CSIR NET Maths previous year question papers

Let p, q be non-negative integers. Consider the following statements: (A) There is an integer k ≥ 1 such that p + k = q. (B) There is an integer k ≥ 1 such that q + k = p. Which of the following statements is true?

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?

Let $f : \mathbb{R}$ \ $\mathbb{Q} \to \mathbb{R}$ \ $\mathbb{Q}$ be the function defined as f(x) = (3x + 2)/(4x + 3). Let $x_{1} \in \mathbb{R}$ \ $\mathbb{Q}$. For n ≥ 1, define $x_{n}$₊$_{1} = f(x_{n})$. Suppose that the sequence $(x_{n})_{n}\ge_{1}$ converges to a real number $\ell$. Which of the following statements is true?

Let $f(x) = x log_{e}(1 + 1/x)$ for $x \in (0, \infty)$. Which of the following statements is true?

For each n ≥ 1, let $f_{n} : [0, 1] \to \mathbb{R}$ be defined as $f_{n}(x) =$ nx if $x \in [0, 1/n], f_{n}(x) = 2 -$ nx if x ∈ (1/n, 2/n), and $f_{n}(x) = 0$ if x ∈ [2/n, 1]. Which of the following statements is true?

Which of the following polynomials is the characteristic polynomial of a real 2 × 2 matrix A such that trace(A) = 7 and trace$(A^{2}) = 29$?

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 𝔽$_{5}$ denote the field with 5 elements. How many 2 × 2 matrices with entries in 𝔽$_{5}$ have rank one?

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 $C[0, \pi]$ 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 $f : \mathbb{C} \to \mathbb{C}$ be a polynomial map. For R > 0, let $\gamma_R : [0, 1] \to \mathbb{C}$ be the map t ↦ $Re^{2\pi{}it}$. Suppose that there exists $c \in \mathbb{R}$ such that $\int_{0}^{1}$ |(f ∘ $\gamma_R)(t) \gamma'_R(t)$| dt → c as $R \to \infty$. Which of the following statements is FALSE?

Let f be an entire function such that $f(\mathbb{C}) \subset$ {x + iy | y = x + 1}. Which of the following statements is true?

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?

Which of the following statements is true?

Which of the following statements is true?

Let A be a subring of the field of rationals $\mathbb{Q}$ such that for any nonzero rational $r \in \mathbb{Q}, r \in A$ or 1/r ∈ A. Which of the following statements is FALSE?

Which of the following statements is true?

Let S = {1, 2, 3, 4, 5} be equipped with the topology $\tau =$ {∅, {1}, S}. What is the number of homeomorphisms of S onto itself?

If $\varphi(x) = x$ is a solution of the ordinary differential equation (ODE$) d^{2}y/$dx$^{2} - (2/x^{2} + 1/x)(x$ dy/dx $- y) = 0, 0 < x < \infty$, then the general solution of the ODE is given by

Let $(\lambda_{n})_{n}\in\mathbb{N}$ be the sequence of eigenvalues of the Sturm-Liouville problem (d/dx)(x dy/dx$) + (\lambda/x)y = 0, 1 < x <$ $e^{2\pi}$, y(1) = 0, $y(e^{2\pi})$ = 0. Then $\sum_{n}$₌$_{1}^\infty 1/\lambda_{n}$ is equal to

Let u = u(x, y) be the solution to the Cauchy problem $(y + u)\partial{}u/\partial{}x + y \partial{}u/\partial{}y = x - y, x \in \mathbb{R}, y > 0, u(x, 1) = 1 + x, x \in \mathbb{R}$. Then which of the following statements is true?

Let u = u(x, t) be the solution of $\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0, u(x, 0) = 1 + x^{2}, x \in \mathbb{R}, \partial{}u/\partial{}t(x, 0) = x + 1, x \in \mathbb{R}$. Then the value of u(1, 1) is

If the function $s : [0, 4] \to \mathbb{R}$ defined by $s(x) = a(x - 2)^{2} + b(x - 1)^{2}$ for $0 \le x \le 1, s(x) = (x - 2)^{2}$ for 1 < x ≤ 3, and $s(x) = 2c(x - 2)^{2} + (x - 3)^{3}$ for 3 < x ≤ 4, is a cubic spline, then the value of 2a + b + 2c is