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 $J(y)={\int }_0^4(y^{\prime }{)}^2/y^2$ 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 $z^{\prime }=(1-z^4{)}^{100}/(1+z^2),z(0)=0$, 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 $y^{\prime \prime }+(e^{(}{x}^2)+\cos x)y=0$ such that $y_1(0)=3,y_1^{\prime }(0)=-1,y_2(0)=-5,y_2^{\prime }(0)=a$. 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 ${\sum }_{n\ge 4}$ $1/[n(lo{g}_{e}n{)}^{\alpha }(lo{g}_e(lo{g}_{e}n){)}^{\beta }]$. 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 ${\gamma }_R(t)=R{e}^{(}2\pi$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 $li{m}_{|z|\to \infty }$ 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 ${\gamma }_R(t)=2+i+R{e}^{(}2\pi$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 $Y_1={\beta }_1+{\beta }_2+{\epsilon }_1,Y_2={\beta }_1+2{\beta }_2+{\epsilon }_2,Y_3={\beta }_1+c{\beta }_2+{\epsilon }_3$, where ${\beta }_1,{\beta }_2\in \mathbb{R}$ are unknown parameters, and the uncorrelated errors ${\epsilon }_i,i=1,2,3$ 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?