CSIR NET Maths previous year question papers

Let X be the image of the interval [0, 1] under the Möbius transformation f(z) = (z − i)/(z + i). Which of the following statements is true?

Let $M, N \in M_{3}(\mathbb{C})$ be such that Trace(Mᵏ) = Trace(Nᵏ), 1 ≤ k ≤ 3. Which of the following statements is necessarily true?

Let A be a 3 × 3 complex matrix such that $A^{3}$ is the identity matrix. Which of the following statements is true?

Let A be a 3×3 real matrix whose characteristic polynomial p(T) is divisible by $T^{2}$. Which of the following statements is true?

Let T be a linear operator on $\mathbb{R}^{3}$. Let $f(X) \in \mathbb{R}[X]$ denote its characteristic polynomial. Consider the following statements. (a) Suppose T is non-zero and 0 is an eigenvalue of T. If we write f(X) = X·g(X) in $\mathbb{R}[X]$, then the linear operator g(T) is zero. (b) Suppose 0 is an eigenvalue of T with at least two linearly independent eigenvectors. If we write f(X) = X·g(X) in $\mathbb{R}[X]$, then the linear operator g(T) is zero. Which of the following is true?