CSIR NET Maths previous year question papers

Let X be the $\mathbb{R}-$vector space of all twice differentiable real valued functions on [0, 1]. Consider the linear map $\varphi : X \to \mathbb{R}^{3}$ defined by $\varphi(f) = (f(1), f'(1), f''(1))$. Which of the following statements is true?

Let V be a vector space over $\mathbb{R}$. Let $T_{1}, T_{2} : V \to V$ be two $\mathbb{R}-$linear transformations such that $T_{1} + T_{2}$ and $T_{1} - T_{2}$ are linearly independent over $\mathbb{R}$. Consider the following statements: (A) The transformations $T_{1}$ and $T_{2}$ are linearly independent over $\mathbb{R}. (B)$ There exist $\mathbb{R}-$linear transformations $T_{3}, T_{4} : V \to V$ such that {$T_{1} + T_{2}, T_{1} - T_{2}, T_{3}, T_{4}$} is linearly independent over $\mathbb{R}$. Which of the following statements is true?

Let U denote the span of {eᵗ, $e^{2}$ᵗ, $e^{3}$ᵗ} in the real vector space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Consider the $\mathbb{R}-$vector spaces V = {$f : U \to \mathbb{R}$ | f is an $\mathbb{R}-$linear transformation} and W = {f ∈ V | $f(e^{3}$ᵗ) = 0}. Which of the following statements is true?

Let V be the real vector space of 2 × 2 matrices with entries in $\mathbb{R}$. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where A = ((2, 0), (0, 1)). What is the characteristic polynomial of T?

Suppose S is an infinite set. Assuming that the axiom of choice holds, which of the following is true?

Let $I_{n}$ denote the n × n identity matrix. Which one of the following statements is true?