CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Partial Differential Equations

1. A

   $u(x_{0}, y_{0}) = \sqrt{2} e^\pi$

2. B

   $u(x_{0}, y_{0}) = e^\pi$

3. C

   $u(x_{0}, y_{0}) = -1$

4. D

   $u(x_{0}, y_{0}) = -e^\pi$

Multi-select · full marks only for the exact set · Estimation

1. A

   $T_{n}$ is an unbiased estimator of $\theta$.

2. B

   $\lim(n\to\infty) E_\theta(T_{n}) = \theta$ for all $\theta \in \mathbb{R}$.

3. C

   $T_{n}$ is a consistent estimator of $\theta$.

4. D

   max{$X_{1}, X_{2}, \dots, X_{n}$} is a consistent estimator of $\theta$.

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   The sum of squares residuals is strictly positive with probability 1.

2. B

   For every $\beta_{i}, 1 \le i \le n$, there are infinitely many linear unbiased estimators.

3. C

   $Var(\beta$̂$_{1}) = \sigma^{2}$

4. D

   $Y_{3} - Y_{2}$ is the BLUE of $\beta_{3}$.

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   MSE$(\sigma$̂$^{2}($MLE)) < MSE$(\sigma$̂$^{2})$ if r = 2, n = 12

2. B

   $Var(\sigma$̂$^{2}($MLE$)) > Var(\sigma$̂$^{2})$ if 2 ≤ r ≤ n − 2, n ≥ 12

3. C

   $Var(\sigma$̂$^{2}($MLE$)) < Var(\sigma$̂$^{2})$ if 1 ≤ r ≤ n − 2, n ≥ 3

4. D

   MSE$(\sigma$̂$^{2}($MLE)) > MSE$(\sigma$̂$^{2})$ if r = 7, n = 12

Multi-select · full marks only for the exact set · The Real Line

1. A

   S is an unbounded subset of $\mathbb{R}$.

2. B

   S is a bounded subset of $\mathbb{R}$ and sup(S) = 1.

3. C

   There exists u ∈ X such that ‖u‖ = 1 and T(u) = 1.

4. D

   S is a closed subset of $\mathbb{R}$.

Multi-select · full marks only for the exact set · Ordinary Differential Equations

1. A

   There exists an $\alpha \in (0, \infty)$ such that for every $b \in \mathbb{R}$ with $b > \alpha$, the solution y_b is bounded above on I_b

2. B

   There exists an $\alpha \in (0, \infty)$ such that for every $b \in \mathbb{R}$ with $b > \alpha$, the solution y_b is bounded below on I_b

3. C

   There exists an $\alpha \in (-\infty, 0)$ such that for every $b \in \mathbb{R}$ with $b < \alpha$, the solution y_b is bounded above on I_b

4. D

   There exists an $\alpha \in (-\infty, 0)$ such that for every $b \in \mathbb{R}$ with $b < \alpha$, the solution y_b is bounded below on I_b

Multi-select · full marks only for the exact set · Calculus of Variations

1. A

   S(2) has exactly two elements

2. B

   S(1/2) has exactly one element

3. C

   S(2) is empty

4. D

   S(1/2) has exactly two elements

Multi-select · full marks only for the exact set · Calculus of Variations

1. A

   $19\sqrt{2}/8$

2. B

   $19\sqrt{2}$

3. C

   19/8

4. D

   $19/(2\sqrt{2})$

Multi-select · full marks only for the exact set · Estimation

1. A

   Maximum likelihood estimator of $\theta$ is min{$X_{(1)}, X_{(n)}/2$}

2. B

   Maximum likelihood estimator of $\theta$ is max{$-X_{(1)}, X_{(n)}/2$}

3. C

   Method of moments estimator of $\theta$ is 2X̄

4. D

   Method of moments estimator of $\theta$ is 2X̄/3

Multi-select · full marks only for the exact set · Estimation

1. A

   Bayes estimate of $\beta$ tends to $\beta_{0}$ as $\tau^{2} \to 0$

2. B

   Bayes estimate of $\beta$ tends to ȳ/x̄ as $\tau^{2} \to 0$

3. C

   Bayes estimate of $\beta$ tends to the BLUE of $\beta$ as $\tau^{2} \to \infty$

4. D

   Bayes estimate of $\beta$ tends to MLE of $\beta$ as $\tau^{2} \to \infty$

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f is continuous on $\mathbb{R}$

2. B

   f is differentiable on $\mathbb{R}$

3. C

   f is differentiable only at 0

4. D

   f is not differentiable at 0

Multi-select · full marks only for the exact set · Integration

1. A

   The integral I is convergent

2. B

   The integral I is not convergent

3. C

   The integral $I_{a}$ converges for a = 1/2 but not for a = 0

4. D

   The integral $I_{a}$ converges for all a ≥ 0

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   The power series converges for every $z \in \mathbb{C}$

2. B

   The power series converges for every $z \in \mathbb{R}$

3. C

   The power series converges for every z ∈ {$2^{n} : n \in \mathbb{N}$}

4. D

   The power series converges for every z ∈ {$1/5^{n} : n \in \mathbb{N}$}

Multi-select · full marks only for the exact set · Ordinary Differential Equations

1. A

   There is a positive $y_{0}$ such that the solution of the IVP is unbounded

2. B

   There is a negative $y_{0}$ such that the solution of the IVP is bounded

3. C

   For every $y_{0} \in \mathbb{R}$, every solution of the IVP is bounded

4. D

   For every $y_{0} \in \mathbb{R}$, there is a solution to the IVP for all $x \in \mathbb{R}$

Multi-select · full marks only for the exact set · Linear Integral Equations

1. A

   $R(x, t, \lambda) = 3$xt$/(3 - 8\lambda) - 5x^{2}t^{2}/(5 - 32\lambda)$

2. B

   $R(x, t, \lambda) = 3$xt$/(3 - 8\lambda) + 5x^{2}t^{2}/(5 - 32\lambda)$

3. C

   $u(1) = -5/(5 - 32\lambda)$

4. D

   $u(1) = 3/(3 - 8\lambda)$

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   P{$Y_{n} \in [0, 2n]$}

2. B

   P{$Y_{n} \in [n, 3n]$}

3. C

   P{$Y_{n} \in [2n, 4n]$}

4. D

   P{$Y_{n} \in [3n, 5n]$}

Multi-select · full marks only for the exact set · Integration

1. A

   There exists $\lambda$ such that $I_\lambda$ converges, but $K_\lambda$ does not converge.

2. B

   There exists $\lambda$ such that $K_\lambda$ converges, but $I_\lambda$ does not converge.

3. C

   There exists $\lambda$ such that $I_\lambda, K_\lambda$ both converge.

4. D

   There exists $\lambda$ such that neither $I_\lambda$ nor $K_\lambda$ converges.

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   $(f_{n})$ converges pointwise on [0, 1].

2. B

   $(f_{n})$ converges uniformly on compact subsets of [0, 1) but not on [0, 1).

3. C

   $(f_{n})$ converges uniformly on [0, 1) but not on [0, 1].

4. D

   $(f_{n})$ converges uniformly on [0, 1].

Multi-select · full marks only for the exact set · Probability

1. A

   E(Y) = 0

2. B

   P(Y > 0) < P(Y < 0)

3. C

   P(Y < −1) < P(Y > 1)

4. D

   $E(Y^{2}) = 1$

Multi-select · full marks only for the exact set · Estimation

1. A

   X̄ is the method of moments estimator of $\theta$

2. B

   $X_{(1)}$ is the maximum likelihood estimator of $\theta$

3. C

   $X_{(1)} - 1/n$ is the uniformly minimum variance unbiased estimator of $\theta$

4. D

   $X_{(1)}$ is a sufficient statistic for $\theta$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   $({\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.05}$, ${\bar{X}} + (S/\sqrt{n})$ $t_{n-1,0.05})$

2. B

   $({\bar{X}} - (\sigma/\sqrt{n})$ $z_{0.05}$, ${\bar{X}} + (\sigma/\sqrt{n})$ $z_{0.05})$

3. C

   $[{\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9}$, $\infty)$

4. D

   $(-\infty, {\bar{X}} - (S/\sqrt{n})$ $t_{n-1,0.9})$