MCQ GA A 2 MCQ GA C 2 MCQ GA B 2 MCQ GA A 2 MCQ MA C 1 MCQ MA C 1 MCQ MA B 1 MCQ MA B 1 MCQ MA D 1 NAT MA 0.24 : 0.26 1 NAT MA 1.39 : 1.43 1 MCQ MA A 1 NAT MA -0.1 : 0.1 1 NAT MA 5.2 : 5.3 1 NAT MA 0.9 : 1.1 1 NAT MA 1.95 : 2.05 1 MCQ MA A 1 MCQ MA B 1 NAT MA 0.22 : 0.28 1 MCQ MA B 1 MCQ MA B 1 NAT MA 0.9 : 1.1 1 NAT MA 0.9 : 1.1 1 MCQ MA B;D 1 MCQ MA B 1 MCQ MA D 1 MCQ MA B 1 MCQ MA C 1 NAT MA -0.1 : 0.1 1 NAT MA 0.039 : 0.043 2 NAT MA 9.9 : 10.1 2 NAT MA 15.9 : 16.1 2 MCQ MA A 2 NAT MA 0.65 : 0.71 2 MCQ MA A 2 NAT MA 1.9 : 2.1 2 NAT MA 0.45 : 0.55 2 MCQ MA A 2 NAT MA 0.9 : 1.1 2 NAT MA -0.1 : 0.1 2 NAT MA 6.9 : 7.1 2 NAT MA 5.4 : 5.6 2 MCQ MA B 2 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 NAT NAT MCQ MCQ NAT NAT NAT NAT NAT MCQ MCQ NAT MCQ NAT NAT MCQ MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA MA 1.9 : 2.1 1.9 : 2.1 A C 1.9 : 2.1 1.2 : 1.3 1.9 : 2.1 0.7 : 0.8 0.4 : 0.6 A C 0.69 : 0.73 B 1.9 : 2.1 49.9 : 50.1 C 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

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