CSIR NET Maths previous year question papers

Consider the following statements: $S_{1}$: There exists an $\varepsilon > 0$ such that $\forall{}x_{0} \in (2 - \varepsilon, 2 + \varepsilon)$, the iterative sequence defined by $x(n+1) = 6/(5 - x_{n}), n = 0, 1, 2, 3, \dots$, converges to $2. S_{2}$: There exists an $\varepsilon > 0$ such that $\forall{}x_{0} \in (3 - \varepsilon, 3 + \varepsilon)$, the iterative sequence defined by $x(n+1) = (x_{n}^{2} + 6)/5, n = 0, 1, 2, 3, \dots$, converges to $3. S_{3}$: There exists an $\varepsilon > 0$ such that $\forall{}x_{0} \in (3 - \varepsilon, 3 + \varepsilon)$, the iterative sequence defined by $x(n+1) = (5x_{n} - 6)/x_{n}, n = 0, 1, 2, 3, \dots$, converges to 3. Then which of the following statements are true?

Let $f : [0, 1] \times \mathbb{R} \to \mathbb{R}$ be a function such that f and its partial derivatives of orders less than or equal to 3 are continuous and bounded. Let $y : [0, 1] \to \mathbb{R}$ be the solution of dy/dt $= f(t, y(t)), 0 \le t \le 1, y(0) = y_{0}$, where $y_{0} \in \mathbb{R}$. For h > 0, denote t_j = jh. Let Y_j be an approximation of y(t_j), defined by Y_(j+1) = Y_j + ahf(t_j, Y_j) + bhf(t_(j+1), Y_j + chf$(t_j, Y_j)), 0 < (j+1)h < 1, Y_{0} = y_{0}$, where $a, b, c \in \mathbb{R}$. If there exists an M > 0 such that |y(t_(j+1)) − y(t_j) − ahf(t_j, y(t_j)) − bhf(t_(j+1), y(t_j) + chf(t_j, y(t_j)))| ≤ Mh$^{3}$ for every h > 0, and every j with 0 ≤ (j+1)h < 1, then

The integral equation $u(x) = f(x) + (2/\pi)\int_{0}^\pi \cos(x + t)u(t)$dt has infinitely many solutions if

Let R(x, t) and u(x) denote the resolvent kernel and the solution, respectively, of the Volterra integral equation u(x) = eˣ $+ 2\int_(\ln 6)^x e^(2(t-x))u(t)$dt. Then which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 3)$ be a random sample from a population with absolutely continuous cumulative distribution function F(·). The corresponding order statistics are X(1:n) < X(2:n) < ⋯ < X(r:n) < ⋯ < X(n:n). Define, for r = 2, 3, Y(r,n) = nF(X(r:n)). Suppose that Y(r,n) converges in distribution to a random variable Y_r as $n \to \infty, r = 2, 3$. Then, which of the following statements are true?

Let X, Y, Z be independent and identically distributed (i.i.d.) random variables, each following Bernoulli$(\theta), 0 < \theta < 1$. Then, which of the following statements are true?

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{n}, Y_{n})$ be a random sample of size n from a bivariate distribution F(X,Y) with absolutely continuous marginal distribution functions F_X and F_Y of X and Y, respectively. Let r_S be the Spearman's rank correlation coefficient defined with ranks of $X_{i}$ and ranks of $Y_{i}, i = 1, 2, \dots, n$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_m$ and $Y_{1}, Y_{2}, \dots, Y_{n}$ be two mutually independent random samples from populations with absolutely continuous distribution functions F_X and F_Y, respectively. For N = m + n, define $T_N = \sum(i=1..N)$ iZ$_{i}$ where $Z_{i} = 1$ if the i-th observation in the combined ordered arrangement of N observations is from F_X; and $Z_{i} = 0$, otherwise. Then, which of the following statements are true?

Consider a probability proportional to size without replacement sample involving two draws from a population of N (> 6) units with normed size measures $p_{i}'s (p_{i} > 0, i = 1, 2, \dots, N; \sum(i=1..N)p_{i} = 1)$. Then, which of the following statements are true?

Consider maximizing the objective function $P = x_{1} + x_{2}$ subject to $x_{1} + x_{2} + 2x_{3} \le 5, 2x_{1} - 3x_{3} \ge 1, x_{2} + x_{3} \le 0, x_{1} \ge 0, x_{2} \ge 0, x_{3} \ge 0$. Then, which of the following statements are true?

For each positive integer n, define $f_{n} : [0, 1] \to \mathbb{R}$ by $f_{n}(x) =$ nx$(1 - x)^{n}$. Which of the following statements are true?

Let $g : \mathbb{R} \to \mathbb{R}$ be a continuous function. Define $f(x) = \int_{0}$ˣ(x − t)g(t)dt, $x \in \mathbb{R}$. Which of the following statements are true?

Let $f : [0, 1] \to \mathbb{R}$ be a monotonic function. Which of the following statements are true?

For a variable x, consider the $\mathbb{Q}-$vector space V = {ax$^{3} +$ bx$^{2} +$ cx + d | $a, b, c, d \in \mathbb{Q}$}. Further, let A = {$f : V \to \mathbb{Q}$ | f is $a \mathbb{Q}-$linear transformation} and B = {f ∈ A | f(1) = 0}. Which of the following statements are true?

For a variable x, consider the real vector space V = {ax$^{3} +$ bx$^{2} +$ cx + d | $a, b, c, d \in \mathbb{R}$}. Let D : V → V be the linear transformation where D(f) is the derivative of f with respect to x, and M : V → V be the linear transformation M(f) = xD(f). Which of the following statements are true?

Let A, B be 2 × 2 matrices with real entries, and M = AB − BA. Let $I_{2}$ denote the 2 × 2 identity matrix. Which of the following statements are necessarily true?

Let $M_{2}(\mathbb{R})$ denote the $\mathbb{R}-$vector space of 2 × 2 matrices with real entries. Let A = [[1, 2], [0, 3]] and B = [[−1, 0], [1, 5]]. Define a linear transformation $T : M_{2}(\mathbb{R}) \to M_{2}(\mathbb{R})$ by T(X) = AXBᵗ, where Bᵗ denotes the transpose of the matrix B. Which of the following statements are true?

For a 4 × 4 positive definite real symmetric matrix A and real numbers a, b, c, d, consider the 5 × 5 matrix B whose first row is (0, a, b, c, d), whose first column is (0, a, b, c, d)ᵗ, and whose lower-right 4 × 4 block is A. Which of the following statements are necessarily true?

Let R be a nonzero ring with unity such that $r^{2} = r$ for all r ∈ R. Which of the following statements are true?

Let $f \in \mathbb{R}[x]$ be a product of distinct monic irreducible polynomials $P_{1}, P_{2}, \dots, P_{n}$, where n ≥ 2. Let (f) denote the ideal generated by f in the ring $\mathbb{R}[x]$. Which of the following statements are true?

Consider the non-homogeneous ordinary differential equation (ODE$) d^{2}y/$dx$^{2} + 5$dy/dx + 6y = sin(e^(−5x)), x > 0. Then which of the following statements are true?

If x = x(t), y = y(t) is the solution of the initial value problem dx/dt = x − 4e^(−2t)y, dy/dt = e^(2t)x − y, x(0) = 1, y(0) = 1, then which of the following statements are true?

Let u = u(x, y) be the solution of the boundary value problem $\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} = 0$ on (0, 1) × (0, 1), with $u(x, 0) = e^(\pi{}x), u(x, 1) = -e^(\pi{}x)$ for $x \in [0, 1], u(0, y) = \cos(\pi{}y) + \sin(\pi{}y)$ and $u(1, y) = e^\pi(\cos(\pi{}y) + \sin(\pi{}y))$ for y ∈ [0, 1]. Then there exists a point $(x_{0}, y_{0}) \in (0, 1) \times (0, 1)$ such that

Let u = u(x, t) be the solution of the initial-boundary value problem $\partial{}u/\partial{}t = \partial^{2}u/\partial{}x^{2}$ on $(0, 1) \times (0, \infty)$, with u(x, 0) = 4x(1 − x) for x ∈ [0, 1] and u(0, t) = u(1, t) = 0 for t ≥ 0. Then which of the following statements are true?

Consider the system of two particles with total kinetic energy T = (5/2)ẋ$^{2} + l^{2}\theta$̇$^{2} + 2l$ẋ$\theta$̇$\cos \theta$, and Lagrangian L = (5/2)ẋ$^{2} + l^{2}\theta$̇$^{2} + 2l$ẋ$\theta$̇$\cos \theta + 2$gl $\cos \theta$, where $x, \theta$ are generalized coordinates, and g, l are positive constants. Then the non-zero frequency of the normal mode of the system with small oscillations (|$\theta$| ≪ 1) is