CSIR NET Maths previous year question papers

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 · The Real Line

1. A

   S is the empty set

2. B

   S has exactly one element

3. C

   S has at least two elements

4. D

   S has to be a finite set

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 · Functions of Several Variables

1. A

   $\partial{}f/\partial{}x(0, 0)$ exists

2. B

   $\partial{}f/\partial{}y(0, 0)$ exists

3. C

   f is not continuous at (0, 0)

4. D

   f is not differentiable at (0, 0)

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   f is injective in a neighbourhood of (0, 0)

2. B

   There exists an open neighbourhood U of (0, 0) in $\mathbb{R}^{2}$ such that $f_{3}$ is a function of $f_{1}$ and $f_{2}$

3. C

   f maps an open neighbourhood of (0, 0) in $\mathbb{R}^{2}$ onto an open subset of $\mathbb{R}^{3}$

4. D

   (0, 0) is an isolated point of f⁻$^{1}(${f(0,0)})

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   There exists a polynomial p(x) ∈ W such that $x^{4} - p(x) \in W^\perp$

2. B

   W^⊥ = {0}

3. C

   W and W^⊥ have the same dimension over $\mathbb{R}$

4. D

   W^⊥ is an infinite dimensional vector space over $\mathbb{R}$

Multi-select · full marks only for the exact set · Singularities and Residues

1. A

   f has an essential singularity at 0

2. B

   g has an essential singularity at 0

3. C

   f has a removable singularity at 0

4. D

   g has a removable singularity at 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 · Rings and Fields

1. A

   Every quotient ring of R is a principal ideal domain

2. B

   There exists a quotient ring S of R and an ideal I ⊆ S which is not principal

3. C

   R has countably many ideals

4. D

   Every quotient ring S(≠ {0}) of R has a unique maximal ideal which is principal

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   S is a principal ideal domain

2. B

   $S/(y^{2} + x^{2})$ is a unique factorization domain

3. C

   S is a unique factorization domain

4. D

   S/(x) is a principal ideal domain

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   $q = 2^{18}$

2. B

   $q = 2^{32}$

3. C

   $q = 2^{12}$

4. D

   $q = 2^{243}$

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

1. A

   X × [0, 1] is closed in X × Y with respect to the product topology

2. B

   X × [0, 1] is compact with respect to the product topology

3. C

   X is compact

4. D

   X × Y is compact with respect to the product topology

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

1. A

   $\beta$ is a basis for topology $\tau$

2. B

   $\mathbb{R}$ is compact in the topology $\tau$

3. C

   Topology $\tau$ is the same as the Euclidean topology

4. D

   Topology $\tau$ is Hausdorff

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 · Ordinary Differential Equations

1. A

   B = −2, C = −1, D = 1

2. B

   B = −2, C = 1, D = −1

3. C

   B = 2, C = 1, D = 1

4. D

   B = 2, C = −1, D = −1

Multi-select · full marks only for the exact set · Numerical Analysis

1. A

   [[5, 8], [1, 2]] ∈ S

2. B

   [[3, 2], [1, 2]] ∈ S

3. C

   [[−3, 1], [2, 3]] ∈ S

4. D

   [[2, 2], [4, 3]] ∈ S

Multi-select · full marks only for the exact set · Numerical Analysis

1. A

   c = 13

2. B

   g(5) = 6

3. C

   g(1) = 14

4. D

   c = 15

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

1. A

   $(e/(1 + e^{2}))(e$ˣ + e⁻ˣ$) + x^{2} - 1$

2. B

   $(e/(1 + e^{2}))(e$ˣ + e⁻ˣ$) + 1 - x^{2}$

3. C

   $(e/(1 + e^{2}))(e$ˣ + e⁻ˣ)

4. D

   $(e/(1 + e^{2}))(e$ˣ + e⁻ˣ$) + \sin(2\pi{}x)$

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

   $\lim(n\to\infty) P((\sqrt{n}\cdot\sum_{i}$₌$_{1}^{n}X_{i})/(\sum_{i}$₌$_{1}^{n}X_{i}^{2}) \le 0) = 1/2$

2. B

   $(\sum_{i}$₌$_{1}^{n}X_{i})/(\sum_{i}$₌$_{1}^{n}X_{i}^{2})$ converges in probability to 0 as $n \to \infty$

3. C

   $(1/n)\sum_{i}$₌$_{1}^{n}X_{i}^{2}$ converges in probability to 1 as $n \to \infty$

4. D

   $\lim(n\to\infty) P((\sum_{i}$₌$_{1}^{n}X_{i})/\sqrt{n} \le 0) = 1/2$

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

1. A

   $Cov(X_{(n)}/X_{(1)}, X_{(n)}) = 0$

2. B

   $E(X_{(1)}/X_{(n)}) = E(X_{(1)})/E(X_{(n)})$

3. C

   $Cov(X_{(1)}/X_{(n)}, X_{(n)}) = 0$

4. D

   $Cov(\ln(X_{(1)}) - \ln(X_{(1)} + X_{(n)}), X_{(n)}) < 0$

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

1. A

   Uniformly minimum variance unbiased estimator of $\theta$ is (n − 1)/(nT$_{n})$

2. B

   Cramer-Rao lower bound for the variance of any unbiased estimator of $\theta$ is $\theta^{2}/n$

3. C

   Uniformly minimum variance unbiased estimator of $\theta$ attains the Cramer-Rao lower bound

4. D

   $(1 - e^(-1/T_{n}))$ is a consistent estimator of $P_\theta(X_{1} \le 1)$

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

1. A

   $H_{0}$ is rejected at 5% level of significance

2. B

   $H_{0}$ is rejected at 1% level of significance

3. C

   $H_{0}$ is not rejected at 5% level of significance

4. D

   Observed value of the test statistic is 12.5

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

1. A

   p-value of the sign test is 0.04

2. B

   $H_{0}$ is NOT rejected at 5% level of significance by the sign test

3. C

   The observed value of Wilcoxon signed rank test statistic W⁺ is equal to 10

4. D

   If $P_(H_{0})(W$⁺ ≥ 14) = 0.06, then $H_{0}$ is rejected at 5% level of significance by the Wilcoxon signed rank test

Multi-select · full marks only for the exact set · Sampling and Design of Experiments

1. A

   $\alpha = 20$

2. B

   $\beta = 6$

3. C

   m = 10

4. D

   $\gamma = 2$