CSIR NET Maths previous year question papers

Let {$A_{n}$}$_{n\ge1}$ be a collection of non-empty subsets of $\mathbb{Z}$ such that $A_{n} \cap A_{m} = \emptyset$ for m ≠ n. If $\mathbb{Z} = \cup_{n\ge1}$ $A_{n}$, then which of the following statements are necessarily true?

Let $q_{1}, q_{2}$ be generalized coordinates and $p_{1}, p_{2}$ their conjugate momenta. Let a and b be such that $Q_{1} = q_{1}, P_{1} = a p_{1} + 16 p_{2}, Q_{2} = p_{2}, P_{2} = 2q_{1} + b q_{2}$ is a canonical transformation. Which of the following statements are true?

Consider the one-way fixed effects ANOVA model $Y_{ij} = \mu + \alpha_{i} + \varepsilon_{ij}, j = 1, \dots, n_{i}; i = 1, \dots, k$, with uncorrelated errors of mean 0 and variance $\sigma^{2}$. Let Ȳ$_{i}$ be the i-th group mean. Which of the following statements are true?

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), (X_{3}, Y_{3})$ be i.i.d. bivariate normal with mean (0, 0) and correlation matrix $[[1, \rho], [\rho, 1]]$, |$\rho$| < 1. Let $S_\rho = 3E($sgn$(X_{1} - X_{2})(Y_{1} - Y_{3}))$. Which of the following statements are true?