Γενεραλ Απτιτυδε ΓΑ Σετ 4. Σελεχτ τηε mοστ συιταβλε σεντενχε ωιτη ρεσπεχτ το γραmmαρ ανδ υσαγε. (Α) Τηε πριχε οφ αν αππλε ισ γρεατερ τηαν αν ονιον.

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1 ΓΑΤΕ 2016 Γενεραλ Απτιτυδε ΓΑ Σετ 4 Q. 1 Q. 5 carry one mark each. Θ.1 Αν αππλε χοστσ Ρσ. 10. Αν ονιον χοστσ Ρσ. 8. Σελεχτ τηε mοστ συιταβλε σεντενχε ωιτη ρεσπεχτ το γραmmαρ ανδ υσαγε. (Α) Τηε πριχε οφ αν αππλε ισ γρεατερ τηαν αν ονιον. (Β) Τηε πριχε οφ αν αππλε ισ mορε τηαν ονιον. (Χ) Τηε πριχε οφ αν αππλε ισ γρεατερ τηαν τηατ οφ αν ονιον. (D) Αππλεσ αρε mορε χοστλιερ τηαν ονιονσ. Θ.2 Τηε Βυδδηα σαιδ, Ηολδινγ ον το ανγερ ισ λικε γρασπινγ α ηοτ χοαλ ωιτη τηε ιντεντ οφ τηροωινγ ιτ ατ σοmεονε ελσε; ψου αρε τηε ονε ωηο γετσ βυρντ. Σελεχτ τηε ωορδ βελοω ωηιχη ισ χλοσεστ ιν mεανινγ το τηε ωορδ υνδερλινεδ αβοϖε. (Α) βυρνινγ (Β) ιγνιτινγ (Χ) χλυτχηινγ (D) φλινγινγ Θ.3 M ηασ α σον Q ανδ α δαυγητερ R. Ηε ηασ νο οτηερ χηιλδρεν. E ισ τηε mοτηερ οφ P ανδ δαυγητερ ιν λαω οφ M. Ηοω ισ P ρελατεδ το M? (Α) P ισ τηε σον ιν λαω οφ M. (Β) P ισ τηε γρανδχηιλδ οφ M. (Χ) P ισ τηε δαυγητερ ιν λαω οφ M. (D) P ισ τηε γρανδφατηερ οφ M. Θ.4 Τηε νυmβερ τηατ λεαστ φιτσ τηισ σετ: (324, 441, 97 ανδ 64) ισ. (Α) 324 (Β) 441 (Χ) 97 (D) 64 Θ.5 Ιτ τακεσ 10 σ ανδ 15 σ, ρεσπεχτιϖελψ, φορ τωο τραινσ τραϖελλινγ ατ διφφερεντ χονσταντ σπεεδσ το χοmπλετελψ πασσ α τελεγραπη ποστ. Τηε λενγτη οφ τηε φιρστ τραιν ισ 120 m ανδ τηατ οφ τηε σεχονδ τραιν ισ 150 m. Τηε mαγνιτυδε οφ τηε διφφερενχε ιν τηε σπεεδσ οφ τηε τωο τραινσ (ιν m/σ) ισ. (Α) 2.0 (Β) 10.0 (Χ) 12.0 (D) /3

2 ΓΑΤΕ 2016 Q. 6 Q. 10 carry two marks each. Γενεραλ Απτιτυδε ΓΑ Σετ 4 Θ.6 Τηε ϖελοχιτψ ς οφ α ϖεηιχλε αλονγ α στραιγητ λινε ισ mεασυρεδ ιν m/σ ανδ πλοττεδ ασ σηοων ωιτη ρεσπεχτ το τιmε ιν σεχονδσ. Ατ τηε ενδ οφ τηε 7 σεχονδσ, ηοω mυχη ωιλλλ τηε οδοmετερ ρεαδινγ ινχρεασε βψ (ιν m)? (Α) 0 (Β) 3 Θ.7 Τηε οϖερωηελmινγ νυmβερ οφ πεοπλε ινφεχτεδ ωιτη ραβιεσ ιν Ινδια ηασ βεεν φλαγγεδ βψ τηε Wορλδ Ηεαλτη Οργανιζατιον ασ α σουρχε οφ χονχερν. Ιτ ισ εστιmατεδ τηαττ ινοχυλατινγ 70% οφ πετσ ανδ στραψ δογσ αγαινστ ραβιεσ χαν λεαδ το α σιγνιφιχαντ ρεδυχτιον ιν τηε νυmβερ οφ πεοπλε ινφεχτεδ ωιτη ραβιεσ. Wηιχη οφ τηε φολλοωινγ χαν βε λογιχαλλψ ινφερρεδ φροm τηε αβοϖε σεντενχεσ? (Α) Τηε νυmβερ οφ πεοπλε ιν Ινδια ινφεχτεδ ωιτη ραβιεσ ισ ηιγη. (Β) Τηε νυmβερ οφ πεοπλε ιν οτηερ παρτσ οφ τηε ωορλδ ωηο αρε ινφεχτεδ ωιτη ραβιεσ ισ λοω. (Χ) Ραβιεσ χαν βε εραδιχατεδ ιν Ινδια βψ ϖαχχινατινγ 70% οφ στραψ δογσ. (D) Στραψ δογσ αρε τηε mαιν σουρχε οφ ραβιεσ ωορλδωιδε. Θ.8 Α φλατ ισ σηαρεδ βψ φουρ φιρστ ψεαρ υνδεργραδυατε στυδεντσ. Τηεψ αγρεεδ το αλλοω τηε ολδεστ οφ τηεm το ενϕοψ σοmε εξτρα σπαχε ιν τηε φλατ. Μανυ ισ τωο mοντησ ολδερ τηαν Σραϖαν, ωηο ισ τηρεε mοντησ ψουνγερ τηαν Τριδεεπ. Παϖαν ισ ονε mοντη ολδερ τηαν Σραϖαν. Wηο σηουλδ οχχυπψ τηε εξτρα σπαχε ιν τηε φλατ? (Χ) 4 (D) 5 (Α) Μανυ (Β) Σραϖαν (Χ) Τριδεεπ (D) Παϖαν Θ.9 Φινδ τηε αρεα βουνδεδ βψ τηε λινεσ 3ξ+2ψ=14, 2ξ 3ψ=5 ιν τηε φιρστ θυαδραντ. (Α) (Β) (Χ) (D) /3

3 ΓΑΤΕ 2016 Γενεραλ Απτιτυδε ΓΑ Σετ 4 Θ.10 Α στραιγητ λινε ισ φιτ το α δατα σετ (λν x, y). Τηισ λινε ιντερχεπτσ τηε αβσχισσα ατ λν x = 0.1 ανδ ηασ α σλοπε οφ Wηατ ισ τηε ϖαλυε οφ y ατ x = 5 φροm τηε φιτ? (Α) (Β) (Χ) (D) END OF THE QUESTION PAPER 3/3

4 List of Symbols Notations and Data iid 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 ν Set of real numbers Set of complex numbers ν dimensional Euclidean space Usual metric δ on is given by Normed linear space of all squaresummable real sequences Set of all real valued continuous functions on the interval Conjugate transpose of the matrix Μ Transpose of the matrix Μ Ιδ )dentity matrix of appropriate order Range space of Μ Null space of Μ Orthogonal complement of the subspace W MA 1/16

5 Θ. 1 Θ. 25 χαρρψ ονε mαρκ εαχη. Q Let be a basis of Consider the following statements P and Q P is a basis of Q is a basis of Which of the above statements hold TRUE A both P and Q C only Q B only P D Neither P nor Q Q Q Consider the following statements P and Q P )f then M is singular. Q Let Σ be a diagonalizable matrix )f Τ is a matrix such that Σ + 5 Τ = Ιδ then Τ is diagonalizable Which of the above statements hold TRUE A both P and Q C only Q Consider the following statements P and Q B only P D Neither P nor Q P )f Μ 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 C only Q B only P D Neither P nor Q MA 2/16

6 Q Consider a real vector space ς of dimension ν and a nonzero linear transformation )f dimension and for some then which of the following statements is TRUE A determinant B There exists a nontrivial subspace of ς such that for all C T is invertible 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 Then Q lim is equal to Maximum is equal to Q Let such that Then the Cauchy problem has a unique solution if A C on B D MA 3/16

7 Q Let be the dalemberts solution of the initial value problem for the wave equation where χ is a positive real number and are smooth odd functions Then is equal to Q Let the probability density function of a random variable Ξ be otherwise Then the value of χ is equal to Q Let ς be the set of all solutions of the equation satisfying where are positive real numbers Then dimensionς is equal to Q Let where and are continuous functions )f and are two linearly independent solutions of the above equation then is equal to Q Let be the Legendre polynomial of degree and is a nonnegative integer Consider the following statements P and Q P if Q if is an odd integer Which of the above statements hold TRUE A both P and Q where κ C only Q B only P D Neither P nor Q MA 4/16

8 Q Consider the following statements P and Q P has two linearly independent Frobenius series solutions near Q has two linearly independent Frobenius series solutions near Which of the above statements hold TRUE A both P and Q C only Q B only P D Neither P nor Q Q Let the polynomial be approximated by a polynomial of degree which interpolates at and Then the maximum absolute interpolation error over the interval is equal to Q Let be a sequence of distinct points in with lim Consider the following statements P and Q P There exists a unique analytic function φ on such that for all ν Q There exists an analytic function φ on such that if ν is even and if ν is odd Which of the above statements hold TRUE A both P and Q C only Q B only P D Neither P nor Q Q Let be a topological space with the cofinite topology Every infinite subset of is A Compact but NOT connected B Both compact and connected C NOT compact but connected D Neither compact nor connected MA 5/16

9 Q Let and Then dimension is equal to Q Consider where maximum Let be defined by and the norm preserving linear extension of to Then is equal to Q is called a shrinking map if for all and a contraction if there exists an such that for all 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

10 Q Let be a random sample from the following probability density function for 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 exist D Maximum likelihood estimator of Neither nor exists Q Suppose Ξ and Ψ are two random variables such that is a normal random variable for all Consider the following statements P Q R and S P Ξ is a standard normal random variable Q The conditional distribution of Ξ given Ψ is normal R The conditional distribution of Ξ given is normal S has mean Which of the above statements ALWAYS hold TRUE A both P and Q C both Q and S Q Consider the following statements P and Q B both Q and R D both P and S P )f is a normal subgroup of order of the symmetric group then is abelian Q )f is the quaternion group then is abelian Which of the above statements hold TRUE A both P and Q C only Q B only P D Neither P nor Q Q Let be a field of order Then the number of nonzero solutions of the equation is equal to MA 7/16

11 Θ. 26 Θ. 55 χαρρψ τωο mαρκσ εαχη. Q Let be oriented in the counterclockwise direction Let Then the value of is equal to Q Let be a harmonic function and its harmonic conjugate )f then is equal to Q Let be the triangular path connecting the points and in the counter clockwise direction in Then is equal to Q Let ψ be the solution of Then is equal to A C B D Q Let Ξ be a random variable with the following cumulative distribution function Then is equal to MA 8/16

12 Q Let be the curve which passes through and intersects each curve of the family orthogonally Then also passes through the point A C B D Q Let be the Fourier series of the periodic function defined by Then is equal to Q Let be a continuous function on )f Q then Let and A ln C ln is equal to Q For any let Q distance Then is equal to B ln D ln inimum Then is equal to Let and Then is equal to MA 9/16

13 Q Let be a real matrix with eigenvalues and )f the eigenvectors corresponding to and are and respectively then the value of is equal to Q Let and )f then is equal to Q Let the integral where Consider the following statements P and Q P )f is the value of the integral obtained by the composite trapezoidal rule with two equal subintervals then is exact Q )f is the value of the integral obtained by the composite trapezoidal rule with three equal subintervals then is exact Which of the above statements hold TRUE A both P and Q C only Q B only P D Neither P nor Q Q The difference between the least two eigenvalues of the boundary value problem is equal to Q The number of roots of the equation cos in the interval is equal to MA 10/16

14 Q For the fixed point iteration consider the following statements P and Q P )f then the fixed point iteration converges to for all Q )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 B only P D Neither P nor Q Q Let be defined by Then A B but bounded C D is unbounded Q Minimize subject to Then the minimum value of is equal to Q Maximize subject to Then the maximum value of is equal to MA 11/16

15 Q Let be a sequence of iid random variables with mean )f Ν is a geometric random variable with the probability mass function and it is independent of the s then is equal to Q Let be an exponential random variable with mean and a gamma random variable with mean and variance )f and are independently distributed then is equal to Q Let be a sequence of iid uniform random variables Then the value of is equal to lim ln ln Q Let Ξ be a standard normal random variable Then is equal to A C B D Q Let be a random sample from the probability density function otherwise where 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 for some for some D Uniformly Most Powerful test is of the form for some MA 12/16

16 Q Let be a sequence of iid random variables Then is equal to lim Q Let be a random sample from uniform for some )f Maximum then the UMVUE of is A C B D Q Let be a random sample from a Poisson random variable with mean where Then the maximum likelihood estimator of is equal to Q The remainder when is divided by is equal to Q Let be a group whose presentation is Then is isomorphic to A C B D MA 13/16

17 ΕΝD ΟΦ ΤΗΕ ΘΥΕΣΤΙΟΝ ΠΑΠΕΡ MA 14/16

18 MA 15/16

19 MA 16/16