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 $\sqrt{1 + (y'(t))^{2}}$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 $a U(-\theta, 2\theta)$ 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 $Y_{ij} = \mu + \alpha_{i} + \beta_{j} + \varepsilon_{ij}, i = 1, 2; j = 1, 2$, 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 $\varepsilon_{ij}$ is the corresponding error with $E(\varepsilon_{ij}) = 0$ 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 $lim_{n}\to\infty a_{n}$ 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 $\sum_(n\ge0) a_{n}z^{n}$ 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$) y'(x) = \sin(y(x))/(1 + y^{4}(x))$ 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 $\min[(1/2)\int$₋$_{1}^{1} ((y'(x))^{2} + (y(x))^{2})$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 $\chi^{2}_{5};_{0}._{05} = 11.07, \chi^{2}_{6};_{0}._{05} = 12.59, \chi^{2}_{5};_{0}._{01} = 15.09, \chi^{2}_{6};_{0}._{01} = 16.81$. 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?