CSIR NET December 2023 Mathematical Sciences — all questions solved

Consider the following subset of $\mathbb{R}:U=$ {$x\in \mathbb{R}:x^2-9x+18\le 0,x^2-7x+12\le 0$}. Which one of the following statements is true?

Consider the following infinite series: $(a){\sum }_{n\ge 1}$ $\sin (n\pi /2)/\sqrt{n},(b){\sum }_{n\ge 1}$ $\log (1+1/n^2)$. Which one of the following statements is true?

Let f(x) be a cubic polynomial with real coefficients. Suppose that f(x) has exactly one real root and that this root is simple. Which one of the following statements holds for ALL antiderivatives F(x) of f(x)?

Let X be a non-empty finite set and Y = {$f^{-1}(0)$ : f is a real-valued function on X}. Which one of the following statements is true?

Let A be an n × n matrix with complex entries. If n ≥ 4, which one of the following statements is true?

Let $A=(a_{ij})$ be the n × n real matrix with $a_{ij}=$ ij for all 1 ≤ i, j ≤ n. If n ≥ 3, which one of the following is an eigenvalue of A?

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 $\mathbb{Z}/89\mathbb{Z}$?

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?

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}?

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

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 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

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

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 $1/\theta$?

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

Let x be a real number. Which of the following statements are true?

Let $f:[0,\infty )\to \mathbb{R}$ be the periodic function of period 1 given by f(x) = 1 − |2x − 1| for x ∈ [0, 1], and define $g:[0,\infty )\to \mathbb{R}$ by $g(x)=f(x^2)$. Which of the following statements are true?

For a real number $\lambda$, consider the improper integrals $I_{\lambda }={\int }_0^1$ dx$/(1-x{)}^{\lambda }$ and $K_{\lambda }={\int }_1^{\infty }$ dx$/x^{\lambda }$. Which of the following statements are true?

For real numbers a, b, c, d, e, f, consider the function $F:{\mathbb{R}}^2\to {\mathbb{R}}^2$ given by F(x, y) = (ax + by + c, dx + ey + f). Which of the following statements are true?

Let A be an n × n real symmetric matrix. Which of the following statements are necessarily true?

Let $n\in \mathbb{Z}$ be such that n ≡ 1 (mod 7) and n ≡ 4 (mod 15). Which of the following statements are true?

Let $f(X)=X^3-2\in \mathbb{Q}[X]$ and let $K\subset \mathbb{C}$ be the splitting field of f(X) over $\mathbb{Q}$. Let $\omega =$ $e^{2\pi i/3}$. Which of the following statements are true?

For an integer k, consider the contour integral $I_k={\int }_{|z|=1}$ eᶻ/zᵏ dz. Which of the following statements are true?

Consider the initial value problem $x^2{y}^{\prime \prime }-2x^2{y}^{\prime }+(4x-2)y=0,y(0)=0$. Suppose $y=\phi (x)$ is a polynomial solution satisfying $\phi (1)=1$. Which of the following statements are true?

Suppose U ~ Uniform(0,1) and $X=\tan (\pi (U-$ ½)). Then which of the following statements are true?

Consider a population of 3 units having values 2, 4 and 6. A simple random sample (without replacement) of 2 units is drawn. Let M be the sample mean. Which of the following statements are true?