CSIR NET Maths previous year question papers

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