CSIR NET Maths previous year question papers

For n≥3, consider the space $M_{n}(\mathbb{C})$ of n×n complex matrices endowed with the inner product ⟨A,B⟩ = Trace(A*B). For 0≤k≤n, let $W_{k} =$ {$A \in M_{n}(\mathbb{C})$ : ⟨A,B⟩=0 for all $B \in M_{n}(\mathbb{C})$ with rank k}. Which of the following statements are necessarily true?

There are six persons. On each day of the year 2024 at least one of them borrowed money from another. A person P is labeled troublesome in a calendar month M of 2024 if P borrowed from the same person at least twice during month M. Which of the following statements are necessarily true?

Let V be a real vector space. Suppose that {u,v,w,x,y} ⊆ V is a spanning set of V and that {u,v,x,y} is linearly independent. Which of the following statements are necessarily true?

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{1} x^{2}(y')^{2}$ dx subject to $y(0)=0, y(1)=1, \int_{0}^{1}y^{2}$dx = 1/7. Then which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{20}$ be a random sample from $N_{12}(\mu,\sum)$, where $\mu\in\mathbb{R}^{12}$ and $\sum$ is positive definite. Suppose ${\bar{X}} = (1/20)\sum_{i}$₌$_{1}^{20}X_{i}$ and $S = (1/19)\sum_{i}$₌$_{1}^{20}(X_{i}-{\bar{X}})(X_{i}-{\bar{X}})^{T}$. Then which of the following statements are true?

Let u(x,t) be the solution of the initial-boundary value problem $u_{tt} - u_{xx} + u_{t} = 0, 0<x<\pi, t>0, u(0,t)=0, u(\pi,t)=0, t>0, u(x,0)=\sin x, u_{t}(x,0)=0, 0<x<\pi$. For t≥0, define $E(t) = \int_{0}^\pi (u_{t}^{2}+u_{x}^{2})$ dx. Then which of the following statements are true?

Let S be the set of all 2×2 matrices A such that the iterative sequence generated by the Gauss-Seidel method converges for every initial guess, when employed to solve the system of equations $A(x_{1},x_{2})^{T}=(1,2)^{T}$. Then which of the following statements are true?

Consider a commutative ring R with unity with at least five elements such that for any two elements a,b∈R, there exists c∈R such that a=bc or b=ac. Which of the following statements are necessarily true?

Consider the following statements: (P) The initial value problem $y'+y=e^(-x^{2}), y(0)=0$ has a Taylor series solution about the point x=0. (Q) The initial value problem y′+y=r(x), y(0)=0, where $r(x)=e^(-1/x^{2})$ for x≠0 and r(0)=0, has a Taylor series solution about the point x=0. Then which of the following statements are true?

Let $F : \mathbb{R}^{3} \to \mathbb{R}$ be a continuously differentiable function such that $F(0,0,0)=-1, \partial{}F/\partial{}x(0,0,0)=2, \partial{}F/\partial{}y(0,0,0)=0$, and $\partial{}F/\partial{}z(0,0,0)=3$. For $\varepsilon>0$, define $\Omega_\varepsilon=(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon)\subset\mathbb{R}^{2}$. Which of the following statements are necessarily true?

For each positive integer n, let $f_{n}:[-1,1]\to\mathbb{R}$ be given by $f_{n}(x)=-1$ if $-1\le{}x\le-1/n, f_{n}(x)=$nx if −1/n<x<1/n, and $f_{n}(x)=1$ if 1/n≤x≤1. On which of the following intervals does the sequence {$f_{n}$}$_{n}\ge1$ converge uniformly?

Let (X,d) be a metric space. For a non-empty subset A of X, and x∈X, define d(x,A) = $inf_{a\in{}A}$ d(x,a). Which of the following statements are necessarily true?

Consider the following subset of real numbers A = {$(1+(-1)^{n})n - 1/n : n$ is a positive integer}. Which of the following statements are true?

Let $f : (0,\infty) \to (0,\infty)$ be the function defined by $f(x) = \int_{0}$ˣ $\sqrt{t}/(1+t^{2})$ dt, where $\sqrt{t}$ denotes the positive square root for t>0. Which of the following statements are true?

Consider the following real-valued functions $F_{1}$ and $F_{2}$ defined on $\mathbb{R}$, given by $F_{1}(x)=1$ if x≥1, 0 otherwise, and $F_{2}(x)=\int_{0}$ˣexp(−t)dt if x≥0, 0 otherwise. Define another function $F:\mathbb{R}\to[0,1]$ by $F(x)=(2/3)F_{1}(x)+(1/3)F_{2}(x)$ for all $x\in\mathbb{R}$. Which of the following statements are true?

Consider the following $2^{5}-$factorial design with 8 blocks (Block 1: (1), acd, bce, abde; Block 2: e, acde, bc, abd; and six further blocks). Which of the following statements are true?

Let $\mathbb{N}$ denote the set of all positive integers. For $n\in\mathbb{N}$, let $f_{n}:[0,1]\to\mathbb{R}$ be given by $f_{n}(x)=n(1-$nx) if 0≤x≤1/n, 0 if x>1/n. Which of the following statements are necessarily true?

Let f be an entire function. Consider the function g given by g(z) = f(z) − 1/z for $z\in\mathbb{C}${0}. Which of the following statements are necessarily true?

Let $\ell^{2}$ denote the vector space of all square summable sequences {$a_{n}$}$_{n}\ge1$ of real numbers with the inner product ⟨{$a_{n}$},{$b_{n}$}⟩ $= \sum_{n}$₌$_{1}^\infty a_{n}b_{n}$. Let H = {{$a_{n}$}$\in\ell^{2}$ : |$a_{n}$|≤1/n for all positive integers n}. Which of the following statements are true?

Let {$X_{n} : n\ge1$} be a sequence of independent and identically distributed random variables, where $X_{1}$ has an Exponential distribution with mean 1. Define $T_{n} = \max${$X_{1},X_{2},\dots,X_{n}$} − ln n, n≥1. Suppose $T_{n} \to$ᵈ Y as $n\to\infty$. Then which of the following statements are true?

Let f:[0,1]→[0,1] be a monotonically increasing function, that is, a≤b implies f(a)≤f(b). For any $\alpha\in(0,1)$, let $L_\alpha$⁺=$lim_{x\to\alpha^{+}}f(x)$ and $L_\alpha$⁻=$lim_{x\to\alpha^{-}}f(x)$ denote the right hand and left hand limits respectively, provided they exist. For $\alpha\in(0,1)$, if $L_\alpha$⁺ and $L_\alpha$⁻ exist, define $U_\alpha=(L_\alpha$⁻,$L_\alpha$⁺) if $L_\alpha$⁻$<L_\alpha$⁺, and ∅ if $L_\alpha$⁺$\le{}L_\alpha$⁻. Which of the following statements are true?

Suppose that the transition probability matrix of a homogeneous Markov chain with state space {1,2,3,4} is given by P, with row 1 = (1/4, 3/4, 0, 0), row 2 = (1, 0, 0, 0), row 3 = (1/8, 0, 7/8, 0), row 4 = (0, 0, 1/9, 8/9). Which of the following statements are true?

Let X and Y be two independent random variables such that the moment generating functions of X and Y are $M_X(t)=e^(3(e^t-1)), t\in\mathbb{R}$, and $M_Y(t)=((1/2)e^(-3t)+(1/2)e^(3t))^{2}, t\in\mathbb{R}$, respectively. Then which of the following statements are true?

Let A be a non-zero 3×3 matrix with integer entries. Let $\lambda_{i}\in\mathbb{C}, 1\le{}i\le3$ be all the eigenvalues of A (not necessarily distinct). Which of the following statements are necessarily true?

A mechanical system is described using generalized position q and generalized momentum p. Let Q and P denote new generalized position and generalized momentum variables respectively, generated by the generating function $F(q,P)=q^{2}e^P$, and Q,P are canonical coordinates. Let G(p,Q) be a function such that G(2,e)=0, and it generates the same canonical coordinates Q,P. Then which of the following statements are true?