CSIR NET Maths previous year question papers

Consider the following statements: (P) The initial value problem y′ = f(y), where f(y) = y·cos(1/y) for y ≠ 0 and f(0) = 0, y(0) = 0, has at most one solution. (Q) If z(x) is the solution of the initial value problem $z' = (1-z^{4})^{100}/(1+z^{2}), z(0) = 0$, then for each $\alpha \in (0,2)$, there exists $x_\alpha \in \mathbb{R}$ such that $z(x_\alpha) = 1-\alpha$. Then which of the following statements is true?

For $\alpha \ge 0$, consider the functional $J_\alpha[y] = \int_{1}^{2} (y')^{2}/x^\alpha$ dx defined for all continuously differentiable functions defined on the interval [1, 2] satisfying the conditions y(1) = 1, y(2) = 2. Then which of the following statements are true?

Let S denote the set of all 2 × 2 matrices A such that the iterative sequence generated by the Gauss-Seidel method applied to the system of linear equations $A(x_{1}, x_{2})^{T} = (2, 3)^{T}$ converges for every initial guess. Then which of the following statements are true?