CSIR NET Maths previous year question papers

What is the cardinality of the set of real solutions of eˣ + x = 1?

Let S be a dense subset of $\mathbb{R}$ and $f : \mathbb{R} \to \mathbb{R} a$ given function. Define $g : S \to \mathbb{R}$ by g(x) = f(x). Which of the following statements is necessarily true?

For each n ≥ 1 define $f_{n} : \mathbb{R} \to \mathbb{R}$ by $f_{n}(x) = x^{2} / \sqrt{x^{2} + 1/n}, x \in \mathbb{R}$, where $\surd$ denotes the non-negative square root. Wherever $lim_{n}\to\infty f_{n}(x)$ exists, denote it by f(x). Which of the following statements is true?

Let $A : \mathbb{R}^{m} \to \mathbb{R}^{n}$ be a non-zero linear transformation. Which of the following statements is true?

Let A be a 10 × 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true?

Let ((2, a), (b, c)) be a 2 × 2 real matrix for which 6 is an eigenvalue. Which of the following statements is necessarily true?

Let V be the real vector space of 2 × 2 matrices with entries in $\mathbb{R}$. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where A = ((2, 0), (0, 1)). What is the characteristic polynomial of T?

Let A = ((0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 1, 1), (0, 0, 1, 1)), and consider the symmetric bilinear form on $\mathbb{R}^{4}$ given by ⟨v, w⟩ = vᵗAw, for $v, w \in \mathbb{R}^{4}$. Which of the following statements is true?

For a quadratic form $f(x, y, z) \in \mathbb{R}[x, y, z]$, we say that $(a, b, c) \in \mathbb{R}^{3}$ is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)?

Let f be an entire function. Which of the following statements is FALSE?

Consider the contour $\gamma$ given by $\gamma(\theta) = e^(2i\theta)$ for $\theta \in [0, \pi/2]; \gamma(\theta) = 1 + 2e^(2i\theta)$ for $\theta \in [\pi/2, 3\pi/2]; \gamma(\theta) = e^(2i\theta)$ for $\theta \in [3\pi/2, 2\pi]$. Then what is the value of $\int_\gamma$ dz / (z(z − 2))?

Let a, b be two real numbers such that a < 0 < b. For a positive real number r, define $\gamma_r(t) =$ re^(it) (where $t \in [0, 2\pi])$ and $I_r = (1/2\pi{}i) \int_(\gamma_r) (z^{2} + 1) / ((z - a)(z - b))$ dz. Which of the following statements is necessarily true?

For a complex number a such that 0 < |a| < 1, which of the following statements is true?

How many arrangements of the digits of the number 1234567 are there, such that exactly three of them occur in their original position? (E.g., in the arrangement 5214763, exactly the digits 2, 4 and 6 are in their original positions. In the arrangement 1243576, exactly the digits 1, 2 and 5 are in their original positions.)

The number of group homomorphisms from $\mathbb{Z}/150\mathbb{Z}$ to $\mathbb{Z}/90\mathbb{Z}$ is

Consider the ring R = { $\sum_(n\in\mathbb{Z}) a_{n}X^{n}$ | $a_{n} \in \mathbb{Z}$, and $a_{n} \ne 0$ only for finitely many $n \in \mathbb{Z}$ }, with the usual addition and multiplication of such sums. Which of the following statements is true?

Consider the initial value problem (IVP$) y'(x) = \surd$|$y(x) + \varepsilon$|, $x \in \mathbb{R}, y(0) = y_{0}$. Consider the following statements: S1: There is an $\varepsilon > 0$ such that for all $y_{0} \in \mathbb{R}$, the IVP has more than one solution. S2: There is $a y_{0} \in \mathbb{R}$ such that for all $\varepsilon > 0$, the IVP has more than one solution. Then

Let $\varphi$ denote the solution to the boundary value problem (BVP) (xy′)′ − 2y′ + y/x = 1 for $1 < x < e^{4}$, with y(1) = 0 and $y(e^{4}) = 4e^{4}$. Then the value of $\varphi(e)$ is

Let u = u(x, t) be the solution of the following initial value problem: $u_{t} + 2024u_{x} = 0$ for $x \in \mathbb{R}, t > 0$, and $u(x, 0) = u_{0}(x)$ for $x \in \mathbb{R}$, where $u_{0} : \mathbb{R} \to \mathbb{R}$ is an arbitrary $C^{1}$ function. Consider the following statements: $S_{1}$: If $A_{t} :=$ {$x \in \mathbb{R} : u(x, t) < 1$} and |$A_{t}$| denotes the Lebesgue measure of $A_{t}$ for every t ≥ 0, then |$A_{t}$| = |$A_{0}$|, $\forall{}t > 0. S_{2}$: If $u_{0}$ is Lebesgue integrable, then for every t > 0, the function x ↦ u(x, t) is Lebesgue integrable. Then

If u = u(x, t) is the solution of the initial value problem $u_{t} = u_{xx}$ for $x \in \mathbb{R}, t > 0$, with u(x, 0) = sin(4x) + x + 1 for $x \in \mathbb{R}$, satisfying |u(x, t)| $< 3e^(x^{2})$ for all $x \in \mathbb{R}$ and t > 0, then

If the value of the approximate solution of the initial value problem $y'(x) = x(y(x) + 1), x \in \mathbb{R}, y(0) = \beta$ at x = 0.2 using the forward Euler method with step size 0.1 is 1.02, then the value of $\beta$ is

Let B(0,1) = {$(x, y) \in \mathbb{R}^{2}$ | $x^{2} + y^{2} < 1$} be the open unit disc in $\mathbb{R}^{2}, \partial{}B(0,1)$ denote the boundary of B(0,1), and $\nu$ denote the unit outward normal to $\partial{}B(0,1)$. Let $f : \mathbb{R}^{2} \to \mathbb{R}$ be a given continuous function. The Euler–Lagrange equation of the minimization problem min { ½∬_B(0,1) |$\nabla{}u$|$^{2}$ dxdy + ½∬$_B(0,1) e^(u^{2})$ dxdy $+ \int_\partial{}B(0,1) f u$ ds } subject to $u \in C^{1}($closure of B(0,1)) is

Let u be the solution of the Volterra integral equation $\int_{0}$ᵗ [½ $+ \sin(t - \tau)] u(\tau) d\tau = \sin t$. Then the value of u(1) is

Consider a solid circular cylinder of radius 2 meters and height 3 meters of uniform density. If the density of the cylinder is $\rho$ kg/meter$^{3}$, then the moment of inertia (in kg meter$^{2})$ of the cylinder about a diameter of its base is

Let $A_{1}, A_{2}, A_{3}$ be events satisfying $0 < P(A_{i}) < 1$ for i = 1, 2, 3. Which of the following statements is true?