CSIR NET Maths previous year question papers

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?

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 $C[0, \pi]$ 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 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?

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?

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?

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

Let h be a holomorphic function on $\mathbb{C}${0} such that $lim_{|z|\to\infty}$ h(z) = 0. For every n ≥ 1, consider $f_{n}(z) = \sum_{k=1}^n$ h(zᵏ). Let 𝔻 denote the open unit disc {$z \in \mathbb{C}$ : |z| < 1}. Which of the following statements is necessarily true?

Let S be a subset of the open interval (0,1) that consists of all the real numbers $\alpha \in (0,1)$ whose infinite decimal expansion $\alpha = 0.a_{1}a_{2}a_{3}\cdots$ is such that $a_{i} \in$ {0,2,4} for all i ≥ 1. Which of the following statements is true?

Consider the quadratic form f(x,y,z) = [x y z] A [x;y;z], where A is an invertible 3×3 symmetric matrix over $\mathbb{Q}$. Assume that there exists $(\alpha,\beta,\gamma) \in \mathbb{C}^{3}${(0,0,0)} such that $f(\alpha,\beta,\gamma) = 0$. Which of the following statements is necessarily true?

Let $X_{1}$ and $X_{2}$ be a random sample from Uniform$[0,\theta]$ distribution, where $\theta > 0$. For testing the hypothesis $H_{0}: \theta=1$ against $H_{1}: \theta=2$, consider a test which rejects $H_{0}$ if $X_{1}+X_{2} > 4/5$. Then, the probability of type-I error is

Which of the following statements is necessarily true?

Let A, B be two non-empty subsets of $\mathbb{R}$. Let S = {$f : A \to \mathbb{R}$ | f is continuous}, T = {$f : B \to \mathbb{R}$ | f is continuous}, U = {$f : A \times A \to \mathbb{R}$ | f is continuous}. Which of the following statements is true?

Let $T : \mathbb{R}^{4} \to \mathbb{R}^{4}$ be an $\mathbb{R}-$linear transformation such that $(T^{2} + T + I)(T - 2I) = 0$ and $(T^{2} + T + 2I)(T^{2} - 4I) = 0$. Which of the following statements is FALSE?

Consider the real matrix A = [[0, 0, 1], [0, 1, 0], [1, 0, 0]]. Define $B : \mathbb{R}^{3} \times \mathbb{R}^{3} \to \mathbb{R}$ by B(v, w) = vᵗAw. Which of the following statements is true?

For z = x + iy $\in \mathbb{C}$, let f(z) = u(x, y) + iv(x, y) define an entire function. Consider the function g(z) = u(x, −y) − iv(x, −y), for z = x + iy $\in \mathbb{C}$. Suppose that v(x, 0) = 0 for all $x \in \mathbb{R}$. Define E = {$z \in \mathbb{C}$ | f(z) = g(z)}. Which of the following statements is true?

Let p be a prime number. An element a of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})$ˣ is said to be a primitive root in $(\mathbb{Z}/p\mathbb{Z})$ˣ if the order of a in $(\mathbb{Z}/p\mathbb{Z})$ˣ is p − 1. Let S_p be the number of primitive roots in $(\mathbb{Z}/p\mathbb{Z})$ˣ and $\varphi$ denote the Euler $\varphi-$function. Which of the following statements is true?

Let G be a group of order n > 3 and H be a subgroup with 1 < |H| < n. Consider the set X = ⋃(g∈G) gHg⁻$^{1}$. Which of the following statements is true?

If u is a solution of the integral equation $u(x) = \lambda\int_{0}^{1}K(x, t)u(t)$dt, where K(x, t) := x(1 − t) for 0 ≤ x ≤ t ≤ 1 and t(1 − x) for 0 ≤ t ≤ x ≤ 1, then

Let X be a random variable with the probability density function $f(x) = 2\theta{}x + 2(1 - \theta)(1 - x)$ if 0 < x < 1, and 0 otherwise, where $\theta \in [0, 1]$. Based on single observation x, the critical region of the most powerful test for testing null hypothesis $H_{0} : \theta = 1/2$ against alternative hypothesis $H_{1} : \theta = 1$, at level of significance $\alpha = 0.25$, is

For a variable x, consider the $\mathbb{R}-$vector space V = {$a_{0} + a_{1}x + a_{2}x^{2}$ | $a_{1}, a_{2}, a_{3} \in \mathbb{R}$}. Let T : V → V be the linear transformation defined by T(f) = f + df/dx, where df/dx denotes the derivative of f with respect to x. Which of the following statements is true?

Let V be the $\mathbb{R}-$vector space of 5 × 5 real matrices. Let S = {AB − BA | A, B ∈ V} and W denote the subspace of V spanned by S. Let $T : V \to \mathbb{R}$ be the linear transformation mapping a matrix A to its trace. Which of the following statements is true?

Let 𝔻 = {z = x + iy $\in \mathbb{C}$ : |z| < 1} be the open unit disc and f : 𝔻 $\to \mathbb{C} a$ holomorphic function such that f(0) = 0. Let $\psi(z) =$ |f(z)|$^{2}$, and $\partial^{2}\psi/\partial{}x^{2} + \partial^{2}\psi/\partial{}y^{2} \equiv 0$. Which of the following statements is FALSE?