Mathematics (MA) 2012

The straight lines L1: x = 0, L2: y = 0 and L3: x + y = 1 are mapped by the transformation w = z2 into the curves C1, C2 and C3 respectively. The angle of intersection between the curves at w = 0 is

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

Consider the following statements: P: The family of subsets An = (-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; 1, x ≠ y. The metric space (X, d) is compact. R: In a Frechet (T1) topological space, every finite set is closed. S: If f: R → X is continuous, where R is given the usual topology and (X, τ) is a Hausdorff (T2) space, then f is a one-one function. Which of the above statements are correct?

Let H be a Hilbert space and S⊥ denote the orthogonal complement of a set S ⊆ H. Which of the following is INCORRECT?

Let H be a complex Hilbert space, T: H → H be a bounded linear operator and let T* denote the adjoint of T. Which of the following statements are always TRUE? P: ∀x, y ∈ H, ⟨Tx, y⟩ = ⟨x, T*y⟩ Q: ∀x, y ∈ H, ⟨x, Ty⟩ = ⟨T*x, y⟩ R: ∀x, y ∈ H, ⟨Tx, T*y⟩ = ⟨x, y⟩ S: ∀x, y ∈ H, ⟨x, T*y⟩ = ⟨T*x, y⟩

Let X = a, b, c and let τ = φ, a, b, a, b, X be a topology defined on X. Then which of the following statements are TRUE? P: (X, τ) is a Hausdorff space. Q: (X, τ) is a regular space. R: (X, τ) is a normal space. S: (X, τ) is a connected space.