GATE 20172018 :: GATE Mathematics
 In a topological space, which of the following statements is NOT always true :

Consider the following statements: P: The family of subsets {A_{n} = (1/n, 1/n), n = 1, 2, ...} satisfies the finite intersection property.Q: On an infinite set X , a metric d : X * X > R is defined as d(x,y) = [0 , x = y and 1, x â‰ y] The metric space (X,d) is compact.R: In a Frechet (T_{1}) topological space, every finite set is closed.S: If f : R > X is continuous, where R is given the usual topology and (X, t) is a Hausdorff (T_{2}) space, then f is a oneone function.Which of the above statements are correct?
 A simple random sample of size 10 from 2 N(Î¼,Ïƒ^{2}) gives 98% confidence interval (20.49, 23.51). Then the null hypothesis H_{0} : Î¼ = 20.5 against H_{A} : Î¼ â‰ 20.5

For the linear programming problem Maximize z = x_{1} + 2x_{2} + 3x_{3}  4x_{4}Subject to 2x_{1} + 3x_{2}  x_{3}  x_{4} = 156x_{1} + x_{2} + x_{3}  3x_{4} = 218x_{1} + 2x_{2} + 3x_{3}  4x_{4} = 30x_{1}, x_{2}, x_{3}, x_{4} â‰¥ 0,x_{1} = 4, x_{2} = 3, x_{3} = 0, x_{4} = 2 is
 Which one of the following statements is TRUE?
 Let V = â„‚^{2} be the vector space over the field of complex numbers and Bï€½{(1, i), (i,1)}be a given ordered basis of V. Then for which of the following, B* = {f_{1}, f_{2}}is a dual basis of B over â„‚?
 Let R = â„¤*â„¤*â„¤ and I = â„¤*â„¤*{0}. Then which of the following statement is correct?

The function u(r, Î¸) satisfying the Laplace equationsubject to the conditions u(e, Î¸) = 1, u(e^{2} ,Î¸) = 0 is

If a transformation y = uv transforms the given differential equation f(x)y^{''}  4f^{'}(x)y^{'} + g(x)y = 0 into the equation of the form v^{''} + h(x)v = 0, then u must be
A.
f_{1}(z_{1}, z_{2}) = 1/2 (z_{1}  iz_{2}), f_{2}(z_{1}, z_{2}) = 1/2 (z_{1} + iz_{2})

B.
f_{1}(z_{1}, z_{2}) = 1/2 (z_{1} + iz_{2}), f_{2}(z_{1}, z_{2}) = 1/2 (iz_{1} + z_{2})

C.
f_{1}(z_{1}, z_{2}) = 1/2 (z_{1}  iz_{2}), f_{2}(z_{1}, z_{2}) = 1/2 (iz_{1} + z_{2})

D.
f_{1}(z_{1}, z_{2}) = 1/2 (z_{1} + iz_{2}), f_{2}(z_{1}, z_{2}) = 1/2 (iz_{1}  z_{2})
