CSIR NET December 2025 — Part B

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?

Consider a finite population of size N = 100. Let $T_{1}$ be the sample mean of a study variable based on a sample of size n (1 < n < N) under simple random sampling with replacement scheme. Let $T_{2}$ be the sample mean of the same study variable based on a sample of size n under simple random sampling without replacement scheme. If $Var(T_{1}) = 9Var(T_{2})$, then the sample size n equals

Let $\gamma_R(t) = 2 + i + Re^(2\pi$it) for t ∈ [0,1] and R = 1, 2. Which of the following statements is true?

Let B be a 4×4 positive-definite real symmetric matrix which is not the identity matrix. Consider the inner product on $\mathbb{R}^{4}$ given by ⟨v,w⟩ = vᵀBw, where vᵀ denotes the transpose of v. Which of the following statements is FALSE?

Let u(x,t) be the solution of the initial value problem $u_{tt} - u_{xx} = e^(-t), x \in \mathbb{R}, t > 0, u(x,0) = \cos x, u_{t}(x,0) = 0, x \in \mathbb{R}$. Then which of the following statements is true?

Let $f : (0,\infty) \to (0,\infty)$ be a uniformly continuous function. Define $g : (0,\infty) \to (0,\infty)$ by $g(x) = x^{2}$ if 0 < x ≤ 1, and g(x) = x if x > 1. Which of the following statements is true?

If p(x) is the interpolating polynomial for the data x = −2,−1,0,1,2 with y = −1,3,1,−1,3, then the value of p(1/2) is

Let S be a subset of the open interval (0,1) that consists of all the real numbers $\alpha \in (0,1)$ whose infinite decimal expansion $\alpha = 0.a_{1}a_{2}a_{3}\cdots$ is such that $a_{i} \in$ {0,2,4} for all i ≥ 1. Which of the following statements is true?

Consider the quadratic form f(x,y,z) = [x y z] A [x;y;z], where A is an invertible 3×3 symmetric matrix over $\mathbb{Q}$. Assume that there exists $(\alpha,\beta,\gamma) \in \mathbb{C}^{3}${(0,0,0)} such that $f(\alpha,\beta,\gamma) = 0$. Which of the following statements is necessarily true?

Let $X_{1}$ and $X_{2}$ be a random sample from Uniform$[0,\theta]$ distribution, where $\theta > 0$. For testing the hypothesis $H_{0}: \theta=1$ against $H_{1}: \theta=2$, consider a test which rejects $H_{0}$ if $X_{1}+X_{2} > 4/5$. Then, the probability of type-I error is

Let {$Y_{n} : n \ge 1$} be a sequence of independent and identically distributed random variables, where $Y_{1}$ ~ Bernoulli(1/2). Define $Z = \sum_{n=1}^\infty$ $4Y_{n}/5^{n}$. Then, which of the following statements is true?

Let A be a 3×3 matrix over complex numbers with trace 1 and determinant 1. Suppose, further, that one of the eigenvalues of A is 1. Which of the following statements is necessarily true?

Let X be a single sample from an absolutely continuous distribution with probability density function f(x∣$\theta) = (2/\theta^{2})(\theta-x)$ if $0 < x < \theta$, and 0 otherwise, where $\theta > 0$ is unknown. Which of the following intervals is a 95% confidence interval for $\theta$?

Which of the following statements is necessarily true?

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?