CSIR NET June 2025 — Part C

Let f be a bounded, twice continuously differentiable real-valued function on $(0,\infty )$ such that f″(x) ≥ 0 for all $x\in (0,\infty )$. Which of the following statements are true?

Let $(f_n{)}_n{\ge }_1$ be a sequence of real-valued functions on $\mathbb{R}$. Which of the following statements are true?

For each n ≥ 1, let $f_n:\mathbb{R}\to \mathbb{R}$ be a function defined by $f_n(x)=$ $e^{-n^2{x}^2}/n$. Which of the following statements are true?

Let A, B be distinct 2 × 2 real matrices. Which of the following statements are true?

Let $S,T:{\mathbb{C}}^2\to {\mathbb{C}}^2$ be $\mathbb{C}-$linear transformations and I denote the identity transformation on ${\mathbb{C}}^2$. Which of the following statements are necessarily true?

Let V be the $\mathbb{R}-$vector space of all polynomials with real coefficients. Let $f(x)=x^2+x+1$. Which of the following subsets of V are linearly independent?

Consider the field 𝔽${}_3$ consisting of 3 elements. Let V be an 𝔽${}_3-$vector space of dimension 3 and W an 𝔽${}_3-$vector space of dimension 2. Which of the following statements are true?

Let V be a finite dimensional complex inner product space. For a linear map T : V → V, let T* denote its adjoint. Which of the following statements are true?

Let B(v, w) be a nondegenerate symmetric bilinear form on ${\mathbb{R}}^2$ and let q(v) = B(v, v) be the corresponding quadratic form. Suppose there exist vectors $v,w\in {\mathbb{R}}^2$ such that B(v, v) = 0 and B(v, w) ≠ 0. Which of the following statements are necessarily true?

Let f be an entire function. Which of the following statements are true?

Let 𝔻^× = {$z\in \mathbb{C}:0<$ |z| < 1} be the punctured unit disk and f be a bijective holomorphic map of 𝔻^× onto itself. Which of the following statements are true?

Let f be an entire function which is not a polynomial. Let A = {$\alpha \in \mathbb{C}$ | f⁽${}^n$⁾$(\alpha )\ne 0$ for all n ≥ 0}. Which of the following statements are true?

Consider the ordinary differential equation (ODE$)d^{2}y/$dx${}^2+\cos (x)$ dy/dx + sin(x) y = 0. Let ${\phi }_1(x),{\phi }_2(x)$ be solutions of the ODE, satisfying ${\phi }_1(0)=1,d{\phi }_1/$dx(0) = 0, and ${\phi }_2(0)=0,d{\phi }_2/$dx(0) = 1. Then which of the following statements are true?

For a continuous function q defined on $\mathbb{R}$, consider the ordinary differential equation (ODE$)d^{2}y/$dx${}^2+q(x)y=0,x\in \mathbb{R}$. Then which of the following statements are FALSE?

Consider the Cauchy problem (CP$)x\partial u/\partial x+\partial u/\partial y=1,u(0,y)=e^y,y\in \mathbb{R}$. Then which of the following statements are true?

Suppose u = u(x, y) is the solution of the boundary value problem ${\partial }^{2}u/\partial x^2+{\partial }^{2}u/\partial y^2+\partial u/\partial x=0$ in {$(x,y)\in {\mathbb{R}}^2:x^2+y^2<1$}, $u(x,y)=1+2x^2{y}^2$ on {$(x,y)\in {\mathbb{R}}^2:x^2+y^2=1$}. Then which of the following statements are true?

If $\alpha ,\beta \in \mathbb{R}$ are such that the equation ${\int }_0^{3}f(x)$ dx $=(3/2)[f(\alpha )+f(\alpha +\beta )]$ holds for all polynomials f(x) of degree less than or equal to 2, then which of the following statements are true?

For $\alpha \ge 0$, consider the functional $J_{\alpha }[y]={\int }_1^2(y^{\prime }{)}^2/x^{\alpha }$ dx defined for all continuously differentiable functions defined on the interval [1, 2] satisfying the conditions y(1) = 1, y(2) = 2. Then which of the following statements are true?

For $a,b,c\in \mathbb{R}$, consider the variational problem: Minimize $J[y]={\int }_0^2[a(y^{\prime }{)}^2+2$byy′ + cy${}^2]$ dx, subject to y(0) = 10, y(1) = 100. Then which of the following statements are true?

Let X be a random variable with the cumulative distribution function F_X(x) = 0 if x < 0, F_X(x) = (x + 2)/5 if 0 ≤ x < 2, and F_X(x) = 1 if x ≥ 2. Then, which of the following statements are true?

Let $X_1,X_2,\dots ,X_n$ be a random sample from $N(\theta ,1),\theta \in \mathbb{R}$. If $\theta$̂ is the Bayes estimator of $\theta$ with respect to some prior $\pi (\theta )$ and loss function $L(\theta ,d)$. Then, which of the following statements are true?

Consider a Markov chain {$X_n:n\ge 1$} on state space {1, 2, 3, 4, 5} with the transition probability matrix whose rows are (0, 1/2, 1/2, 0, 0), (0, 0, 1, 0, 0), (0, 1/3, 0, 1/3, 1/3), (1, 0, 0, 0, 0) and (0, 0, 0, 0, 1). Then, which of the following statements are true?

Consider the M/M/1 queue in which customers arrive according to a Poisson process with rate 3 and successive service times are independent exponential random variables having mean 1/9. Let $P_n$ be the long run probability that there are exactly n customers in the system. Then, which of the following statements are true?

Let X and Y be independent Poisson random variables with means 4 and 2, respectively. Then, which of the following statements are true?

A system has two components $C_1$ and $C_2$ put in parallel. The components $C_1$ and $C_2$ have independent lifetimes $X_1$ and $X_2$, respectively. The probability distribution of X_j is exponential with mean 1/j, j = 1, 2. Suppose that R(t) and h(t) are the reliability and the hazard rate functions of the system, respectively. Then, which of the following statements are true?

Let X and Y be independent and identically distributed N(0, 1) random variables. Let $S=X^2+Y^2$ and T = $e^{-(X^2+Y^2)/2}$. Then, which of the following statements are true?

Let $X_1,X_2,\dots ,X_n$ be independent and identically distributed random variables with the probability density function f(x|$\theta )=$ $e^{-(x-\theta )}$ if $x\ge \theta$, and 0 otherwise, where $\theta \in \mathbb{R}$ is the unknown parameter. Further, let $Y=\min (X_1,X_2,\dots ,X_n)$. Which of the following are confidence intervals for $\theta$ with the confidence coefficient $(1-\alpha )$, where $\alpha \in (0,1)$?

Consider a paired data $(x_i,y_i);i=1,2,3,4,5$, where $(x_1,x_2,x_3,x_4,x_5)=(-2,-1,0,1,2)$ and $y_i=x_i^2$ for all i = 1, 2, 3, 4, 5. On this data, a simple linear regression model with an intercept term and a simple linear regression model without an intercept term are fitted using the method of least squares. Which of the following statements are true?

An analyst fits a multiple linear regression model $Y_i={\beta }_0+{\beta }_1{X}_{1i}+\cdots +{\beta }_4{X}_{4i}+{\epsilon }_i,i=1,2,\dots ,n$, using the method of least squares. However, his coordinator insists that the intercept and two regressors $Z_1=X_1+X_3$ and $Z_2=X_2-X_4$ are enough to represent the model. Suppose the coordinator's claim can be tested in the form of a general linear hypothesis, viz., $H_0:L^T\beta =0$ against $H_A:H_0$ is not true, where ${\beta }^T=({\beta }_0,{\beta }_1,\dots ,{\beta }_4)$. Suppose we have n observations on the response Y and each regressor. Further, assume that the errors with or without restrictions are independent $N(0,{\sigma }^2)$ variables. Then, which of the following statements are true?

Let $(X_1,Y_1),(X_2,Y_2),\dots ,(X_{10},Y_{10})$ be a random sample from a bivariate normal distribution BVN$({\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2,\rho )$ with ${\mu }_1=5,{\mu }_2=6,{\sigma }_1^2=4,{\sigma }_2^2=9$ and $\rho =1/2$. Then, which of the following statements are true?

Let the random vector $(X_1,X_2,X_3{)}^T$ have the positive definite dispersion matrix with rows $(1,\rho ,\rho ),(\rho ,1,\rho )$ and $(\rho ,\rho ,1)$. Then, which of the following statements are true?

Consider the sequences $(s_n{)}_n{\ge }_1$ and $(t_n{)}_n{\ge }_1$ defined by $s_n={\sum }_{k=0}^n$ $1/(k!{)}^2$ and $t_n={\sum }_{k=0}^n$ C(n, k)(−1)ᵏ/nᵏ. Which of the following statements are true?

What is the value of the limit $li{m}_{n\to \infty }$ (1/n)[(n + 1)(n + 2)⋯(n + n)]${}^{1/n}$?

Let f and g be real-valued Riemann integrable functions on [a, b] such that g([a, b]) ⊆ [a, b]. Which of the following statements are necessarily true?

Let $\mu$ denote the Lebesgue measure on $\mathbb{R}$. Suppose that f is a non-negative Lebesgue measurable function on $\mathbb{R}$. Let $0=a_0<a_1<a_2<\cdots$ be an unbounded sequence such that $a_{n+1}$ ≤ ca${}_n$ for some real number c and for all n ≥ 1. Let A_k = {$x\in \mathbb{R}$ | a_k ≤ f(x) < $a_{k+1}$} for each k ≥ 0. Which of the following statements are true?

Let $f:{\mathbb{R}}^2\to \mathbb{R}$ be a twice continuously differentiable non-zero function such that f(tx${}_1$, tx${}_2)=t^{3}f(x_1,x_2)$ for all t > 0 and $(x_1,x_2)\in {\mathbb{R}}^2$. Which of the following statements are necessarily true?

Let $M_2(\mathbb{R})$ denote the space of real 2 × 2 matrices. Let S be the vector subspace of $M_2(\mathbb{R})$ comprising of all symmetric matrices. Let $F:M_2(\mathbb{R})\to S$ be the map defined by F(X) = XXᵀ. Let DF${}_A:M_2(\mathbb{R})\to S$ be the derivative of F at $A\in M_2(\mathbb{R})$. Which of the following statements are true?

Let C[0, 1] be the $\mathbb{R}-$vector space of real valued continuous functions equipped with the norm ‖f‖ = $su{p}_{x\in [0,1]}$ |f(x)|. Let T : C[0, 1] → C[0, 1] be defined as $T(f)(x)={\int }_0$ˣ f(t)dt, for x ∈ [0, 1]. Let $T^n=T$ ∘ T ∘ ⋯ ∘ T (n times). Which of the following statements are true?

Let $T:{\mathbb{C}}^7\to {\mathbb{C}}^7$ be $a\mathbb{C}-$linear operator with eigenvalues 2, 3 and 5. Consider the subspace W := {$v\in {\mathbb{C}}^7:(T-5I{)}^{k}v=0$ for some integer k > 0} of ${\mathbb{C}}^7$. Suppose that $(T-2I{)}^2(T-3I{)}^2(T-5I{)}^2=0$. Which of the following statements are necessarily true?

Which of the following matrices are similar over $\mathbb{R}$ to the matrix A whose rows are (−1, 1, 0, 0), (0, −1, 0, 0), (0, 0, 1, 1) and (0, 0, 0, 1)?

Let $\gamma :[0,1]\to \mathbb{C}$ be the function t ↦ $e^{2\pi it}$ and $I={\int }_{\gamma }{e}^z$ $e^{1/z}$ dz. Which of the following statements are true?

Let G be a group, H a subgroup of G, and T = {gH | g ∈ G}, the set of all left cosets of H in G. Let S_T be the set of all permutations of T and $\pi :G\to S_T$ be the map defined by $\pi (g)(g_{1}H)=$ gg${}_{1}H$. For a prime number p, let 𝔽_p denote the field with p elements. In which of the following cases is $\ker \pi$ trivial?

For a group G, let Aut(G) denote the group (under composition) of all bijective group homomorphisms from G onto itself. Which of the following statements are true?

Let $G_1$ and $G_2$ be subgroups of a group G. Which of the following statements are true?

Let $f:\mathbb{Q}[X]\to \mathbb{Q}[X]$ be a ring homomorphism with f(1) = 1. For n ≥ 1, let $f^n=f$ ∘ ⋯ ∘ f (n times). Which of the following statements are true?

Let $f(X)=X^5+X+1\in \mathbb{Q}[X]$ and $g(X)=X^5-X+1\in \mathbb{Q}[X]$. Which of the following statements are true?

Let p > 2 be a prime number. Let 𝔽_p denote the field with p elements and 𝔽̄_p an algebraic closure of 𝔽_p. Which of the following statements are true?

Let $p:\mathbb{R}\to \mathbb{R}$ be a nonconstant polynomial. Which of the following statements are true?

Consider $\mathbb{R}$ with the usual topology and S = {$a+b\sqrt{2}$ | $a,b\in \mathbb{Q}$} $\subset \mathbb{R}$ with the subspace topology. Which of the following statements are true?

Let D = {$(x,y)\in {\mathbb{R}}^2:-1\le x\le 1,-1\le y\le 1$}, and $f:D\to \mathbb{R}$ be the function defined by $f(x,y)=1+\surd (y$₊), where y₊ = max{y, 0}. Consider the initial value problem (IVP) dy/dx = f(x, y), y(0) = 0. Then which of the following statements are true?

Consider the initial value problem (IVP) y′ + y = 0, y(0) = 1. Let $(y_n)$ be the iterates of forward Euler method, applied to the IVP, with step size h where 0 < h < 1. Then which of the following statements are true?

Let u(x) be the solution to the Volterra integral equation $u(x)=x^2+4{\int }_0$ˣ $(t-x{)}^{2}u(t)$ dt. Then which of the following statements are true?

Let f and K be such that the solution of the initial value problem y″ − 3y′ + 2y = 4sin(x), y(0) = 1, y′(0) = −2 satisfies the Volterra integral equation $y(x)=f(x)+{\int }_0$ˣ K(x, t)y(t) dt. Then which of the following statements are true?

Suppose f, g are smooth functions of generalized coordinates $q_1,q_2,\dots ,q_n$, the associated conjugate momenta $p_1,p_2,\dots ,p_n$, and time t. Let [f, g] denote the Poisson bracket of f and g. Suppose H is a Hamiltonian of the system. Then which of the following statements are true?

Let $X_1,X_2,\dots$ be a sequence of independent and identically distributed random variables with $E(X_1)=0,E(X_1^2)=1,E(X_1^3)=0,E(X_1^4)=3$. Let $S_n={\sum }_i$₌${}_1^n{X}_i,T_n={\sum }_i$₌${}_1^n{X}_i^2,U_n={\sum }_i$₌${}_1^n{X}_i^3$ and $V_n={\sum }_i$₌${}_1^n{X}_i^4$. Then, which of the following statements are true?

Let $X_1,X_2,\dots ,X_n(n\ge 3)$ be a random sample from the uniform distribution on the interval $({\theta }_1-{\theta }_2,{\theta }_1+{\theta }_2)$, where ${\theta }_1\in \mathbb{R}$ and ${\theta }_2>0$ are unknown parameters. Let $X_{(j)}$ be the jᵗʰ order statistic, j = 1, 2, …, n, and let $\bar{X}=(1/n){\sum }_i$₌${}_1^n{X}_i$. Here, $(X_{(1)},X_{(n)})$ is a complete and sufficient statistic for $({\theta }_1,{\theta }_2)$. Then, which of the following statements are true?

Let $X_1,X_2,\dots ,X_n(n\ge 2)$ be a random sample from a continuous distribution with the probability density function f(x|$\theta )=$ $(1/2\theta )e^{-|x-\theta |/\theta }$, $x\in \mathbb{R}$, where $\theta (>0)$ is an unknown parameter. Let $U_n=(1/n){\sum }_i$₌${}_1^n{X}_i,V_n=(1/n){\sum }_i$₌${}_1^n{X}_i^2$, and $S_n^2=(1/(n-1)){\sum }_i$₌${}_1^n(X_i-U_n{)}^2$. Then, which of the following statements are true?

Let X be a random sample of size one from the probability density function f(x|$\theta )=$ $\theta e^{-\theta (x-1)}$ if x > 1, and 0 otherwise, where $\theta (>0)$ is the unknown parameter. Suppose we want to test the null hypothesis $H_0:\theta =1$ against the alternative hypothesis $H_1:\theta \ne 1$, based on the observed value x of X. Then, which of the following statements are true?

Let $(X_1,X_2,\dots ,X_7)$ and $(Y_1,Y_2,\dots ,Y_9)$ be two independent random samples from the continuous distribution functions $F(x-\mu )$ and $F(x-\theta )$, respectively, where $F,\mu$ and $\theta$ are all unknown. Further, let $\mu$ be the unique median of $F(x-\mu )$ and $\theta$ be the unique median of $F(x-\theta )$. Let $R_i$ be the rank of $Y_i$ in the combined sample, i = 1, 2, …, 9. For testing $H_0:\mu =\theta$ against $H_1:\mu >\theta$, the test statistic $T={\sum }_i$₌${}_1^9{R}_i$ is proposed. Then, which of the following statements are true?

If the incidence matrix of a block design is given by N with rows (1, 1, 1, 0), (1, 1, 0, 1), (1, 0, 1, 1) and (0, 1, 1, 1), then which of the following statements are true?