CSIR NET Maths previous year question papers

Let V be a real vector space and L(V) denote the space of linear operators on V. Let T∈L(V) be a non-zero operator such that $T^{2}=T$. Consider the subspace W of L(V) spanned by {$I, T^{n} : n$ is a positive integer}. Which of the following statements are necessarily true?

Let G be a finite non-abelian group. Which of the following statements are necessarily true?

Consider a series system comprising four components, having independent and identically distributed lifetimes with hazard rate $\lambda(t)=1/(1+t), t>0$, and survival function S(t), t>0. If Y denotes the lifetime of the series system, then which of the following statements are true?

Let a,b be distinct positive real numbers. Consider the function $f:\mathbb{R}^{2}\to\mathbb{R}$ given by f(x,y)=(ax+by$)^{2}/($ax$^{2}+$by$^{2})$ if (x,y)≠(0,0), and 0 if (x,y)=(0,0). Which of the following statements are necessarily true?

Let $X_{1},X_{2},\dots,X_{n} (n>5)$ be independent random variables such that $X_{t}=\alpha+\alpha^{2}t+\varepsilon_{t}$, for t=1,…,n, where $\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{n}$ are independent and identically distributed $N(0,\sigma^{2})$ random variables. Here $\alpha\in\mathbb{R}$ and $\sigma>0$ are unknown parameters. Which of the following statements are true?

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

Let $Y=X\beta+\varepsilon$ be a multiple linear regression model with p regressors and an intercept, where $\beta=(\beta_{0},\beta_{1},\dots,\beta_{p})^{T}$ and the random error $\varepsilon$~$N_{n}(0,\sigma^{2}I), \sigma>0$ and n>p+1. The least squares method provides a unique estimator $\beta$̂. Let the total sum of squares (corrected), sum of squares due to regression, and sum of squares due to error, based on $\beta$̂, be denoted YᵀAY, YᵀBY and YᵀCY respectively, so YᵀAY=YᵀBY+YᵀCY. Which of the following statements are always true?

Let $\lambda\in\mathbb{R}$ be such that the integral equation $y(x) = \lambda\int$₋$_{1}^{1}(5$xt$^{3}+4x^{2}t+3$xt)y(t)dt admits a non-trivial solution y(x) such that y(1)=5/2. Then which of the following statements are true?

Let {$x_{n}$} be a convergent iterative sequence generated by Newton-Raphson method for solving the equation sin x−1=0 such that $x_{n}\to\pi/2$ as $n\to\infty$. For $n\in\mathbb{N}$, let $e_{n}=x_{n}-\pi/2$. Let p>0 be such that $\lim(n\to\infty)$|$e_{n}$₊$_{1}$|/|$e_{n}$|^p exists and is non-zero. Then which of the following statements are true?

Consider the boundary value problem (BVP$) u_{xx}+u_yy=0$ in $\Omega=${$(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1$}, u(x,y)=e^(x+y) on $\partial\Omega=${$(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}=1$}. Then which of the following statements are true?

Consider the simple linear regression model $Y_{i}=\beta{}x_{i}+\varepsilon_{i}, i=1,2,\dots,n$, where $\beta>0, \sum{}x_{i}^{2}>0$, and the uncorrelated errors $\varepsilon_{i}$ have zero mean and finite variance $\sigma^{2}(>0)$. Let $\beta$̃$_{1}=\sum{}a_{i}$$Y_{i}$, where $a_{i}$'s minimize $E(\sum{}a_{i} Y_{i}-\beta)^{2}$ with respect to scalars $a_{1},a_{2},\dots,a_{n}$. Let $\beta$̃$_{2}$ be the ordinary least squares estimator of $\beta$. Which of the following statements are true?

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=\sin^{2}z+\cos^{2}$|z|. Which of the following statements are true?