CSIR NET December 2025 — Part C

Let $X_1,X_2,\dots ,X_{15}$ be a random sample from an Exponential distribution with the probability density function f(x∣$\sigma )=(1/\sigma )\exp (-x/\sigma )$ if x>0, 0 elsewhere, where the unknown parameter $\sigma$ is positive. Let $\bar{X}=(1/15){\sum }_1^{15}{X}_i$. Suppose that $\phi$ denotes the likelihood ratio test for testing $H_0:\sigma \le 1$ against $H_1:\sigma >1$ at level $\alpha =0.1$. It is given that ${\chi }_{15}^2{,}_0{.}_1=22.307,{\chi }_{15}^2{,}_0{.}_9=8.547,{\chi }_{30}^2{,}_0{.}_1=40.256,{\chi }_{30}^2{,}_0{.}_9=20.599$, where $P(W>{\chi }_n^2{,}_a)=\alpha$ and W~${\chi }_n^2$. Then which of the following statements are true?

Which of the following statements are true?

Let $X_1,X_2,\dots ,X_n(n\ge 2)$ be a random sample from an Exponential distribution with the probability density function f(x∣$\mu ,\sigma )=(1/\sigma )\exp ((\mu -x)/\sigma )$ if $x>\mu ,0$ otherwise, where parameters $\mu$ and $\sigma$ are unknown and positive. Let ${\bar{X}}_n,S_n^2$ and $X_1{:}_n$ denote the sample mean, the sample variance and the sample smallest order statistic, respectively, and let $\theta =\sigma /\mu$. Then which of the following statements are true?

Consider a dataset of n observations given by A = {$x_1,x_2,\dots ,x_n$}. Create a new dataset, given by B = {$x_1,x_2,\dots ,x_n,-x_1,-x_2,\dots ,-x_n$}. Which of the following statements are always true? (Variance is calculated with divisor as the number of observations in the corresponding dataset.)

Let G be a group of order 8. Which of the following statements are necessarily true?

Let $\alpha ={\sum }_{l=1}^{2025}$ $a_l/10$ˡ with $a_l\in$ {0,1,…,9} for all 1≤l≤2025 and $a_{2025}\ne 0$. Let $({\beta }_n{)}_n\ge 1$ be a strictly increasing sequence of positive real numbers in (0,1) that converges to $\alpha$. For each n≥1, write ${\beta }_n={\sum }_{l=1}^{\infty }$ $b_{n,l}/10$ˡ with $b_{n,l}$∈{0,1,…,9}, for the infinite decimal expansion of ${\beta }_n$. Which of the following statements are necessarily true?

Let F be the set of all functions $f:[0,\infty )\to [0,\infty )$ that are Riemann integrable on [0,t] for all $t\in [0,\infty )$. Which of the following statements are necessarily true?

Consider the ordinary differential equation (ODE) y″ + r(x)y = 0, where $r(x)=x^3\sin (1/x)$ for x≠0, and r(0)=0. Then which of the following statements are true?

Consider $X={\prod }_{c\in (0,\infty )}$ [0,c] together with the product topology, where [0,c] is equipped with the euclidean topology. Which of the following statements are necessarily true?

Let $X_1,X_2,\dots ,X_n(n\ge 2)$ be a random sample from Uniform$[-3\theta /2,\theta /2]$ distribution, where $\theta >0$ is an unknown parameter. Let $\bar{X},X_{(1)}$ and $X_{(n)}$ denote respectively the mean, the smallest order statistic and the largest order statistic of the sample. Then which of the following statements are true?

Let ${\tau }_E$ denote the euclidean topology on $\mathbb{R}$. Which of the following statements are necessarily true?

Let y(x) be the solution of the integral equation (IE) y(x) = eˣ $+e+1+(1/3){\int }_0^{1}y(t)$dt, and R(x,t,1/3) be the resolvent kernel associated to IE. Then which of the following statements are true?

Suppose X∣$\theta$ ~ Binomial$(7,\theta ),0<\theta <1$, and the prior distribution of $\theta$ is Beta$(\alpha ,\beta )$ where $\alpha >0$ and $\beta >0$ are known. Then which of the following statements MAY NOT be true?

Let $A\in M_4(\mathbb{C})$ be such that $A^2=I$ and Trace(A) = 0. Which of the following statements are necessarily true?

Consider an M/M/2 queuing system with the birth rate $\lambda =4$ per minute, the death rate $\mu =1$ per minute, and the total capacity of 3 customers (including the ones that are being served). Let $p_0$ and $p_3$ denote the long-run probabilities that the system will be empty (i.e. without customers) and will be blocked (i.e. full), respectively. Which of the following statements are true?

For n≥3, consider the space $M_n(\mathbb{C})$ of n×n complex matrices endowed with the inner product ⟨A,B⟩ = Trace(A*B). For 0≤k≤n, let $W_k=$ {$A\in M_n(\mathbb{C})$ : ⟨A,B⟩=0 for all $B\in M_n(\mathbb{C})$ with rank k}. Which of the following statements are necessarily true?

There are six persons. On each day of the year 2024 at least one of them borrowed money from another. A person P is labeled troublesome in a calendar month M of 2024 if P borrowed from the same person at least twice during month M. Which of the following statements are necessarily true?

Let V be a real vector space. Suppose that {u,v,w,x,y} ⊆ V is a spanning set of V and that {u,v,x,y} is linearly independent. Which of the following statements are necessarily true?

Suppose y(x) is the extremal of the variational problem $J(y)={\int }_0^1{x}^2(y^{\prime }{)}^2$ dx subject to $y(0)=0,y(1)=1,{\int }_0^1{y}^2$dx = 1/7. Then which of the following statements are true?

Let $X_1,X_2,\dots ,X_{20}$ be a random sample from $N_{12}(\mu ,\sum )$, where $\mu \in {\mathbb{R}}^{12}$ and $\sum$ is positive definite. Suppose $\bar{X}=(1/20){\sum }_i$₌${}_1^{20}{X}_i$ and $S=(1/19){\sum }_i$₌${}_1^{20}(X_i-\bar{X})(X_i-\bar{X}{)}^T$. Then which of the following statements are true?

Let u(x,t) be the solution of the initial-boundary value problem $u_{tt}-u_{xx}+u_t=0,0<x<\pi ,t>0,u(0,t)=0,u(\pi ,t)=0,t>0,u(x,0)=\sin x,u_t(x,0)=0,0<x<\pi$. For t≥0, define $E(t)={\int }_0^{\pi }(u_t^2+u_x^2)$ dx. Then which of the following statements are true?

Let S be the set of all 2×2 matrices A such that the iterative sequence generated by the Gauss-Seidel method converges for every initial guess, when employed to solve the system of equations $A(x_1,x_2{)}^T=(1,2{)}^T$. Then which of the following statements are true?

Consider a commutative ring R with unity with at least five elements such that for any two elements a,b∈R, there exists c∈R such that a=bc or b=ac. Which of the following statements are necessarily true?

Consider the following statements: (P) The initial value problem $y^{\prime }+y=e^{(}-x^2),y(0)=0$ has a Taylor series solution about the point x=0. (Q) The initial value problem y′+y=r(x), y(0)=0, where $r(x)=e^{(}-1/x^2)$ for x≠0 and r(0)=0, has a Taylor series solution about the point x=0. Then which of the following statements are true?

Let $F:{\mathbb{R}}^3\to \mathbb{R}$ be a continuously differentiable function such that $F(0,0,0)=-1,\partial F/\partial x(0,0,0)=2,\partial F/\partial y(0,0,0)=0$, and $\partial F/\partial z(0,0,0)=3$. For $\epsilon >0$, define ${\Omega }_{\epsilon }=(-\epsilon ,\epsilon )\times (-\epsilon ,\epsilon )\subset {\mathbb{R}}^2$. Which of the following statements are necessarily true?

For each positive integer n, let $f_n:[-1,1]\to \mathbb{R}$ be given by $f_n(x)=-1$ if $-1\le x\le -1/n,f_n(x)=$nx if −1/n<x<1/n, and $f_n(x)=1$ if 1/n≤x≤1. On which of the following intervals does the sequence {$f_n$}${}_n\ge 1$ converge uniformly?

Let (X,d) be a metric space. For a non-empty subset A of X, and x∈X, define d(x,A) = $in{f}_{a\in A}$ d(x,a). Which of the following statements are necessarily true?

Consider the following subset of real numbers A = {$(1+(-1{)}^n)n-1/n:n$ is a positive integer}. Which of the following statements are true?

Let $f:(0,\infty )\to (0,\infty )$ be the function defined by $f(x)={\int }_0$ˣ $\sqrt{t}/(1+t^2)$ dt, where $\sqrt{t}$ denotes the positive square root for t>0. Which of the following statements are true?

Consider the following real-valued functions $F_1$ and $F_2$ defined on $\mathbb{R}$, given by $F_1(x)=1$ if x≥1, 0 otherwise, and $F_2(x)={\int }_0$ˣexp(−t)dt if x≥0, 0 otherwise. Define another function $F:\mathbb{R}\to [0,1]$ by $F(x)=(2/3)F_1(x)+(1/3)F_2(x)$ for all $x\in \mathbb{R}$. Which of the following statements are true?

Consider the following $2^5-$factorial design with 8 blocks (Block 1: (1), acd, bce, abde; Block 2: e, acde, bc, abd; and six further blocks). Which of the following statements are true?

Let $\mathbb{N}$ denote the set of all positive integers. For $n\in \mathbb{N}$, let $f_n:[0,1]\to \mathbb{R}$ be given by $f_n(x)=n(1-$nx) if 0≤x≤1/n, 0 if x>1/n. Which of the following statements are necessarily true?

Let f be an entire function. Consider the function g given by g(z) = f(z) − 1/z for $z\in \mathbb{C}${0}. Which of the following statements are necessarily true?

Let ${\ell }^2$ denote the vector space of all square summable sequences {$a_n$}${}_n\ge 1$ of real numbers with the inner product ⟨{$a_n$},{$b_n$}⟩ $={\sum }_n$₌${}_1^{\infty }{a}_n{b}_n$. Let H = {{$a_n$}$\in {\ell }^2$ : |$a_n$|≤1/n for all positive integers n}. Which of the following statements are true?

Let {$X_n:n\ge 1$} be a sequence of independent and identically distributed random variables, where $X_1$ has an Exponential distribution with mean 1. Define $T_n=\max${$X_1,X_2,\dots ,X_n$} − ln n, n≥1. Suppose $T_n\to$ᵈ Y as $n\to \infty$. Then which of the following statements are true?

Let f:[0,1]→[0,1] be a monotonically increasing function, that is, a≤b implies f(a)≤f(b). For any $\alpha \in (0,1)$, let $L_{\alpha }$⁺=$li{m}_{x\to {\alpha }^{+}}f(x)$ and $L_{\alpha }$⁻=$li{m}_{x\to {\alpha }^{-}}f(x)$ denote the right hand and left hand limits respectively, provided they exist. For $\alpha \in (0,1)$, if $L_{\alpha }$⁺ and $L_{\alpha }$⁻ exist, define $U_{\alpha }=(L_{\alpha }$⁻,$L_{\alpha }$⁺) if $L_{\alpha }$⁻$<L_{\alpha }$⁺, and ∅ if $L_{\alpha }$⁺$\le L_{\alpha }$⁻. Which of the following statements are true?

Suppose that the transition probability matrix of a homogeneous Markov chain with state space {1,2,3,4} is given by P, with row 1 = (1/4, 3/4, 0, 0), row 2 = (1, 0, 0, 0), row 3 = (1/8, 0, 7/8, 0), row 4 = (0, 0, 1/9, 8/9). Which of the following statements are true?

Let X and Y be two independent random variables such that the moment generating functions of X and Y are $M_X(t)=e^{(}3(e^t-1)),t\in \mathbb{R}$, and $M_Y(t)=((1/2)e^{(}-3t)+(1/2)e^{(}3t){)}^2,t\in \mathbb{R}$, respectively. Then which of the following statements are true?

Let A be a non-zero 3×3 matrix with integer entries. Let ${\lambda }_i\in \mathbb{C},1\le i\le 3$ be all the eigenvalues of A (not necessarily distinct). Which of the following statements are necessarily true?

A mechanical system is described using generalized position q and generalized momentum p. Let Q and P denote new generalized position and generalized momentum variables respectively, generated by the generating function $F(q,P)=q^2{e}^P$, and Q,P are canonical coordinates. Let G(p,Q) be a function such that G(2,e)=0, and it generates the same canonical coordinates Q,P. Then which of the following statements are true?

Let 𝔻={$z\in \mathbb{C}$:|z|<1} and f:𝔻→𝔻 be a holomorphic function which satisfies f(−1/2)=0. Which of the following statements are necessarily true?

For $t\in [0,2\pi ]$, let ${\gamma }_1(t)=e^{(}$it) and ${\gamma }_2(t)=1+i+e^{(}$it). Which of the following statements are true?

Let V be a finite-dimensional $\mathbb{R}-$vector space and T:V→V a linear operator such that $T^2$ is diagonalizable over $\mathbb{R}$. Which of the following statements are necessarily true?

Let $X_1,X_2,\dots ,X_7$ be a random sample drawn from a continuous distribution with unknown unique median M. The null hypothesis $H_0:M=2$ is tested against the alternative $H_1:M>2$ at level of significance 0.05 using the right-tailed test based on the Sign test statistic K, which is the number of observations in the sample greater than 2. If the observed sample is −3,−6,1,9,4,10,12, which of the following statements are true?

Suppose $y_1(x)$ and $y_2(x)$ are two linearly independent solutions of the differential equation $x^2{y}^{\prime \prime }+[(1+\sin x)x/2]y^{\prime }-(3/2)(\cos x)y=0,x>0$, satisfying $y_2(0)=0$. Then which of the following statements are true?

Suppose two fair dice are thrown independently at random. Let X and Y be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?

Let V be a 4-dimensional complex vector space and A a linear operator on V. Which of the following statements are necessarily true?

Let p≥3 be a prime number and $f(x)\in \mathbb{Q}[x]$ an irreducible polynomial of degree p. Suppose that $a_1,\dots ,a_p\in \mathbb{C}$ are the roots of f and that $a_1\notin \mathbb{R},a_2\notin \mathbb{R}$, and $a_i\in \mathbb{R}$ for all 3≤i≤p. Let $K=\mathbb{Q}(a_1,\dots ,a_p)$ be the subfield of $\mathbb{C}$ generated by the roots of f. Consider the Galois group G of K over $\mathbb{Q}$ as a subgroup of $S_p$, the group of permutations of {$a_1,\dots ,a_p$}. Which of the following statements are true?

Let V be a real vector space and L(V) denote the space of linear operators on V. Let T∈L(V) be a non-zero operator such that $T^2=T$. Consider the subspace W of L(V) spanned by {$I,T^n:n$ is a positive integer}. Which of the following statements are necessarily true?

Let G be a finite non-abelian group. Which of the following statements are necessarily true?

Consider a series system comprising four components, having independent and identically distributed lifetimes with hazard rate $\lambda (t)=1/(1+t),t>0$, and survival function S(t), t>0. If Y denotes the lifetime of the series system, then which of the following statements are true?

Let a,b be distinct positive real numbers. Consider the function $f:{\mathbb{R}}^2\to \mathbb{R}$ given by f(x,y)=(ax+by${)}^2/($ax${}^2+$by${}^2)$ if (x,y)≠(0,0), and 0 if (x,y)=(0,0). Which of the following statements are necessarily true?

Let $X_1,X_2,\dots ,X_n(n>5)$ be independent random variables such that $X_t=\alpha +{\alpha }^{2}t+{\epsilon }_t$, for t=1,…,n, where ${\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n$ are independent and identically distributed $N(0,{\sigma }^2)$ random variables. Here $\alpha \in \mathbb{R}$ and $\sigma >0$ are unknown parameters. Which of the following statements are true?

Suppose y(x) is the extremal of the variational problem $J(y)={\int }_0^1((y^{\prime }{)}^2\sin x+(2cosx)y)$ dx subject to y(0)=0, y(1)=1. Then which of the following statements are true?

Let $Y=X\beta +\epsilon$ be a multiple linear regression model with p regressors and an intercept, where $\beta =({\beta }_0,{\beta }_1,\dots ,{\beta }_p{)}^T$ and the random error $\epsilon$~$N_n(0,{\sigma }^{2}I),\sigma >0$ and n>p+1. The least squares method provides a unique estimator $\beta$̂. Let the total sum of squares (corrected), sum of squares due to regression, and sum of squares due to error, based on $\beta$̂, be denoted YᵀAY, YᵀBY and YᵀCY respectively, so YᵀAY=YᵀBY+YᵀCY. Which of the following statements are always true?

Let $\lambda \in \mathbb{R}$ be such that the integral equation $y(x)=\lambda \int$₋${}_1^1(5$xt${}^3+4x^{2}t+3$xt)y(t)dt admits a non-trivial solution y(x) such that y(1)=5/2. Then which of the following statements are true?

Let {$x_n$} be a convergent iterative sequence generated by Newton-Raphson method for solving the equation sin x−1=0 such that $x_n\to \pi /2$ as $n\to \infty$. For $n\in \mathbb{N}$, let $e_n=x_n-\pi /2$. Let p>0 be such that $\lim (n\to \infty )$|$e_n$₊${}_1$|/|$e_n$|^p exists and is non-zero. Then which of the following statements are true?

Consider the boundary value problem (BVP$)u_{xx}+u_{y}y=0$ in $\Omega =${$(x,y)\in {\mathbb{R}}^2:x^2+y^2<1$}, u(x,y)=e^(x+y) on $\partial \Omega =${$(x,y)\in {\mathbb{R}}^2:x^2+y^2=1$}. Then which of the following statements are true?

Consider the simple linear regression model $Y_i=\beta x_i+{\epsilon }_i,i=1,2,\dots ,n$, where $\beta >0,\sum x_i^2>0$, and the uncorrelated errors ${\epsilon }_i$ have zero mean and finite variance ${\sigma }^2(>0)$. Let $\beta$̃${}_1=\sum a_i\ast Y_i$, where $a_i{\ast }^{\prime }s$ minimize $E(\sum a_i{Y}_i-\beta {)}^2$ with respect to scalars $a_1,a_2,\dots ,a_n$. Let $\beta$̃${}_2$ be the ordinary least squares estimator of $\beta$. Which of the following statements are true?

Let $f:\mathbb{C}\to \mathbb{C}$ be defined by $f(z)={\sin }^{2}z+{\cos }^2$|z|. Which of the following statements are true?