CSIR NET December 2024 — Part C

Consider maximizing the objective function $P=x_1+2x_2+3x_3$ subject to $x_1+x_2\le 5,x_2+x_3\le 2,x_1+x_2+x_3\le 6,x_1\ge 0,x_2\ge 0,x_3\ge 0$. Then, which of the following statements are true?

For a real number x, let [x] denote the largest integer ≤ x. Which of the following sets are uncountable?

Let A = {$1/(\sqrt{n+\sqrt{n}}-\sqrt{n})$ | $n\in \mathbb{Z},n>0$}. Which of the following statements are true?

Let $f:[0,\infty )\to [0,\infty )$ be a continuous function such that $\lim (x\to \infty )f(x)/x=\alpha <1$. Let S = {$c\in [0,\infty )$ | f(c) = c}. Which of the following statements are true?

Let $f:\mathbb{R}\to \mathbb{R}$ be a uniformly continuous function. For each positive integer n and $x\in \mathbb{R}$, let $g_n$ and $h_n$ be defined by $g_n(x)=f(x+1/n)$ and $h_n(x)=f($nx). Which of the following statements are necessarily true?

Consider the following functions on the interval $[0,1]:g(x)=\sin (\pi x),h(x)=\cos (\pi (x-1))$. Which of the following statements are true?

For each n > 1, let V denote the $\mathbb{C}-$vector space of all n × n complex matrices and A ∈ V. Which of the following statements are necessarily true?

For each n > 1, let V be the $\mathbb{R}-$vector space of all n × n real matrices and A ∈ V be invertible. Consider the $\mathbb{R}-$linear transformation $\phi :\mathbb{R}[x]\to V$ such that $\phi (x^k)=A^k$ for all k ≥ 1 and $\phi (1)$ is the identity matrix of order n. Which of the following statements are necessarily true?

Let $T:{\mathbb{C}}^2\to {\mathbb{C}}^2$ be $a\mathbb{C}-$linear transformation. For any ordered basis ℬ of ${\mathbb{C}}^2$, let [T]ℬ denote the matrix of T with respect to ℬ. Suppose that ℬ${}_1$ and ℬ${}_2$ are two ordered bases of ${\mathbb{C}}^2$ such that the matrix [T](ℬ${}_1)$ is upper-triangular and [T]_(ℬ${}_2)=([T{]}_{(}$ℬ${}_1){)}^2$. Which of the following statements are FALSE?

Let $T:{\mathbb{R}}^4\to {\mathbb{R}}^4$ be an $\mathbb{R}-$linear transformation. Assume that the characteristic polynomial of T has two distinct monic irreducible quadratic factors $q_1(x)$ and $q_2(x)$. Which of the following statements are true?

Let (V, ⟨ , ⟩) be an inner product space over $\mathbb{C}$ and T : V → V be $a\mathbb{C}-$linear transformation. Let $v_1$ and $v_2$ be non-zero vectors in V such that Tv${}_1=c_1{v}_1$ and Tv${}_2=c_2{v}_2$ for some $c_1,c_2\in \mathbb{C}$. Let $v_3=v_2-($⟨$v_2,v_1$⟩/‖$v_1$‖${}^2)v_1$. Suppose that $v_3$ is non-zero and Tv${}_3=c_3{v}_3$ for some $c_3\in \mathbb{C}$. Which of the following statements are true?

Let V be a two-dimensional real vector space with a basis {$v_1,v_2$}. Let $f:V\times V\to \mathbb{R}$ be a symmetric bilinear form. Let $r=f(v_1,v_1),s=f(v_1,v_2),t=f(v_2,v_2)$. Which of the following statements are necessarily true?

For z = x + iy $\in \mathbb{C}$, let f(z) = u(x, y) + iv(x, y) be an entire function such that u(x, y) + v(x, y) = 1. Which of the following statements are FALSE?

Consider the disc 𝔻 = {$z\in \mathbb{C}$ : |z| < 1} and a non-constant holomorphic function f : 𝔻 → 𝔻. Suppose that f(0) = 0. For each n ≥ 1 and z ∈ 𝔻, define $f_n(z)=f(z^n)$. Which of the following statements are true?

Let G be a finite group. For any prime number p, let E(p) = {g ∈ G | g^p = 1}. Which of the following statements are true?

Which of the following quotient rings are fields?

Consider ${\mathbb{R}}^2$ with the Euclidean metric d. Let $O=(0,0),P_0=(0,1)\in {\mathbb{R}}^2$, and for any integer $n\ge 1,P_n=(1/n,1)\in {\mathbb{R}}^2$. Let X = ⋃$(n\ge 0)L_n$, where $L_n$ is the line segment joining $P_n$ and O. Define $d_X:X\times X\to \mathbb{R}$ as follows: d_X(a, b) = d(a, b) when $a,b\in L_n$ for some n, and d_X(a, b) = d(a, O) + d(b, O) otherwise. Let $\tau$ be the smallest topology such that the sets $B_{(}a,\epsilon )=$ {$b\in X:d_X(a,b)<\epsilon$} are open in $\tau$ for all a ∈ X and $\epsilon >0$. Which of the following statements are true?

Consider the ordinary differential equation (ODE$)d^{2}y/$dx${}^2+2(e^{(}2x)-1)($dy/dx$)+\alpha e^{(}4x)y=0,x\in \mathbb{R}$, where $\alpha \in \mathbb{R}$. Note that the ODE transforms into an equation with constant coefficients under the change of independent variable given by t = e^(2x)/2. Then which of the following statements are true?

If x = x(t), y = y(t) is the solution of the initial value problem dx/dt + dy/dt = 2x − 3y + eᵗ, dx/dt + 2(dy/dt) = 3x − 4y + 2eᵗ, x(0) = −1, y(0) = −1/2, then which of the following statements are true?

Let u = u(x, t) be the solution to the initial value problem ${\partial }^{2}u/\partial t^2-{\partial }^{2}u/\partial x^2=0,x\in \mathbb{R},t>0,u(x,0)=f(x),x\in \mathbb{R},(\partial u/\partial t)(x,0)=1/\sqrt{1+x^2},x\in \mathbb{R}$, where f is a twice continuously differentiable function defined on $\mathbb{R}$ satisfying f(x) → 0 as |x| $\to \infty$. Then which of the following statements are true?

Define S := {$y\in C^1[0,1]:y(0)=y(1)=0$}, ‖y‖$(\infty ):=\max (x\in [0,1])$|y(x)| for every $y\in S,B_0(0,\epsilon ):=$ {y ∈ S : ‖y‖$(\infty )<\epsilon$}, $B_1(0,\epsilon ):=$ {y ∈ S : ‖y‖$(\infty )+$ ‖y′‖$(\infty )<\epsilon$}. Consider the functional $J:S\to \mathbb{R}$ given by $J[y]={\int }_0^1[(y^{\prime }{)}^2-2x(y^{\prime }{)}^4]$dx, then there exists an $\epsilon >0$ such that

Consider the variational problem minimize $J[y]={\int }_0^1[(y^{\prime }{)}^2+2025$ yy′]dx, where y(0) and y(1) are free. Then which of the following statements are true?

Consider the boundary value problem (BVP$)d^{2}y/$dx${}^2+4($dy/dx$)+(\lambda +2)y=0,y(0)=y(\pi )=0$. Then which of the following statements are true?

Consider a particle of mass m which is moving on a surface due to gravity. Suppose the Lagrangian of the particle in the cylindrical coordinates is given by $L=(m/2)[(1+4{\alpha }^2{r}^2)$ṙ${}^2+r^2\theta$̇${}^2]-m\alpha$gr${}^2$, where $\alpha$ is a positive constant, and g is the acceleration due to gravity. If the particle is in circular motion, then

Let $X_1,X_2\dots$ be a sequence of independent random variables with $X_i$ following N(0, 1 + 1/i) distribution for all $i\in \mathbb{N}$. Let X be N(0, 1)-random variable independent of {$X_n:n\ge 1$}. Then, which of the following statements are true?

Let $U_1,U_2,\dots$ be a sequence of independent and identically distributed (i.i.d.) U(0, 1) random variables. Let $G_n=(\prod (i=\mathrm{1..}n)U_i{)}^{(}1/n)$ be the geometric mean of $U_1,U_2,\dots ,U_n$ for $n\in \mathbb{N}$. Let $X_1$ and $X_2$ be degenerate random variables such that $P(X_1=0)=1$ and $P(X_2=1/e)=1$. Then, which of the following statements are true?

Let $X_1,X_2,\dots ,X_n$ be independent and identically distributed (i.i.d.) random variables such that for $x=3,4,\dots ,P(X_1=\pm x)=1/(2$cx${}^2\ln (x))$, where $c=\sum (x=3..\infty )1/(x^2\ln (x))$. Then, which of the following statements are true?

Consider the Markov chain with state space {0, 1, 2, 3, 4} and the transition probability matrix P = [[0.4, 0.3, 0.3, 0, 0], [0, 0.5, 0, 0.5, 0], [0.5, 0, 0.5, 0, 0], [0, 0.5, 0, 0.5, 0], [0, 0.3, 0, 0.3, 0.4]]. Then, which of the following statements are true?

Let X and Y be independent random variables with the moment generating functions M_X(t) = (1/9)(2 + eᵗ${)}^2,t\in \mathbb{R}$ and M_Y(t) = exp[eᵗ $-1],t\in \mathbb{R}$, respectively. Then, which of the following statements are true?

Let $X_1,X_2,\dots ,X_{10}$ be a random sample of size 10 from a Bernoulli distribution with success probability $1/(1+e^{\theta })$, where $\theta \in \mathbb{R}$ is an unknown parameter. If $M=\sum (i=\mathrm{1..10})X_i$, then which of the following statements are true?

Let $X_1,X_2,\dots ,X_n$ be a random sample of size n from $U(\theta ,\theta +1)$ distribution, where $\theta \in \mathbb{R}$ is an unknown parameter. If $T_n=\max${$X_1,X_2,\dots ,X_n$}, n = 1, 2, …, then which of the following statements are true?

Consider a manufacturing unit producing three-component series systems, where component's lifetimes are independent and identically distributed random variables with hazard function $h(x)=\lambda$ if x > 0, and 0 otherwise. Here, $\lambda >0$ is an unknown parameter. Let $X_1,X_2,X_3$ be a random sample of size 3 on lifetimes of systems produced by the manufacturing unit. Let $\theta =P_{\lambda }(X_1>1/2)$. If $\theta$̂ is the maximum likelihood estimate of $\theta$ based on the realization $x_1=1/2,x_2=1/6,x_3=1/3$ of the random sample, then which of the following statements are true?

The life (in hours) of an electrical component is exponentially distributed with mean $\theta >0$. For testing $H_0:\theta =5/(\ln (10)-\ln (9))$ against $H_1:\theta =5/\ln (2)$, three such components are chosen at random. Suppose that we reject the null hypothesis $H_0$ if and only if two or more of these three components survive for less than five hours. Then, which of the following statements are true?

Consider a linear model $Y_i={\beta }_1+{\beta }_2+\cdots +{\beta }_i+{\epsilon }_i,1\le i\le n$, where the errors ${\epsilon }_i^{\prime }s$ are uncorrelated with zero mean and finite variance ${\sigma }^2>0$. Then, which of the following statements are true?

Consider a linear regression model $Y=X\beta +\epsilon$, with r regressors and an intercept. Random error $\epsilon$ ~ $N_n(0,{\sigma }^2{I}_n)$ and X has full column rank. Here $I_n$ denotes the identity matrix of order n. Regression coefficients are estimated by the least squares estimation method. Let $\sigma$̂${}^2$ and $\sigma$̂${}^2($MLE), respectively, be the mean squares residuals and the maximum likelihood estimator of ${\sigma }^2$. Then, which of the following statements are true?

If $(X_1,X_2)$ᵗ ~ $N_2((0,0)$ᵗ, [[3, 1], [1, 3]]), then which of the following statements are true?

Let $f:[0,1]\to \mathbb{R}$ be a Riemann integrable function. Define $F(x)={\int }_0$ˣ f(t)dt, ∀x ∈ [0, 1]. Which of the following statements are necessarily true?

Let $\mu$ denote the Lebesgue measure on $\mathbb{R}$ and $f:\mathbb{R}\to [0,\infty )$ be a Lebesgue measurable function. For n ≥ 1, let $E_n=$ {$x\in \mathbb{R}:n-1\le f(x)<n$}. Suppose that $\sum (n\ge 1)n^2\mu (E_n)<\infty$. Which of the following statements are true?

Consider the real vector space X = {$f:[0,1]\to \mathbb{R}$ | f is continuous}, along with the norms ‖·‖${}_1$ and ‖·‖${}_2$ defined by ‖f‖${}_1={\int }_0^1$|f(x)|dx and ‖f‖${}_2=({\int }_0^1$|f(x)|${}^2$dx)^(1/2). For n ≥ 1 and x ∈ [0, 1], let $f_n(x)=$ nx${}^n$. Which of the following statements are true?

Let ‖·‖ denote the Euclidean norm on ${\mathbb{R}}^2$ and $\alpha \ge 0$. Let $f:{\mathbb{R}}^2\to \mathbb{R}$ be a function such that |f(x)| ≤ ‖x‖${}^{\alpha }$. Which of the following statements are necessarily true?

Let g be a real-valued continuous function on the set {$(x,y)\in {\mathbb{R}}^2$ | $x^2+y^2=1$} such that g(0, 1) = g(1, 0) = 0 and g(−x, −y) = −g(x, y). Define $f:{\mathbb{R}}^2\to \mathbb{R}$ by $f(x,y)=\sqrt{x^2+y^2}\cdot g(x/\sqrt{x^2+y^2},y/\sqrt{x^2+y^2})$ if (x, y) ≠ (0, 0), and f(x, y) = 0 if (x, y) = (0, 0). For each $(a,b)\in {\mathbb{R}}^2$, define $h_{(}a,b):\mathbb{R}\to \mathbb{R}$ by h_(a,b)(t) = f(ta, tb). Which of the following statements are necessarily true?

Which of the following statements are true?

Let n be a positive integer and A, B be n × n complex matrices such that the minimal polynomials of A and B are the same. Which of the following conditions ensure that A is similar to B?

Let $f:\mathbb{C}$ \ {0} $\to \mathbb{C}$ be a non-zero holomorphic function such that |f(z)| ≤ |z|^(5/2) + 1/|z|${}^{(}1/2),z\in \mathbb{C}$ \ {0}. Which of the following statements are true?

Let $p(z)=a_0+a_{1}z+\cdots +a_n{z}^n$ be a polynomial of degree ≥ 3. Suppose that p(0) = 0 and p′(0) ≠ 0. Let $f:\mathbb{C}$ \ {0} $\to \mathbb{C}$ be defined by $f(z)=p(z)/z^2$. Which of the following statements are true?

An abelian group G is said to have property (P) if for any subgroup N of G, there exists a subgroup H of G such that G = N + H and N ∩ H = {0}. Which of the following statements are true?

Which of the following rings are integral domains?

For a finite group G, let S(G) denote the set of all its Sylow subgroups. A finite group G is said to have property (J) if G is isomorphic to the direct product $\prod (H\in S(G))H$. Which of the following statements are true?

Which of the following statements are true?

Let X and Y be topological spaces and f : X → Y be a continuous function. Which of the following statements are true?

Consider the partial differential equation $4({\partial }^{2}u/\partial x^2)+6({\partial }^{2}u/\partial x\partial y)+3({\partial }^{2}u/\partial y^2)=0$. Which of the following partial differential equations can be obtained by a change of independent variables given by $(\xi ,\eta )$ᵗ = A(x, y)ᵗ for some 2 × 2 invertible matrix A with real entries?

Consider the following statements: $S_1$: There exists an $\epsilon >0$ such that $\forall x_0\in (2-\epsilon ,2+\epsilon )$, the iterative sequence defined by $x(n+1)=6/(5-x_n),n=0,1,2,3,\dots$, converges to $2.S_2$: There exists an $\epsilon >0$ such that $\forall x_0\in (3-\epsilon ,3+\epsilon )$, the iterative sequence defined by $x(n+1)=(x_n^2+6)/5,n=0,1,2,3,\dots$, converges to $3.S_3$: There exists an $\epsilon >0$ such that $\forall x_0\in (3-\epsilon ,3+\epsilon )$, the iterative sequence defined by $x(n+1)=(5x_n-6)/x_n,n=0,1,2,3,\dots$, converges to 3. Then which of the following statements are true?

Let $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be a function such that f and its partial derivatives of orders less than or equal to 3 are continuous and bounded. Let $y:[0,1]\to \mathbb{R}$ be the solution of dy/dt $=f(t,y(t)),0\le t\le 1,y(0)=y_0$, where $y_0\in \mathbb{R}$. For h > 0, denote t_j = jh. Let Y_j be an approximation of y(t_j), defined by Y_(j+1) = Y_j + ahf(t_j, Y_j) + bhf(t_(j+1), Y_j + chf$(t_j,Y_j)),0<(j+1)h<1,Y_0=y_0$, where $a,b,c\in \mathbb{R}$. If there exists an M > 0 such that |y(t_(j+1)) − y(t_j) − ahf(t_j, y(t_j)) − bhf(t_(j+1), y(t_j) + chf(t_j, y(t_j)))| ≤ Mh${}^3$ for every h > 0, and every j with 0 ≤ (j+1)h < 1, then

The integral equation $u(x)=f(x)+(2/\pi ){\int }_0^{\pi }\cos (x+t)u(t)$dt has infinitely many solutions if

Let R(x, t) and u(x) denote the resolvent kernel and the solution, respectively, of the Volterra integral equation u(x) = eˣ $+2{\int }_{(}\ln 6{)}^x{e}^{(}2(t-x))u(t)$dt. Then which of the following statements are true?

Let $X_1,X_2,\dots ,X_n(n\ge 3)$ be a random sample from a population with absolutely continuous cumulative distribution function F(·). The corresponding order statistics are X(1:n) < X(2:n) < ⋯ < X(r:n) < ⋯ < X(n:n). Define, for r = 2, 3, Y(r,n) = nF(X(r:n)). Suppose that Y(r,n) converges in distribution to a random variable Y_r as $n\to \infty ,r=2,3$. Then, which of the following statements are true?

Let X, Y, Z be independent and identically distributed (i.i.d.) random variables, each following Bernoulli$(\theta ),0<\theta <1$. Then, which of the following statements are true?

Let $(X_1,Y_1),(X_2,Y_2),\dots ,(X_n,Y_n)$ be a random sample of size n from a bivariate distribution F(X,Y) with absolutely continuous marginal distribution functions F_X and F_Y of X and Y, respectively. Let r_S be the Spearman's rank correlation coefficient defined with ranks of $X_i$ and ranks of $Y_i,i=1,2,\dots ,n$. Then, which of the following statements are true?

Let $X_1,X_2,\dots ,X_m$ and $Y_1,Y_2,\dots ,Y_n$ be two mutually independent random samples from populations with absolutely continuous distribution functions F_X and F_Y, respectively. For N = m + n, define $T_N=\sum (i=\mathrm{1..}N)$ iZ${}_i$ where $Z_i=1$ if the i-th observation in the combined ordered arrangement of N observations is from F_X; and $Z_i=0$, otherwise. Then, which of the following statements are true?

Consider a probability proportional to size without replacement sample involving two draws from a population of N (> 6) units with normed size measures $p_i^{\prime }s(p_i>0,i=1,2,\dots ,N;\sum (i=\mathrm{1..}N)p_i=1)$. Then, which of the following statements are true?

Consider maximizing the objective function $P=x_1+x_2$ subject to $x_1+x_2+2x_3\le 5,2x_1-3x_3\ge 1,x_2+x_3\le 0,x_1\ge 0,x_2\ge 0,x_3\ge 0$. Then, which of the following statements are true?

For each positive integer n, define $f_n:[0,1]\to \mathbb{R}$ by $f_n(x)=$ nx$(1-x{)}^n$. Which of the following statements are true?

Let $g:\mathbb{R}\to \mathbb{R}$ be a continuous function. Define $f(x)={\int }_0$ˣ(x − t)g(t)dt, $x\in \mathbb{R}$. Which of the following statements are true?

Let $f:[0,1]\to \mathbb{R}$ be a monotonic function. Which of the following statements are true?

For a variable x, consider the $\mathbb{Q}-$vector space V = {ax${}^3+$ bx${}^2+$ cx + d | $a,b,c,d\in \mathbb{Q}$}. Further, let A = {$f:V\to \mathbb{Q}$ | f is $a\mathbb{Q}-$linear transformation} and B = {f ∈ A | f(1) = 0}. Which of the following statements are true?

For a variable x, consider the real vector space V = {ax${}^3+$ bx${}^2+$ cx + d | $a,b,c,d\in \mathbb{R}$}. Let D : V → V be the linear transformation where D(f) is the derivative of f with respect to x, and M : V → V be the linear transformation M(f) = xD(f). Which of the following statements are true?

Let A, B be 2 × 2 matrices with real entries, and M = AB − BA. Let $I_2$ denote the 2 × 2 identity matrix. Which of the following statements are necessarily true?

Let $M_2(\mathbb{R})$ denote the $\mathbb{R}-$vector space of 2 × 2 matrices with real entries. Let A = [[1, 2], [0, 3]] and B = [[−1, 0], [1, 5]]. Define a linear transformation $T:M_2(\mathbb{R})\to M_2(\mathbb{R})$ by T(X) = AXBᵗ, where Bᵗ denotes the transpose of the matrix B. Which of the following statements are true?

For a 4 × 4 positive definite real symmetric matrix A and real numbers a, b, c, d, consider the 5 × 5 matrix B whose first row is (0, a, b, c, d), whose first column is (0, a, b, c, d)ᵗ, and whose lower-right 4 × 4 block is A. Which of the following statements are necessarily true?

Let R be a nonzero ring with unity such that $r^2=r$ for all r ∈ R. Which of the following statements are true?

Let $f\in \mathbb{R}[x]$ be a product of distinct monic irreducible polynomials $P_1,P_2,\dots ,P_n$, where n ≥ 2. Let (f) denote the ideal generated by f in the ring $\mathbb{R}[x]$. Which of the following statements are true?