CSIR NET Maths previous year question papers

For any matrix P, the transpose of P is denoted by Pᵗ. Consider the real matrix A = [[1, 1, 0], [1, 2, 1], [1, 1, 2]]. Which of the following statements is true?

Consider $\mathbb{R}^{4}$ with the standard inner product. Let V be the subspace of $\mathbb{R}^{4}$ spanned by the vectors (1, 0, 0, 1), (0, 1, 0, 1), and (0, 0, 1, 0). Which of the following is NOT an orthonormal basis of V?

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?

Consider the half planes R₊ = {z = x + iy $\in \mathbb{C} : x > 0$} and R₋ = {z = x + iy $\in \mathbb{C} : x < 0$}, and the fractional linear transformations $T_{1}(z) = (z - 1)/(z + 1)$ and $T_{2}(z) = (z + 1)/(z - 1)$. Let disc 𝔻 = {$z \in \mathbb{C}$ : |z| < 1}. Which of the following statements is true?

Let $\gamma$ be the circle {$z \in \mathbb{C}$ : |z| = 3} oriented counterclockwise. Let f be an entire function. What is the value of A for which $\int_\gamma (A/(z - 1) - f(z)/(z - 2)^{2})$dz = 0 holds?

Consider the entire function $f(z) = z^{2}(e$ᶻ − e⁻ᶻ) and the meromorphic function $g(z) = z^{2}/(e$ᶻ − e⁻ᶻ) on $\mathbb{C}$. 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?

Consider the ring homomorphism $\psi : \mathbb{C}[x, y] \to \mathbb{C}[t]$ defined by $\psi(f(x, y)) = f(t^{2}, t^{3})$. Which of the following statements is true?

Let $\mathbb{N}$ be the set of positive integers. Consider $\mathbb{R}^{2}$ with the Euclidean topology and the subsets A = {$(n, 1/n) : n \in \mathbb{N}$} and B = {$(n, 1/m) : n, m \in \mathbb{N}$}. Which of the following statements is true?

For any non-zero solution y = y(x) of the differential equation $(2x + 3)^{2}(d^{2}y/$dx$^{2}) + 6(2x + 3)($dy/dx) + 8y = 0, x > 0, denote S := {$x \in (0, \infty) : y(x) = 0$}. Then

The initial value problem dy/dx $= \surd$|x − 1| sin y, y(0) = 1 has

The problem $\partial^{2}u/\partial{}x^{2} + \partial^{2}u/\partial{}y^{2} = 0$ in {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} > 1$}, u(x, y) = 1 on {$(x, y) \in \mathbb{R}^{2} : x^{2} + y^{2} = 1$} has

The problem $\partial{}u/\partial{}x + \partial{}u/\partial{}y = u^{2}, u(x, 0) = x^{2}, \forall{}x \in \mathbb{R}$ has a solution on an open set containing the line {$(x, y) \in \mathbb{R}^{2}$ : ax + by = 0} if

Consider the quadrature formula $\int$₋$_{1}^{1}$ |x|f(x)dx ≈ (1/2)(f(−1) + f(1)). Then the degree of precision (also known as order of exactness) of the quadrature formula is

If $\varphi \in C^{4}[0, 1]$ is the extremal of the variational problem minimize $J[y] = \int_{0}^{1}($yy$' + (y'')^{2})$dx, subject to y(0) = 0, y′(0) = 1, y(1) = 2, y′(1) = 4, then $\varphi(1/2)$ is equal to

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

Consider a particle of mass $m_{1}$ moving along a horizontal line L that is perpendicular to a vertical wall. Let x denote the distance of the particle from the wall. Suppose a simple pendulum of length l having mass $m_{2}, m_{2} \ne m_{1}$, is attached to the particle, hanging below L with $\theta$ measured from the downward vertical. If the pendulum oscillates in a plane containing L, then the equations of motion in terms of the generalized coordinates x and $\theta$ are (g denotes the acceleration due to gravity)

Suppose that {$X_{n}$}$(n\in\mathbb{N})$ is a sequence of independent and identically distributed (i.i.d.) random variables with the common probability density function $f(x) = 1/(\pi(1 + x^{2})), x \in \mathbb{R}$. Then which of the following statements is true?

Let $U_{1}, U_{2}, \dots, U_{6}$ be 6 urns such that urn U_k contains $3k + k^{2}$ balls, out of which 3k are white balls and $k^{2}$ are black balls, k = 1, 2, …, 6. An urn is selected with the probability of selecting urn U_k being proportional to (k + 3). A ball is chosen randomly from the selected urn. Then the probability that urn $U_{6}$ was selected, given that the ball drawn is white, is equal to

Planes take off in a busy airport in accordance with the Poisson process with rate 60 planes per hour. 10% of these planes are cargo planes and 90% are passenger planes. Given that 10 cargo planes have taken off during one hour, what is the expected total number of planes that have taken off in that hour?

Consider an M/M/3 queuing system with arrival rate $\lambda = 2$ and service rate $\mu = 3/2$. Define, for $i = 1, 2, \dots, I_{i} = 1$ if the first transition is from i to i + 1, and $I_{i} = 0$ if the first transition from i is i − 1. Then, $Var(I_{4})$ equals

Suppose X ~ Uniform(5, 10). Define Z = X + 3 if X ≤ 7, and Z = X − 3 otherwise. Then E(Z) is

Let X and Y be independent random variables such that X follows U(0, 1) distribution and Y follows Bernoulli distribution with success probability p ∈ (0, 1). Define Z = X + Y. Let $z_{1} = 0.5, z_{2} = 1.2, z_{3} = 1.3, z_{4} = 0.9, z_{5} = 0.1, z_{6} = 0.7$ be the observed values from the distribution of Z. Then the maximum likelihood estimate of p equals