CSIR NET Maths previous year question papers

Let A = {$p/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n}$ for some $n \in \mathbb{N} \cup$ {0}, gcd(p, q) = 1}, B = {$p/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n}5^{m}$ for some $n, m \in \mathbb{N} \cup$ {0}, gcd(p, q) = 1} and C = {p/q ∈ (0, 1) : p/q has terminating decimal expansion} be subsets of (0, 1). Which of the following statements is true?

Let A, B be non-empty subsets of $\mathbb{N}$ with cardinality |A| ≥ 2. Let $S_{1} =$ {f : A → B | f is one-to-one} and $S_{2} =$ {g : B → A | g is onto}. Which of the following statements is true?

Consider the real matrix A whose rows are (29, 0, 55, 17), (1, 28, 46, 26), (17, 13, 33, 38) and (21, 67, 0, 13). What is the largest real eigenvalue of A?

Let $C[0, \pi]$ be the real vector space of real-valued continuous functions on the closed interval $[0, \pi]$. For positive integers n, define $f_{n} \in C[0, \pi]$ by $f_{n}(x) = \sin($nx)/sin x if $x \in (0, \pi), f_{n}(0) = n$, and $f_{n}(\pi) = (-1)^{n}$⁻$^{1}n$. Let V be the real subspace of $C[0, \pi]$ spanned by {$f_{1}, f_{2}, f_{3}$}. Consider the inner product on V given by ⟨f, g⟩ $= (2/\pi)\int_{0}^\pi f(x)g(x)\sin^{2}x$ dx. Which of the following statements is true?

Let V = {ax$^{3} +$ bx$^{2} +$ cx | $a, b, c \in \mathbb{R}$}. For f ∈ V, define $Q(f) = \int$₋$_{1}^{1} (f'(t))^{2}$ dt, where f′ denotes the derivative of f. Which of the following statements is FALSE?

Let S be a subset of the open interval (0,1) that consists of all the real numbers $\alpha \in (0,1)$ whose infinite decimal expansion $\alpha = 0.a_{1}a_{2}a_{3}\cdots$ is such that $a_{i} \in$ {0,2,4} for all i ≥ 1. Which of the following statements is true?

Consider the quadratic form f(x,y,z) = [x y z] A [x;y;z], where A is an invertible 3×3 symmetric matrix over $\mathbb{Q}$. Assume that there exists $(\alpha,\beta,\gamma) \in \mathbb{C}^{3}${(0,0,0)} such that $f(\alpha,\beta,\gamma) = 0$. Which of the following statements is necessarily true?

Which of the following statements is necessarily true?

Let A, B be two non-empty subsets of $\mathbb{R}$. Let S = {$f : A \to \mathbb{R}$ | f is continuous}, T = {$f : B \to \mathbb{R}$ | f is continuous}, U = {$f : A \times A \to \mathbb{R}$ | f is continuous}. Which of the following statements is true?

Let $T : \mathbb{R}^{4} \to \mathbb{R}^{4}$ be an $\mathbb{R}-$linear transformation such that $(T^{2} + T + I)(T - 2I) = 0$ and $(T^{2} + T + 2I)(T^{2} - 4I) = 0$. Which of the following statements is FALSE?

Consider the real matrix A = [[0, 0, 1], [0, 1, 0], [1, 0, 0]]. Define $B : \mathbb{R}^{3} \times \mathbb{R}^{3} \to \mathbb{R}$ by B(v, w) = vᵗAw. 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?

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?

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?

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)?