CSIR NET Maths previous year question papers

Suppose $(X_{1}, X_{2}, X_{3})^{T}$ ~ $N_{3}(0, I_{3})$, where 0 is the zero mean vector and $I_{3}$ is the 3 × 3 identity matrix, and $(X, Y, Z)^{T} = A(X_{1}, X_{2}, X_{3})^{T}$ where A has rows (3, 0, 0), (2, 2, 0) and (4, 0, 4). Then the partial correlation coefficient $\rho_{YZ.X}$ is

Suppose we want to estimate the population mean Ȳ of a variable for a finite population of size 85, with 34 Statisticians and 51 Biologists. We consider the following sampling scheme: a stratified random sample with 2 strata of Statisticians (Stratum-1) and Biologists (Stratum-2), where 12 Statisticians and 15 Biologists are drawn from Stratum-1 and Stratum-2, respectively, using SRSWOR scheme. Denote ȳ_S, ȳ_B, and ȳ as the mean of the variable among the Statistician sample, Biologist sample, and the combined sample, respectively. Which of the following is an unbiased estimator of Ȳ?

Solve the following linear programming problem: maximize z = x + y subject to 5x + 3y ≤ 30, 2x + 6y ≤ 25, 2x − y ≤ 8, x ≥ 0, y ≥ 0. Then the optimal value of the objective function is

Let X be a Binomial(n, p) random variable, where n ∈ {5, 6} and p ∈ {1/4, 3/4}. If X = 3 is observed, then the maximum likelihood estimate of (n, p) is

Let X = {1, …, 17} and $S_{17}$ be the group of permutations of X. For a subgroup G of $S_{17}$ and x ∈ X, let S_G(x) = {$\sigma \in G$ ∣ $\sigma(x) = x$}. Which of the following statements is true for every subgroup G of $S_{17}$?

Let I denote the 3×3 identity matrix. Given any three distinct matrices $A, B, C \in M_{3}(\mathbb{R})$, which of the following statements is necessarily true?

Let X and Y be topological spaces. Consider the following statement: S: For every open subset U ⊆ X × Y and every x ∈ X such that {x} × Y ⊆ U, there is a neighbourhood W of x in X such that W × Y ⊆ U. Which of the following statements is true?

Let $M, N \in M_{3}(\mathbb{C})$ be such that Trace(Mᵏ) = Trace(Nᵏ), 1 ≤ k ≤ 3. Which of the following statements is necessarily true?

Let {$X_{n} : n \ge 0$} be any homogeneous Markov Chain on the state space S = {1,2,3,4} having the transition probability matrix $P = (p_{ij})$ given by row 1 = (1/4, 0, 2/3, 1/12), row 2 = (0, 1, 0, 0), row 3 = (1/12, 0, 1/4, 2/3), row 4 = (2/3, 0, 1/12, 1/4). Which of the following statements about stationary distributions of any such Markov Chain is true?

Let F_q be a finite field with q elements. For n ≥ 2, let A be a 2n × 2n matrix with entries in F_q such that rank(A) = n. Let W = {v ∈ F_q^(2n) ∣ Av = 0}. Which of the following is necessarily the number of (n+2)-dimensional subspaces of F_q^(2n) that contain W?

Let 𝔻 denote the open unit disc {$z \in \mathbb{C}$ : |z| < 1} and X = {f : 𝔻 $\to \mathbb{C}$ ∣ f is holomorphic and satisfies $f(2z) = f(z)/(1 - f(z)^{2})$ for all |z| < 1/2}. Which of the following statements is true?

Let u(x,t) be the solution of the partial differential equation $u_{t} = 16u_{xx} + 2, 0 < x < 7, t > 0$, satisfying the conditions $u_{x}(0,t) = u(7,t) = 0$ for t > 0, and u(x,0) = 0 for 0 < x < 7. Then which of the following statements is true?

For a finite group G, let S(G) denote the number of subgroups of G. Which of the following statements is necessarily true?

Let X, Y, and Z be independent Normal random variables with means −1, 0, and 1, respectively, and variances 1, 1, and 3, respectively. Which of the following random variables has a Cauchy distribution with location parameter 0 and scale parameter 1?

Suppose y(x) is the extremal of the variational problem $J(y) = \int_{0}^{4} (y')^{2}/y^{2}$ dx subject to $y(0) = 1, y(4) = e^{8}$. Then which of the following statements is true?

For a non-negative real number a, let $\sqrt{a}$ denote its non-negative square-root. Consider the function $f : \mathbb{R} \to \mathbb{R}$ given by $f(x) = x(\sqrt{x^{2}+3} - \sqrt{x^{2}+2})$. Which of the following statements is true?

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?

Consider the initial-boundary value problem (IBVP$) u_{t} + u_{x} = 0, 0 < x < \infty, t > 0, u(x,0) = e$ˣ, $0 < x < \infty, u(0,t) = 1 - \sin t, t > 0$. Then which of the following statements is true?

For a non-negative real number a, let $\sqrt{a}$ denote its non-negative square-root. Consider the sequence {$x_{n}$}$_{n}\ge_{1}$ defined by $x_{1} = 1$ and $x_{n}$₊$_{1} = \sqrt{1 + x_{n}^{2}/n}$ for n ≥ 1. Which of the following statements is true?

Consider the following linear programming problem: Maximize $3x_{1} + 4x_{2}$ subject to $3x_{1} + 2x_{2} \le 12, 3x_{1} + 5x_{2} \le 15, 2x_{1} - x_{2} \ge 0, x_{2} \le 2$, and $x_{1}, x_{2} \ge 0$. Which of the following values is the optimum value of the objective function in the feasible region?

For $a \in \mathbb{R}$, let $y_{1}(x)$ and $y_{2}(x)$ be solutions of the differential equation $y'' + (e^(x^{2}) + \cos x)y = 0$ such that $y_{1}(0) = 3, y_{1}'(0) = -1, y_{2}(0) = -5, y_{2}'(0) = a$. Suppose $W(y_{1},y_{2})(x)$ denotes the Wronskian of $y_{1}$ and $y_{2}$. If $W(y_{1},y_{2})(1/2) = 4$, then the value of a is

Let $\alpha, \beta \in (0, \infty)$. Consider the infinite series $\sum_{n\ge4}$ $1/[n(log_{e} n)^\alpha (log_{e}(log_{e} n))^\beta]$. Which of the following statements is true?

Let $X = (X_{1},X_{2},X_{3})^{T}$ be a 3×1 random vector with E(X) = (3,2,1)ᵀ and $Cov(X) = \sum$, with rows (3,−2,0), (−2,3,−2), (0,−2,3). Suppose that $Y = (Y_{1},Y_{2},Y_{3})^{T} = (1/\sqrt{3})X$. Then the value of the multiple correlation coefficient between $Y_{1}$ and $(Y_{2},Y_{3})$ equals

Let S be a finite subset of $\mathbb{C}$ containing 0. Let f : $\mathbb{C}\S$ $\to \mathbb{C}$ be a holomorphic function which has a simple pole at 0. For R > 0, let $\gamma_R$ denote the path $\gamma_R(t) = Re^(2\pi$it) for t ∈ [0,1]. Which of the following statements is necessarily true?

Let h be a holomorphic function on $\mathbb{C}${0} such that $lim_{|z|\to\infty}$ h(z) = 0. For every n ≥ 1, consider $f_{n}(z) = \sum_{k=1}^n$ h(zᵏ). Let 𝔻 denote the open unit disc {$z \in \mathbb{C}$ : |z| < 1}. Which of the following statements is necessarily true?