CSIR NET June 2024 — Part C

Consider the linear programming problem: max{$x_1+x_2+x_3$} subject to constraints $x_1+x_2-x_3\le 1,x_1+x_3\le 2,0\le x_1\le 1/2,x_2\ge 0$, and $0\le x_3\le 1$. Which of the following statements are true?

Let $(a_n{)}_n{\ge }_1$ be a sequence of positive real numbers. Let $b_n=a_n/\max${$a_1,\dots ,a_n$}, n ≥ 1. Which of the following statements are necessarily true?

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous and one-to-one function. Which of the following statements are necessarily true?

Let $K\subseteq \mathbb{R}$ be non-empty and f : K → K be continuous such that |x − y| ≤ |f(x) − f(y)| for all x, y ∈ K. Which of the following statements are true?

Let $f:[0,1)\to [1,\infty )$ be defined by f(x) = 1/(1 − x). For n ≥ 1, let $p_n(x)=1+x+\cdots +x^n$. Then which of the following statements are true?

Let V be the subspace spanned by the vectors $v_1=(1,0,2,3,1),v_2=(0,0,1,3,5),v_3=(0,0,0,0,1)$ in the real vector space ${\mathbb{R}}^5$. Which of the following vectors are in V?

Consider $\mathbb{R}$ and $\mathbb{Q}[x]$ as vector spaces over $\mathbb{Q}$. Which of the following statements are true?

Let $T:{\mathbb{R}}^4\to {\mathbb{R}}^4$ be a linear map with four distinct eigenvalues and satisfying $T^4-15T^2+10T+24I=0$. Which of the following statements are necessarily true?

Let A be a 4 × 4 real matrix whose minimal polynomial is $x^2+x+1$ and let $B=A+I_4$. Which of the following statements are necessarily true?

Let $M_5(\mathbb{C})$ be the complex vector space of 5 × 5 matrices with entries in $\mathbb{C}$. Let V be a non-zero subspace of $M_5(\mathbb{C})$ such that every non-zero A ∈ V is invertible. Which among the following are possible values for the dimension of V?

Let V (≠ {0}) be a finite dimensional vector space over $\mathbb{R}$ and T : V → V be a linear operator. Suppose that the kernel of T equals the image of T. Which of the following statements are necessarily true?

Let $q_1(x_1,x_2)$ and $q_2(y_1,y_2)$ be real quadratic forms such that there exist $(u_1,u_2),(v_1,v_2)\in {\mathbb{R}}^2$ such that $q_1(u_1,u_2)=1=q_2(v_1,v_2)$. Define $q(x_1,x_2,y_1,y_2)=q_1(x_1,x_2)-q_2(y_1,y_2)$. Which of the following statements are necessarily true?

Suppose that f is an entire function such that |f(z)| ≥ 2024 for all $z\in \mathbb{C}$. Which of the following statements are necessarily true?

Let f be an entire function such that for every integer k ≥ 1 there is an infinite set X_k such that f(z) = 1/k for all z ∈ X_k. Which of the following statements are necessarily true?

Which of the following numbers are order of some element of the symmetric group $S_5$?

Let R and S be non-zero commutative rings with multiplicative identities 1_R, 1_S, respectively. Let f : R → S be a ring homomorphism with f(1_R) = 1_S. Which of the following statements are true?

Let I be an ideal of the ring 𝔽${}_2[t]/(t^2(1-t{)}^2)$. Which of the following are the possible values for the cardinality of I?

If $x_1=x_1(t),x_2=x_2(t)$ is the solution of the initial value problem e^(−t) dx${}_1/$dt $=-x_1+x_2,e^{(}-t)$ dx${}_2/$dt $=-x_1-x_2,x_1(0)=1,x_2(0)=0$, and $r(t)=\sqrt{x_1^2(t)+x_2^2(t)}$, then which of the following statements are true?

Consider the initial boundary value problem (IBVP$)u_t+u_x=2u$ for x > 0, t > 0; u(0, t) = 1 + sin t for t > 0; u(x, 0) = eˣ cos x for x > 0. If u is the solution of the IBVP, then the value of $u(2\pi ,\pi )/u(\pi ,2\pi )$ is

Let B(0,2) = {$(x,y)\in {\mathbb{R}}^2:x^2+y^2<4$}, and $\partial B$ denote the boundary of B(0,2). Assume $(\alpha ,\beta )\ne (0,0),k\in \mathbb{R}$, and u is any solution to $-\Delta u=0$ in $B(0,2),\alpha u(x,y)+\beta (\partial u/\partial \nu )(x,y)=1+(x^2+y^2)k$ on $\partial B$, where $\nu (x,y)$ is the unit outward normal to B(0,2) at $(x,y)\in \partial B$. Consider the following statements: $S_1$: If $\beta =0$, then there exists $a(x_0,y_0)\in B(0,2)$ such that |$u(x_0,y_0)$| = |1 + 4k|/|$\alpha$|. $S_2$: If $\alpha =0$, then k = −1/4. Then

The infimum of the set {${\int }_a$ᵇ $\sqrt{1+(y^{\prime }(t){)}^2}$dt : $y\in C^1[a,b],y(a)=a^2,y(b)=b-5$} is

For $c\in \mathbb{R}$, consider the following Fredholm integral equation y(x) = 1 + x + cx${}^2+2{\int }_0^1(1-3$xt)y(t)dt. Then the values of c for which the integral equation admits a solution are

Consider a solid torus of constant density $\rho$, formed by revolving the disc $(y-b{)}^2+z^2\le a^2,x=0$ about the z-axis, where 0 < a < b. Then the moment of inertia of the solid torus about the z-axis is

Let X and Y be independent random variables with X ~ N(2, 4) and Y ~ N(−4, 9) where $N(\mu ,{\sigma }^2)$ denotes a normal distribution with mean $\mu$ and variance ${\sigma }^2$. Given $\Phi (1)=0.8413,\Phi (2)=0.9772$ and $\Phi (3)=0.9987$ where $\Phi (\cdot )$ is the cumulative distribution function of a standard normal random variable. Which of the following statements are true?

Transition probability matrix of a homogeneous Markov chain with states 0, 1, 2, 3 is P = [[1/4, 3/4, 0, 0], [1, 0, 0, 0], [2/3, 0, 1/3, 0], [0, 0, 2/5, 3/5]], where rows and columns are indexed 0, 1, 2, 3. Which of the following statements are true?

Let X and Y be jointly distributed continuous random variables with joint probability density function f(x, y) = x/y if 0 < x < y < 2, and 0 otherwise. Which of the following statements are true?

Let $X_1,X_2$ denote lifetimes (in years) of 2 components of an electronic system. Let $Y_1=X_1+X_2,Y_2=\max${$X_1,X_2$} and $Y_3=\min${$X_1,X_2$}. Assume that $X_1$ and $X_2$ are independent, each following exponential distribution with probability density function f(x) = (1/2)e^(−x/2) if x > 0, and 0 otherwise. Which of the following statements are true?

Let $X_1,X_2,X_3$ be a random sample from a continuous distribution having cumulative distribution function F(t), probability density function f(t), and failure rate function r(t) = f(t)/(1 − F(t)), t > 0, where F(0) = 0. If r(t) = 1 for all t > 0, then which of the following statements are true?

Let $X_1,\dots ,X_n(n\ge 2)$ be a random sample from $aU(-\theta ,2\theta )$ distribution, where $\theta >0$ is an unknown parameter. Let $\bar{X}=(1/n){\sum }_i$₌${}_1^n{X}_i,X_{(1)}=\min${$X_1,\dots ,X_n$} and $X_{(n)}=\max${$X_1,\dots ,X_n$}. Which of the following statements are true?

Let $Y_1,\dots ,Y_n(n\ge 2)$ be independent observations; $Y_i$ ~ $N(\beta x_i,{\sigma }^2),i=1,\dots ,n$; where $x_1,\dots ,x_n$ and ${\sigma }^2(>0)$ are known constants and $\beta \in \mathbb{R}$ is an unknown parameter. Consider $N({\beta }_0,{\tau }^2)$ prior for the parameter $\beta$, where ${\beta }_0$ and ${\tau }^2(>0)$ are known constants, and $N(\mu ,{\lambda }^2)$ denotes a normal distribution with mean $\mu$ and variance ${\lambda }^2$. Suppose ȳ $=(1/n){\sum }_i$₌${}_1^n{y}_i$ and $\bar{x}=(1/n){\sum }_i$₌${}_1^n{x}_i$ are observed sample means. Under squared error loss function, which of the following statements are true?

Let $X_1,\dots ,X_{12}$ be a random sample from the N(2, 4) distribution and $Y_1,\dots ,Y_{15}$ be a random sample from the N(−2, 5) distribution, where $N(\mu ,{\sigma }^2)$ denotes a normal distribution with mean $\mu$ and variance ${\sigma }^2$. Assume that the two random samples are mutually independent. Let $\bar{X}=(1/12){\sum }_i$₌${}_1^{12}{X}_i,S_1^2=(1/11){\sum }_i$₌${}_1^{12}(X_i-\bar{X}{)}^2$, Ȳ $=(1/15){\sum }_j$₌${}_1^{15}{Y}_j,S_2^2=(1/14){\sum }_j$₌${}_1^{15}(Y_j-$ Ȳ${)}^2$. Which of the following statements are true?

In a standard linear regression model, let $R^2$ and ${\bar{R}}^2$, respectively, denote the coefficient of determination and adjusted coefficient of determination. Which of the following statements are true?

Consider the two-way ANOVA model $Y_{ij}=\mu +{\alpha }_i+{\beta }_j+{\epsilon }_{ij},i=1,2;j=1,2$, where $\mu$ is the overall mean effect, ${\alpha }_i$ is the effect of the i-th level of factor $A,{\beta }_j$ is the effect of j-th level of factor $B,Y_{ij}$ is the response of the (i, j)-th experimental unit and ${\epsilon }_{ij}$ is the corresponding error with $E({\epsilon }_{ij})=0$ for i = 1, 2; j = 1, 2. Which of the following are estimable linear parametric functions?

Let $X=(X_1,X_2{)}^T$ be a bivariate random vector with covariance matrix $\sum =[[1,\sqrt{2}],[\sqrt{2},2]]$. Which of the following statements are true?

Let ${\sum }_n$₌${}_1^{\infty }{a}_n$ be a convergent series of real numbers. For n ≥ 1 define $A_n=a_n$ if $a_n>0$ and 0 otherwise; $B_n=a_n$ if $a_n<0$ and 0 otherwise. Which of the following statements are necessarily true?

Define $f:\mathbb{R}\to \mathbb{R}$ by f(x) = x|x|. Which of the following statements are true?

Let $(a_n{)}_n{\ge }_1$ be a bounded sequence of real numbers such that $li{m}_n\to \infty a_n$ does not exist. Let S = {$l\in \mathbb{R}$ : there exists a subsequence of $(a_n)$ converging to l}. Which of the following statements are necessarily true?

Consider the improper integrals $I={\int }_{(}\pi /2{)}^{\pi }$ dx$/\sqrt{\sin x}$ and, for $a\ge 0,I_a={\int }_a^{\infty }$ dx$/(x\sqrt{1+x^2})$. Which of the following statements are true?

Define $f:{\mathbb{R}}^2\to \mathbb{R}$ by $f(x,y)=y\sqrt{x^2+y^2}/x$ if x ≠ 0, and f(x, y) = 0 if x = 0. Which of the following statements are true?

Let $f:{\mathbb{R}}^2\to {\mathbb{R}}^3$ be a differentiable function such that (Df)(0,0) has rank 2. Write $f=(f_1,f_2,f_3)$. Which of the following statements are necessarily true?

Consider the real vector space $V=\mathbb{R}[x]$ equipped with an inner product. Let W be the subspace of V consisting of polynomials of degree at most 2. Let W^⊥ denote the orthogonal complement of W in V. Which of the following statements are true?

For $z\in \mathbb{C}$ \ {0}, let f(z) = (1/z)sin(1/z) and g(z) = f(z)sin(z). Which of the following statements are true?

Which of the following conditions ensure that the power series ${\sum }_{(}n\ge 0)a_n{z}^n$ defines an entire function?

Let R be a principal ideal domain with a unique maximal ideal. Which of the following statements are necessarily true?

For two indeterminates x, y, let R = 𝔽${}_3[x]$ and S = R[y]. Which of the following statements are true?

For which of the following values of q, does a finite field of order q have exactly 6 subfields?

Let X denote the topological space $\mathbb{R}$ with the cofinite topology (i.e., the finite complement topology) and let Y denote the topological space $\mathbb{R}$ with the Euclidean topology. Which of the following statements are true?

Let $\tau$ be the smallest topology on the set $\mathbb{R}$ containing $\beta =$ {[a, b) | $a<b;a,b\in \mathbb{R}$}. Which of the following statements are true?

Consider the initial value problem (IVP$)y^{\prime }(x)=\sin (y(x))/(1+y^4(x))$ for $x\in \mathbb{R}$, with $y(0)=y_0$. Then which of the following statements are true?

Consider the boundary value problem (BVP) (e^(−5x)y′)′ + 6e^(−5x)y = −f(x), 0 < x < ln 2, with y(0) = 0, y(ln 2) = 0. If $G(x,\xi )=(e^{(}3x)+$ Be^(2x))(Ce${}^{(}2\xi )+$ De${}^{(}3\xi ))$ for $0\le \xi \le x$, and $G(x,\xi )=(e^{(}3\xi )+$ Be${}^{(}2\xi ))($Ce^(2x) + De^(3x)) for $x\le \xi \le \ln 2($Green's function) is such that ${\int }_0^{(}\ln 2)G(x,\xi )f(\xi )d\xi$ is the solution of the BVP, then the values of B, C and D are

Let S denote the set of all 2 × 2 matrices A such that the iterative sequence generated by the Gauss-Seidel method applied to the system of linear equations $A(x_1,x_2{)}^T=(2,3{)}^T$ converges for every initial guess. Then which of the following statements are true?

Let g(x) be the polynomial of degree at most 4 that interpolates the data (x, y) = (−1, −30), (0, 1), (2, c), (3, 10), (6, 19). If g(4) = 5, then which of the following statements are true?

The extremizer of the problem $\min [(1/2)\int$₋${}_1^1((y^{\prime }(x){)}^2+(y(x){)}^2)$dx] subject to $y\in C^1[-1,1],\int$₋${}_1^1$ xy(x)dx = 0 and y(−1) = y(1) = 1 is

For $\lambda \in \mathbb{R}$ such that |$\lambda$| < 5/32, let $R(x,t,\lambda )$ and u denote the resolvent kernel and the solution, respectively, of the Fredholm integral equation $u(x)=x+(\lambda /2)\int$₋${}_2^2($xt $+x^2{t}^2)u(t)$dt. Then which of the following statements are true?

Let {$X_n$}${}_n{\ge }_1$ be a sequence of independent and identically distributed random variables with $E(X_1)=0$ and $Var(X_1)=1$. Which of the following statements are true?

Let $X_1,\dots ,X_n$ be independent and identically distributed $U(0,\theta ),\theta >0$ random variables. Define $X_{(n)}=\max${$X_1,\dots ,X_n$} and $X_{(1)}=\min${$X_1,\dots ,X_n$}. Which of the following statements are true?

Let $X_1,\dots ,X_n(n\ge 3)$ be a random sample from a distribution having probability density function f(x | $\theta )=\theta e^{(}-\theta x)$ if x > 0, and 0 otherwise, where $\theta >0$ is an unknown parameter. Let $T_n=(1/n){\sum }_i$₌${}_1^n{X}_i$. Which of the following statements are true?

Consider a six faced die whose i-th face is marked with i dots, i = 1, 2, …, 6. In a single random throw of the die, let $p_i$ denote the probability that the obtained upper face has i dots, i = 1, 2, …, 6. The die is rolled 240 times independently and the following result is obtained — face observed 1, 2, 3, 4, 5, 6 with frequencies 40, 55, 40, 25, 35, 45 respectively. Suppose we want to test $H_0:p_i=1/6$ for i = 1, 2, …, 6; against $H_1:p_i\ne 1/6$ for at least one i. It is given that ${\chi }_5^2{;}_0{.}_{05}=11.07,{\chi }_6^2{;}_0{.}_{05}=12.59,{\chi }_5^2{;}_0{.}_{01}=15.09,{\chi }_6^2{;}_0{.}_{01}=16.81$. Based on the asymptotic goodness of fit ${\chi }^2$ test for testing $H_0$ against $H_1$, which of the following statements are true?

Observations on the shear strength of concrete from 5 randomly selected structures are given below — structure 1, 2, 3, 4, 5 with shear strength 1718.4, 1787.4, 2562.3, 2356.9, 2153.2 respectively. The null hypothesis $H_0$ that the median shear strength is 2000 units is tested against the alternative hypothesis $H_1$ that the median shear strength is greater than 2000 units at 5% level of significance. Which of the following statements are true?

Consider the following ANOVA table for a randomized block design — Treatments: sum of squares 48, degrees of freedom 4, mean squares 12, F calculated $\beta$; Blocks: sum of squares 72, degrees of freedom 3, mean squares 24, F calculated 12; Error: sum of squares $\alpha$, degrees of freedom m, mean squares $\gamma$; Total: sum of squares 144, degrees of freedom 19. Which of the following statements are true?