CSIR NET Maths previous year question papers

Consider the linear model $Y_{1} = \beta_{1}+\beta_{2}+\varepsilon_{1}, Y_{2} = \beta_{1}+2\beta_{2}+\varepsilon_{2}, Y_{3} = \beta_{1}+c\beta_{2}+\varepsilon_{3}$, where $\beta_{1},\beta_{2} \in \mathbb{R}$ are unknown parameters, and the uncorrelated errors $\varepsilon_{i}, i=1,2,3$ have zero mean and finite variance $\sigma^{2}(>0)$. The constant c is such that $\beta$̂$_{1}$ and $\beta$̂$_{2}$ are uncorrelated, where $\beta$̂$_{1}$ and $\beta$̂$_{2}$ are the best linear unbiased estimators of $\beta_{1}$ and $\beta_{2}$, respectively. Which of the following statements is the correct option for $(Var(\beta$̂$_{1}), Var(\beta$̂$_{2}))$?

Suppose u(x) is the solution of the integral equation $u(x) = 3 + \int_{0}$ˣ (x−t)u(t) dt. Then which of the following statements is true?

Let $X_{1}, X_{2}, \dots$ be a sequence of independent and identically distributed random variables having Binomial(3, 1/4) distribution. For j=1,2,…, define $Y_{j} = 1$ if $X_{j} \le \sqrt{5}$, and 0 otherwise. If $(1/n)\sum_{j=1}^n$ $Y_{j}^{2}$ converges almost surely to a constant c as $n \to \infty$, then the value of c equals

For a finite group G, let H_G = {g ∈ G ∣ $g^{15} = e$}. Which of the following statements is necessarily true?