CSIR NET June 2025 — Part B

Let p, q be non-negative integers. Consider the following statements: (A) There is an integer k ≥ 1 such that p + k = q. (B) There is an integer k ≥ 1 such that q + k = p. Which of the following statements is true?

Let A = {$p/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n}$ for some $n \in \mathbb{N} \cup$ {0}, gcd(p, q) = 1}, B = {$p/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n}5^{m}$ for some $n, m \in \mathbb{N} \cup$ {0}, gcd(p, q) = 1} and C = {p/q ∈ (0, 1) : p/q has terminating decimal expansion} be subsets of (0, 1). Which of the following statements is true?

Let A, B be non-empty subsets of $\mathbb{N}$ with cardinality |A| ≥ 2. Let $S_{1} =$ {f : A → B | f is one-to-one} and $S_{2} =$ {g : B → A | g is onto}. Which of the following statements is true?

Let $f : \mathbb{R}$ \ $\mathbb{Q} \to \mathbb{R}$ \ $\mathbb{Q}$ be the function defined as f(x) = (3x + 2)/(4x + 3). Let $x_{1} \in \mathbb{R}$ \ $\mathbb{Q}$. For n ≥ 1, define $x_{n}$₊$_{1} = f(x_{n})$. Suppose that the sequence $(x_{n})_{n}\ge_{1}$ converges to a real number $\ell$. Which of the following statements is true?

Let $f(x) = x log_{e}(1 + 1/x)$ for $x \in (0, \infty)$. Which of the following statements is true?

For each n ≥ 1, let $f_{n} : [0, 1] \to \mathbb{R}$ be defined as $f_{n}(x) =$ nx if $x \in [0, 1/n], f_{n}(x) = 2 -$ nx if x ∈ (1/n, 2/n), and $f_{n}(x) = 0$ if x ∈ [2/n, 1]. Which of the following statements is true?

Which of the following polynomials is the characteristic polynomial of a real 2 × 2 matrix A such that trace(A) = 7 and trace$(A^{2}) = 29$?

Consider the real matrix A whose rows are (29, 0, 55, 17), (1, 28, 46, 26), (17, 13, 33, 38) and (21, 67, 0, 13). What is the largest real eigenvalue of A?

Let 𝔽$_{5}$ denote the field with 5 elements. How many 2 × 2 matrices with entries in 𝔽$_{5}$ have rank one?

Let X be the $\mathbb{R}-$vector space of all twice differentiable real valued functions on [0, 1]. Consider the linear map $\varphi : X \to \mathbb{R}^{3}$ defined by $\varphi(f) = (f(1), f'(1), f''(1))$. Which of the following statements is true?

Let $C[0, \pi]$ be the real vector space of real-valued continuous functions on the closed interval $[0, \pi]$. For positive integers n, define $f_{n} \in C[0, \pi]$ by $f_{n}(x) = \sin($nx)/sin x if $x \in (0, \pi), f_{n}(0) = n$, and $f_{n}(\pi) = (-1)^{n}$⁻$^{1}n$. Let V be the real subspace of $C[0, \pi]$ spanned by {$f_{1}, f_{2}, f_{3}$}. Consider the inner product on V given by ⟨f, g⟩ $= (2/\pi)\int_{0}^\pi f(x)g(x)\sin^{2}x$ dx. Which of the following statements is true?

Let V = {ax$^{3} +$ bx$^{2} +$ cx | $a, b, c \in \mathbb{R}$}. For f ∈ V, define $Q(f) = \int$₋$_{1}^{1} (f'(t))^{2}$ dt, where f′ denotes the derivative of f. Which of the following statements is FALSE?

Let $f : \mathbb{C} \to \mathbb{C}$ be a polynomial map. For R > 0, let $\gamma_R : [0, 1] \to \mathbb{C}$ be the map t ↦ $Re^{2\pi{}it}$. Suppose that there exists $c \in \mathbb{R}$ such that $\int_{0}^{1}$ |(f ∘ $\gamma_R)(t) \gamma'_R(t)$| dt → c as $R \to \infty$. Which of the following statements is FALSE?

Let f be an entire function such that $f(\mathbb{C}) \subset$ {x + iy | y = x + 1}. Which of the following statements is true?

Let X be the image of the interval [0, 1] under the Möbius transformation f(z) = (z − i)/(z + i). Which of the following statements is true?

Which of the following statements is true?

Which of the following statements is true?

Let A be a subring of the field of rationals $\mathbb{Q}$ such that for any nonzero rational $r \in \mathbb{Q}, r \in A$ or 1/r ∈ A. Which of the following statements is FALSE?

Which of the following statements is true?

Let S = {1, 2, 3, 4, 5} be equipped with the topology $\tau =$ {∅, {1}, S}. What is the number of homeomorphisms of S onto itself?

If $\varphi(x) = x$ is a solution of the ordinary differential equation (ODE$) d^{2}y/$dx$^{2} - (2/x^{2} + 1/x)(x$ dy/dx $- y) = 0, 0 < x < \infty$, then the general solution of the ODE is given by

Let $(\lambda_{n})_{n}\in\mathbb{N}$ be the sequence of eigenvalues of the Sturm-Liouville problem (d/dx)(x dy/dx$) + (\lambda/x)y = 0, 1 < x <$ $e^{2\pi}$, y(1) = 0, $y(e^{2\pi})$ = 0. Then $\sum_{n}$₌$_{1}^\infty 1/\lambda_{n}$ is equal to

Let u = u(x, y) be the solution to the Cauchy problem $(y + u)\partial{}u/\partial{}x + y \partial{}u/\partial{}y = x - y, x \in \mathbb{R}, y > 0, u(x, 1) = 1 + x, x \in \mathbb{R}$. Then which of the following statements is true?

Let u = u(x, t) be the solution of $\partial^{2}u/\partial{}t^{2} - \partial^{2}u/\partial{}x^{2} = 0, x \in \mathbb{R}, t > 0, u(x, 0) = 1 + x^{2}, x \in \mathbb{R}, \partial{}u/\partial{}t(x, 0) = x + 1, x \in \mathbb{R}$. Then the value of u(1, 1) is

If the function $s : [0, 4] \to \mathbb{R}$ defined by $s(x) = a(x - 2)^{2} + b(x - 1)^{2}$ for $0 \le x \le 1, s(x) = (x - 2)^{2}$ for 1 < x ≤ 3, and $s(x) = 2c(x - 2)^{2} + (x - 3)^{3}$ for 3 < x ≤ 4, is a cubic spline, then the value of 2a + b + 2c is

Let y(x) be the extremal of the functional $J[y] = \int_{0}^{\pi/4}$ $((y')^{2} - 4y^{2} + 2$xy) dx subject to $y(0) = 0, y(\pi/4) = 1$. Then y(x) is equal to

If y(x) is the solution of the integral equation $y(x) = x^{2} + 2\int_{0}^{1}$ xt y(t) dt, then which of the following statements is true?

Suppose a dynamical system has the Lagrangian L = (q̇$_{1})^{2} + (q$̇$_{2})^{2} + (q_{1})^{2} + q$̇$_{1}q$̇$_{2}$. If $p_{1}$ and $p_{2}$ are momenta conjugate to $q_{1}$ and $q_{2}$ respectively, then which of the following statements is true?

Suppose that we have a data set consisting of 2n + 1 observations for some $n \in \mathbb{N}$. Value of each observation is either x or x + r, where $x \in \mathbb{N}, r \ge 0$. Then, which of the following statements is always true?

A biased six-faced die is tossed once. Suppose that the probability of any prime number showing up is twice that of any non-prime number showing up. Then, the probability that an odd number will show up is

Suppose the distribution of X given $\theta$ is normal with mean $\theta$ and variance 15. Further, let the prior (improper) distribution of $\theta$ be proportional to $1, -\infty < \theta < \infty$. If the observed value of X is 13, then which of the following statements is true?

Let $Z_{1}, Z_{2}, \dots$ be a sequence of independent and identically distributed random variables having discrete uniform distribution over {1, 2, …, 2024}. Let $Y_{n} = \sum_{i}$₌$_{1}^{n} Z_{i}, n \ge 2$. Further, let $X_{n}$ be the remainder when $Y_{n}$ is divided by 2025. Then, which of the following statements is true?

A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function f(x) = $e^{-x}$ if x > 0 and 0 otherwise, and that of Brand II follows a gamma distribution with the probability density function g(x) = $(x/4)e^{-x/2}$ if x > 0 and 0 otherwise. The company uses the batteries of Brands I and II in proportion of 20% and 80% respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more that 2 years is

Consider a discrete random variable X with the probability mass function $P(X = 0) = \theta/3, P(X = 1) = 1 - \theta/2, P(X = 2) = \theta/6$, where $\theta \in (0, 1)$ is an unknown parameter. In a random sample of size 90 from this distribution, the observed counts for X = 0, 1 and 2 are 20, 60 and 10, respectively. Then, the maximum likelihood estimate of $\theta$ is

Let X be a random sample of size 1 from the probability density function f(x|$\theta) = (3/\theta^{3})(\theta - x)^{2}$ if $0 < x < \theta$, and 0 otherwise. If $(X/(1 - \lambda_{1}), X/(1 - \lambda_{2}))$ is a confidence interval for $\theta$ with confidence coefficient $1 - \alpha$, where $\lambda_{i} \in (0, 1), i = 1, 2, \lambda_{1} < \lambda_{2}$, and $\alpha \in (0, 1)$, then which of the following statements is true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a continuous distribution with the common probability density function f(x|$\theta) = 2^\theta \theta /$ $x^{\theta+1}$ if x > 2, and 0 otherwise, where $\theta (> 0)$ is an unknown parameter. Suppose $P(Y > \chi^{2}_{m,\beta})$ $= \beta$, where Y ~ $\chi^{2}_m$. For testing $H_{0} : \theta = 1$ against $H_{1} : \theta > 1, a$ uniformly most powerful test of size $\alpha, 0 < \alpha < 1$, will reject $H_{0}$ if

Consider the multiple linear regression model $y_{i} = \beta_{0} + \beta_{1}x_{1i} + \dots + \beta_{8}x_{8i} + \varepsilon_{i}, i = 1, 2, \dots, 29$, where $\varepsilon_{1}, \varepsilon_{2}, \dots, \varepsilon_{29}$ are independent and identically normal distributed with mean 0 and variance $\sigma^{2}$. Suppose the model is fitted using the method of least squares. If the calculated value of the F-statistic for testing the significance of regression is 2.50, then the possible values of $R^{2}$ and Adjusted $R^{2}$ are respectively

Suppose $(X_{1}, X_{2}, X_{3})^{T}$ ~ $N_{3}(0, I_{3})$, where 0 is the zero mean vector and $I_{3}$ is the 3 × 3 identity matrix, and $(X, Y, Z)^{T} = A(X_{1}, X_{2}, X_{3})^{T}$ where A has rows (3, 0, 0), (2, 2, 0) and (4, 0, 4). Then the partial correlation coefficient $\rho_{YZ.X}$ is

Suppose we want to estimate the population mean Ȳ of a variable for a finite population of size 85, with 34 Statisticians and 51 Biologists. We consider the following sampling scheme: a stratified random sample with 2 strata of Statisticians (Stratum-1) and Biologists (Stratum-2), where 12 Statisticians and 15 Biologists are drawn from Stratum-1 and Stratum-2, respectively, using SRSWOR scheme. Denote ȳ_S, ȳ_B, and ȳ as the mean of the variable among the Statistician sample, Biologist sample, and the combined sample, respectively. Which of the following is an unbiased estimator of Ȳ?

Solve the following linear programming problem: maximize z = x + y subject to 5x + 3y ≤ 30, 2x + 6y ≤ 25, 2x − y ≤ 8, x ≥ 0, y ≥ 0. Then the optimal value of the objective function is