CSIR NET Maths previous year question papers

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

1. A

   E(Zᵐ) = 1/(m + 1), for all $m \in \mathbb{N}$

2. B

   $E(\Phi$⁻$^{1}(Z)) = 0$, where $\Phi(\cdot)$ is the cumulative distribution function of standard normal random variable.

3. C

   Z is degenerate at 0.

4. D

   If $Z_{1}$ and $Z_{2}$ are independent and identically distributed (i.i.d.) random variables having distribution same as the distribution of Z, then Z has the same distribution as $(Z_{1} + Z_{2})/2$.

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   There exists a non-zero vector $v_{1} \in \mathbb{R}^{4}$ such that Tv$_{1} = 2v_{1}$

2. B

   There exists a non-zero vector $v_{2} \in \mathbb{R}^{4}$ such that Tv$_{2} = v_{2}$

3. C

   For every non-zero vector $v \in \mathbb{R}^{4}$, the set {2v, 3Tv} is linearly independent

4. D

   T is a one-one function

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   There exists an infinite set X such that f(z) = 0 for all z ∈ X

2. B

   There exists a non-empty closed set X such that f(z) = 0 for all z ∈ X

3. C

   The set X_k is unbounded for each k ≥ 1

4. D

   If there exists a bounded sequence (z_k)_(k≥1) such that z_k ∈ X_k for each k ≥ 1, then f has a zero

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

1. A

   3

2. B

   4

3. C

   5

4. D

   6

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

1. A

   1

2. B

   8

3. C

   16

4. D

   24

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

1. A

   state 0 is positive recurrent

2. B

   state 3 is transient

3. C

   state 1 is aperiodic and positive recurrent

4. D

   state 2 is aperiodic and null-recurrent

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

1. A

   $P(Y_{1} > 2) = 2e$⁻$^{1}$

2. B

   $P(Y_{2} > 2) = e$⁻$^{2}$

3. C

   $P(Y_{3} > 2) = e$⁻$^{2}$

4. D

   $Var(Y_{1} + Y_{2} + Y_{3}) = 32$

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

1. A

   P(max{$X_{1}, X_{2}$} < 1) = 1/(2e)

2. B

   P(min{$X_{1}, X_{2}$} > 1) = 1/(2e)

3. C

   P(min{$X_{1}, X_{2}$} $< X_{3}) = 2/3$

4. D

   P(max{$X_{1}, X_{2}$} $< X_{3}) = 1/3$

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 · 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 · 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 · 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 · Rings and Fields

1. A

   R is isomorphic to a subring of $\mathbb{C}$.

2. B

   For any prime number $p \in \mathbb{Z}$, the ideal generated by $\psi(p)$ is a proper ideal of R.

3. C

   R has infinitely many prime ideals.

4. D

   The ideal generated by $\psi(X)$ is a prime ideal in R.

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

1. A

   The sequence $(p_{n})$ converges to an entire function uniformly on compact subsets of $\mathbb{C}$.

2. B

   For all $n \ge 1, p_{n}$ has a zero in the set {$z \in \mathbb{C}$ : |z| ≤ 2023}.

3. C

   There exists a sequence $(z_{n})$ of complex numbers such that lim |$z_{n}$| $= \infty$ and $p_{n}(z_{n}) = 0$ for all n ≥ 1.

4. D

   Let $S_{n}$ be the set of all zeros of $p_{n}$. If $a_{n} =$ $min_{z\in{}S_{n}}$ |z|, then $a_{n} \to \infty$ as $n \to \infty$.

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   f = 0.

2. B

   f is a constant function.

3. C

   There exists a compact subset K of $\mathbb{C}$ such that $f^{-1}(K)$ is not compact.

4. D

   f is a polynomial.

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   There exists a holomorphic surjective map $f : \Omega_{1} \to \Omega_{2}$.

2. B

   There exists a holomorphic surjective map $f : \Omega_{2} \to \Omega_{1}$.

3. C

   There exists a holomorphic injective map $f : \Omega_{1} \to \Omega_{2}$.

4. D

   There exists a holomorphic injective map $f : \Omega_{2} \to \Omega_{1}$.

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

1. A

   f(x) = [x] sin(1/x) for x ≠ 0, f(0) = 0 (where [x] is the integer part) has a discontinuity at 0 which is removable.

2. B

   $f : [0,\infty) \to \mathbb{R}, f(x) = \sin(\log x)$ for x ≠ 0, f(0) = 0, has a discontinuity at 0 which is NOT removable.

3. C

   f(x) = $e^{1/x}$ for x < 0, f(x) = $e^{1/(x+1)}$ for x ≥ 0, has a jump discontinuity at 0.

4. D

   If $f, g : [0,1] \to \mathbb{R}$ are of bounded variation, then fg has at most countably many discontinuities.