CSIR NET Maths previous year question papers

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

Suppose that the probability density function of the random variable X is $f(x) = 2(\theta - x)/\theta^{2}$ if $0 \le x \le \theta$, and 0 otherwise, where $\theta > 0$ is an unknown parameter. Based on a single observation X, the confidence coefficient of the confidence interval [(2/5)X, (5/2)X] for $\theta$ is

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from distribution Poisson$(\theta), \theta > 0$. Let $\pi(\theta) = e^(-\theta), \theta > 0$, be the prior distribution of $\theta$. Under the squared error loss function, which of the following is the Bayes estimator of $e^(-2\theta)$?

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

Consider a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1)$, where errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Here, $Y_{i}$ is the i-th response. Let Ŷ$_{i}$ be the i-th predicted response by the least squares estimation method, and let $\varepsilon$̂$_{i} = Y_{i} -$ Ŷ$_{i}, 1 \le i \le n$. Then, which of the following statements is true?

Let X, Y and Z be independent and identically distributed (i.i.d.) random variables with distribution N(0, 1). Define U = 2X, V = 3X + Y, W = X + 4Z. Then the partial correlation coefficient of V and W, given U is

Consider the following design where the columns represent blocks and the letters represent treatments — block 1: A, B; block 2: C, D; block 3: A, C; block 4: B, D; block 5: A, D; block 6: B, C; block 7: A, E; block 8: B, E; block 9: C, E; block 10: D, E. Then, which of the following statements is NOT true?

Let A, B, and C be sets. Which of the following sets is equal to A \ (B \ C)?

What is the number of injective functions from {1, 2, …, 7} to {1, 2, …, 10}?

For integers n ≥ 0, let $f_{n} : [-1, 0] \to \mathbb{R}$ be defined by $f_{n}(x) = x/(1 - x)^{n}$. Which of the following statements is true about the series $\sum_(n=0)^\infty f_{n}$?

Consider the sequences $(a_{n})_{n}\ge_{1}$ and $(b_{n})_{n}\ge_{1}$ defined by $a_{n} = (e^{n} + e$⁻$^{n})/2$ and $b_{n} = a_{n}$₊$_{1}/a_{n}$. Which of the following statements is true?

Let $f : [0, 1] \to \mathbb{R}$ be defined by $f(x) = \sin(x^{2})$. Let $A = \lim(n\to\infty) (\sum_(k=1)^n f(k/n) - n\int_{0}^{1}f(x)$dx). Which of the following statements is true?

Consider the power series $\sum_(n=1)^\infty [n^(n^{2})/(n + 1)^(n^{2})]x^{n}$ with coefficients in real numbers $\mathbb{R}$. Which of the following statements is true?

Let U denote the span of {eᵗ, $e^{2}$ᵗ, $e^{3}$ᵗ} in the real vector space of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Consider the $\mathbb{R}-$vector spaces V = {$f : U \to \mathbb{R}$ | f is an $\mathbb{R}-$linear transformation} and W = {f ∈ V | $f(e^{3}$ᵗ) = 0}. Which of the following statements is true?

Let $v = (a, b, c) \in \mathbb{R}^{3}$ be a nonzero vector that lies in the orthogonal complement (with respect to the standard inner product) of the row-space of the matrix A = [[2, 2, 7], [3, 1, 4]]. If a, b, c are all integers, then what is the smallest possible value of |a + b + c|?

Let A = [[0, a, 0], [0, 0, b], [c, 0, 0]], where a, b, c are real numbers with abc = 1. If $B = A + A^{2} + A^{3}$, then which of the following statements is true?

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?