CSIR NET Maths previous year question papers

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

Let X be a Binomial(n, p) random variable, where n ∈ {5, 6} and p ∈ {1/4, 3/4}. If X = 3 is observed, then the maximum likelihood estimate of (n, p) is

Let X = {1, …, 17} and $S_{17}$ be the group of permutations of X. For a subgroup G of $S_{17}$ and x ∈ X, let S_G(x) = {$\sigma \in G$ ∣ $\sigma(x) = x$}. Which of the following statements is true for every subgroup G of $S_{17}$?

Let I denote the 3×3 identity matrix. Given any three distinct matrices $A, B, C \in M_{3}(\mathbb{R})$, which of the following statements is necessarily true?

Let X and Y be topological spaces. Consider the following statement: S: For every open subset U ⊆ X × Y and every x ∈ X such that {x} × Y ⊆ U, there is a neighbourhood W of x in X such that W × Y ⊆ U. Which of the following statements is true?

Let $M, N \in M_{3}(\mathbb{C})$ be such that Trace(Mᵏ) = Trace(Nᵏ), 1 ≤ k ≤ 3. Which of the following statements is necessarily true?

Let {$X_{n} : n \ge 0$} be any homogeneous Markov Chain on the state space S = {1,2,3,4} having the transition probability matrix $P = (p_{ij})$ given by row 1 = (1/4, 0, 2/3, 1/12), row 2 = (0, 1, 0, 0), row 3 = (1/12, 0, 1/4, 2/3), row 4 = (2/3, 0, 1/12, 1/4). Which of the following statements about stationary distributions of any such Markov Chain is true?

Let F_q be a finite field with q elements. For n ≥ 2, let A be a 2n × 2n matrix with entries in F_q such that rank(A) = n. Let W = {v ∈ F_q^(2n) ∣ Av = 0}. Which of the following is necessarily the number of (n+2)-dimensional subspaces of F_q^(2n) that contain W?

Let 𝔻 denote the open unit disc {$z \in \mathbb{C}$ : |z| < 1} and X = {f : 𝔻 $\to \mathbb{C}$ ∣ f is holomorphic and satisfies $f(2z) = f(z)/(1 - f(z)^{2})$ for all |z| < 1/2}. Which of the following statements is true?

Let u(x,t) be the solution of the partial differential equation $u_{t} = 16u_{xx} + 2, 0 < x < 7, t > 0$, satisfying the conditions $u_{x}(0,t) = u(7,t) = 0$ for t > 0, and u(x,0) = 0 for 0 < x < 7. Then which of the following statements is true?

For a finite group G, let S(G) denote the number of subgroups of G. Which of the following statements is necessarily true?