CSIR NET Maths previous year question papers

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?

Let $f : \mathbb{C} \to \mathbb{C}$ be the function defined by f(z) = e^((cos(1+i)) sin z). For z = x + iy $\in \mathbb{C}$, write f(z) as u(x, y) + iv(x, y), where u, v are real-valued functions. Which of the following is the value of $(\partial{}u/\partial{}x)(0, 0)$?

Let 𝔻 = {z = x + iy $\in \mathbb{C}$ : |z| < 1} be the open unit disc and f : 𝔻 $\to \mathbb{C} a$ holomorphic function such that f(0) = 0. Let $\psi(z) =$ |f(z)|$^{2}$, and $\partial^{2}\psi/\partial{}x^{2} + \partial^{2}\psi/\partial{}y^{2} \equiv 0$. Which of the following statements is FALSE?

Let ℍ = {z = x + iy $\in \mathbb{C}$ | y > 0} and f : ℍ $\to \mathbb{C}$ be a non-constant holomorphic function satisfying |f(z)| < 1 for all z ∈ ℍ. Which of the following statements is true?

For integers m, n ≥ 1, let $I_(m,n) = (1/2\pi{}i)\int_C z^{m} {\bar{z}}^{n}$ dz, where C is the circle {$z \in \mathbb{C}$ : |z| = 1} oriented counterclockwise. Which of the following statements is true?

For integers n > 1, let G(n) denote the number of groups of order n, up to isomorphism, i.e. G(n) is the number of isomorphism classes of groups of order n. Which of the following statements is true?

We say that a group G has property (A) if every non-trivial homomorphism from G to any group is injective. Which of the following groups has property (A)?

Let $\mathbb{C}[x, y]$ be the polynomial ring in two variables over $\mathbb{C}$. For which of the following ideals I, the quotient ring $\mathbb{C}[x, y]/I$ is NOT an integral domain?

Which of the following statements is true?

Suppose that the differential equation $d^{2}y/$dx$^{2} + P(x)$dy/dx $+ e^(2x)y = 0, x \in \mathbb{R}$ transforms into a second order differential equation with constant coefficients under the change of independent variable given by s = s(x) satisfying (ds/dx)(0) = 1. Then which of the following statements is true?

Given that $y_{1}(x) = e^(2x)$ is a solution of the ordinary differential equation (ODE$) x d^{2}y/$dx$^{2} - (3 + 4x)$dy/dx + (4x + 6)y = 0, x > 0. Let $y_{2} = y_{2}(x)$ be the solution of the ODE satisfying the conditions $y_{2}(1) = e^{2}/4$ and (dy$_{2}/$dx$)(1) = 3e^{2}/2$. Then which of the following statements is true?

Let u = u(x, y) be the solution of the Cauchy problem $x \partial{}u/\partial{}x + y \partial{}u/\partial{}y = u, (x, y) \ne (0, 0)$, with $u(x, 1) = \sqrt{1 + x^{2}}, x \in \mathbb{R}$. Then which of the following statements is true?

Let u = u(x, t) be a solution of the wave equation $\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0$ satisfying the condition u(0, t) = 0, ∀t ≥ 0. Then which of the following statements is true?

Let $f : \mathbb{R} \to \mathbb{R}$ be such that sup(x≠y) |f(x) − f(y)|/|x − y| = L, where $1 < L < \infty$. Let $h : \mathbb{R} \to \mathbb{R}$ be a differentiable function satisfying |h′(x)| ≤ 3/4 for all $x \in \mathbb{R}$. For $\alpha > 0$, define $g(x) = \alpha{}f(x) + h(x)$ for $x \in \mathbb{R}$. Consider the sequence {$x_{k}$}(k≥0) defined by $x_(k+1) = g(x_{k}), k = 0, 1, \dots$, where $x_{0} \in \mathbb{R}$. The sequence {$x_{k}$}(k≥0) converges to the solution of the equation x = g(x) if

Let S denote the set of all solutions of the Euler-Lagrange equation of the variational problem: minimize $J[y] = \int_{0}^{1}(y^{2} + (y')^{2})$dx, subject to $y(0) = 0, y(1) = 0, \int_{0}^{1}y^{2}$dx = 1. Then the set {$\varphi(1/2) : \varphi \in S$} is equal to

Let $\lambda \in \mathbb{R}$, and $K : [0, 1] \times [0, 1] \to \mathbb{R}$ be a function such that every solution of the boundary value problem $(d^{2}u/$dx$^{2})(x) + \lambda{}u(x) = 0; ($du/dx)(0) = u(0), (du/dx)(1) = 0 satisfies the integral equation $u(x) + \lambda\int_{0}^{1}K(x, t)u(t)$dt = 0. Then

Two blocks of equal mass m are connected by a flexible inelastic cord of mass M. One block is placed on a smooth horizontal table, the other block hangs over the edge. The total potential energy of the entire cord is given by (−Mg$/2l)x^{2}$, where x is the distance of the hanging block from the edge of the table, l is the length of the cord, and g is the gravitational acceleration. Then

$\lim(n\to\infty) \int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1} (x_{1}^{2} + \cdots + x_{n}^{2})/(x_{1} + \cdots + x_{n})$ dx$_{1} \dots$ dx$_{n}$ equals

Let $U_{1}, U_{2}, \dots, U_{5}$ be 5 urns such that urn U_k contains $2k + k^{2}$ balls, out of which 2k are white balls and $k^{2}$ are black balls, k = 1, 2, …, 5. An urn is selected with probability of selecting urn U_k being proportional to (k + 2). A ball is chosen randomly from the selected urn. Then, the probability that the urn $U_{5}$ was selected, given that the ball drawn is white, is equal to

In an examination question paper, all questions are 'True' or 'False' type. These are arranged in such a way that three-fourth of times a question with answer 'True' is followed by a question with answer 'True'. Also two-third of times a question with answer 'False' is followed by a question with answer 'False'. If the question paper has 100 questions, the approximate probability that the correct answer of the 100-th question is 'True', is