CSIR NET Maths previous year question papers

Let $\mathbb{N}$ be the set of positive integers. Consider $\mathbb{R}^{2}$ with the Euclidean topology and the subsets A = {$(n, 1/n) : n \in \mathbb{N}$} and B = {$(n, 1/m) : n, m \in \mathbb{N}$}. Which of the following statements is true?

Let A, B, and C be sets. Which of the following sets is equal to A \ (B \ C)?

For integers n ≥ 0, let $f_{n} : [-1, 0] \to \mathbb{R}$ be defined by $f_{n}(x) = x/(1 - x)^{n}$. Which of the following statements is true about the series $\sum_(n=0)^\infty f_{n}$?

Consider the sequences $(a_{n})_{n}\ge_{1}$ and $(b_{n})_{n}\ge_{1}$ defined by $a_{n} = (e^{n} + e$⁻$^{n})/2$ and $b_{n} = a_{n}$₊$_{1}/a_{n}$. Which of the following statements is true?

Let $f : [0, 1] \to \mathbb{R}$ be defined by $f(x) = \sin(x^{2})$. Let $A = \lim(n\to\infty) (\sum_(k=1)^n f(k/n) - n\int_{0}^{1}f(x)$dx). Which of the following statements is true?

Consider the power series $\sum_(n=1)^\infty [n^(n^{2})/(n + 1)^(n^{2})]x^{n}$ with coefficients in real numbers $\mathbb{R}$. 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 $v = (a, b, c) \in \mathbb{R}^{3}$ be a nonzero vector that lies in the orthogonal complement (with respect to the standard inner product) of the row-space of the matrix A = [[2, 2, 7], [3, 1, 4]]. If a, b, c are all integers, then what is the smallest possible value of |a + b + c|?

Let A = [[0, a, 0], [0, 0, b], [c, 0, 0]], where a, b, c are real numbers with abc = 1. If $B = A + A^{2} + A^{3}$, then which of the following statements is true?

For a variable x, consider the $\mathbb{R}-$vector space V = {$a_{0} + a_{1}x + a_{2}x^{2}$ | $a_{1}, a_{2}, a_{3} \in \mathbb{R}$}. Let T : V → V be the linear transformation defined by T(f) = f + df/dx, where df/dx denotes the derivative of f with respect to x. Which of the following statements is true?

Let V be the $\mathbb{R}-$vector space of 5 × 5 real matrices. Let S = {AB − BA | A, B ∈ V} and W denote the subspace of V spanned by S. Let $T : V \to \mathbb{R}$ be the linear transformation mapping a matrix A to its trace. Which of the following statements is true?

Consider the bilinear form $B : \mathbb{R}^{4} \times \mathbb{R}^{4} \to \mathbb{R}$ defined by $B(x, y) = x_{1}y_{3} + x_{2}y_{4} - x_{3}y_{1} - x_{4}y_{2}$, where $x = (x_{1}, x_{2}, x_{3}, x_{4})$ and $y = (y_{1}, y_{2}, y_{3}, y_{4})$ in $\mathbb{R}^{4}$. Let A denote the matrix of B with respect to the standard ordered basis of $\mathbb{R}^{4}$. Which of the following statements is true?

Consider the set A = {$x \in \mathbb{Q} : 0 < (\sqrt{2} - 1)x < \sqrt{2} + 1$} as a subset of $\mathbb{R}$. Which of the following statements is true?

Let S = {$x \in \mathbb{R} : x > 1$ and $(1 - x^{4})/(1 - x^{3}) > 22$}. Which of the following is true about S?

Let C be the collection of all sets S such that the power set of S is countably infinite. Which of the following statements is true?

Let $(a_{n})_{n}\ge1$ be a bounded sequence in $\mathbb{R}$. Which of the following statements is FALSE?

What is the cardinality of the set of real solutions of eˣ + x = 1?

Let S be a dense subset of $\mathbb{R}$ and $f : \mathbb{R} \to \mathbb{R} a$ given function. Define $g : S \to \mathbb{R}$ by g(x) = f(x). Which of the following statements is necessarily true?

For each n ≥ 1 define $f_{n} : \mathbb{R} \to \mathbb{R}$ by $f_{n}(x) = x^{2} / \sqrt{x^{2} + 1/n}, x \in \mathbb{R}$, where $\surd$ denotes the non-negative square root. Wherever $lim_{n}\to\infty f_{n}(x)$ exists, denote it by f(x). Which of the following statements is true?

Let $A : \mathbb{R}^{m} \to \mathbb{R}^{n}$ be a non-zero linear transformation. Which of the following statements is true?

Let A be a 10 × 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true?

Let ((2, a), (b, c)) be a 2 × 2 real matrix for which 6 is an eigenvalue. Which of the following statements is necessarily 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?

Let A = ((0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 1, 1), (0, 0, 1, 1)), and consider the symmetric bilinear form on $\mathbb{R}^{4}$ given by ⟨v, w⟩ = vᵗAw, for $v, w \in \mathbb{R}^{4}$. Which of the following statements is true?

For a quadratic form $f(x, y, z) \in \mathbb{R}[x, y, z]$, we say that $(a, b, c) \in \mathbb{R}^{3}$ is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)?