CSIR NET Maths previous year question papers

For $a \in \mathbb{R}$, let A_a = [[2, −1, 0], [−1, 2, −1], [0, −1, a]]. Which one of the following statements is true?

Let G be any finite group. Which one of the following is necessarily true?

Which one of the following is equal to $1^{37} + 2^{37} + 3^{37} + \cdots + 88^{37}$ in $\mathbb{Z}/89\mathbb{Z}$?

Consider the field $\mathbb{C}$ with the Euclidean topology. Let K be a proper subfield of $\mathbb{C}$ that is not contained in $\mathbb{R}$. Which one of the following statements is necessarily true?

Let $f : \mathbb{C} \to \mathbb{C}$ be a real-differentiable function and define u(x, y) = Re f(x + iy), v(x, y) = Im f(x + iy). Let $\nabla{}u = (u_{x}, u_y)$ denote the gradient. Which one of the following is necessarily true?

Let ℍ = {$z \in \mathbb{C} : Im z > 0$} and f(z) = $e^{iz}$. Which one of the following statements is true?

How many roots does the polynomial $z^{100} - 50z^{30} + 40z^{10} + 6z + 1$ have in the open disc {$z \in \mathbb{C}$ : |z| < 1}?

Let f be a meromorphic function on an open set containing the unit circle C and its interior. Suppose f has no zeros and no poles on C, and let n_p and $n_{0}$ denote the number of poles and zeros of f inside C. Which one of the following is true?

The smallest real number $\lambda$ for which the problem $-y'' + 3y = \lambda{}y, y(0) = 0, y(\pi) = 0$ has a non-trivial solution is

The partial differential equation $x^{2} u_{xx} - 2$xy $u_{x}$ᵧ $- 3y^{2} u_yy + u_{x} - u_y = 0$ is

Consider the Cauchy problem for the wave equation $u_tt - 4u_{xx} = 0$ on $\mathbb{R} \times (0, \infty)$, with u(x, 0) = $e^{-1/x^{2}}$ for x ≠ 0 (and 0 at x = 0), and u_t(x, 0) = x $e^{-x^{2}}$. Which one of the following is true?

Using Euler's method with step size 0.05, the approximate value of the solution of dy/dx $= \sqrt{3x + 2y + 1}, y(1) = 1$, at x = 1.1 (rounded to two decimal places) is

The cardinality of the set of extremals of $J[y] = \int_{0}^{1} (y')^{2}$ dx subject to $y(0) = 1, y(1) = 6, \int_{0}^{1} y$ dx = 3 is

The value of $\lambda$ for which the integral equation $y(x) = \lambda\int_{0}^{1} x^{2}$ $e^{x+t}$ y(t) dt has a non-zero solution is

Let g denote the acceleration due to gravity and a > 0. A particle of mass m glides without friction on the cycloid $x = a(\theta - \sin \theta), y = a(1 + \cos \theta), 0 \le \theta \le 2\pi$. The equation of motion of the particle is

Suppose X ~ Poisson(3/4). Then which of the following statements is true?

Let (X, Y) have joint moment generating function $M(t_{1}, t_{2}) = (3/4 +$ $e^{t_{1}}/4)^{2}$ (5/6 + $e^{t_{2}}/6)^{3}$. Then P(X + 2Y > 1) equals

Let $X_{1}, X_{2}, \dots$ be i.i.d. with CDF F(x) = 0 for x < 5 and 1 − $e^{5-x}$ for x ≥ 5. Define $Y_{n} = \min${$X_{1}, \dots, X_{n}$} and $Z_{n} = \sqrt{n} (Y_{n} - 5)$, and let Z be standard normal. Which one of the following statements is true?

The probability of a head in tossing a coin is p ∈ (0, 1). The coin is independently tossed 25 times and heads appear 10 times. The Bayes estimate of p with respect to the prior Beta(5, 5) and squared error loss is

For n ≥ 2, let $X_{1}, \dots, X_{n}$ be a random sample with density f(x|$\theta) =$ $\theta{}x^{\theta-1}$ on $0 < x < 1, \theta > 0$ unknown. Which of the following is the UMVUE for $1/\theta$?

For n ≥ 2, let $\varepsilon_{1}, \dots, \varepsilon_{n}$ be $i.i.d. N(0, \sigma^{2})$ and $Y_{i} = i\alpha + i^{2}\alpha^{2} + \varepsilon_{i}$, where $\sigma > 0$ and $\alpha \in \mathbb{R}$ are unknown. Which of the following is a jointly minimal sufficient statistic for $(\alpha, \sigma)$?

Let $X_{1}, \dots, X_{6}$ be a random sample from the gamma density f(x|$\lambda) = (\lambda^{4}/6)$ $e^{-\lambda{}x}$ $x^{3}$ for $x > 0, \lambda > 0$ unknown. Let $T = \sum{}X_{i}$ and $\psi$ be the UMP test of size 0.05 for $H_{0}: \lambda = 1$ against $H_{1}: \lambda > 1$. With $\chi^{2}_{\nu,\alpha}$ the $(1 - \alpha)-$th quantile of $\chi^{2}_\nu$, the test $\psi$ rejects $H_{0}$ iff

Consider $Y = \alpha + \beta{}x + \varepsilon$ with $E(\varepsilon) = 0$. From 10 observations the OLS fit is ŷ = 1.5 + 0.8x. Suppose $\sum(y_{i} -$ ȳ$)^{2} = 5$ and $\sum(x_{i} - {\bar{x}})^{2} = 6$. The adjusted $R^{2} ($to two decimals) equals

For n ≥ p + 1, let $X_{1}, \dots, X_{n}$ be a random sample from $N_{p}(\mu, \sum)$ with $\sum$ positive definite. Define X̄ and $A = \sum(X_{i} - {\bar{X}})(X_{i} - {\bar{X}})^{T}$. The distribution of Trace$(A\sum$⁻$^{1})$ is

In a Latin square design, the degrees of freedom for the error sum of squares is 42. Then the degrees of freedom for the sum of squares due to treatments is