CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Zeros and Mappings

1. A

   S_R is an open subset of $\mathbb{C}$.

2. B

   S_R is a bounded subset of $\mathbb{C}$.

3. C

   |P(z)| = R for every z on the boundary of S_R.

4. D

   Every connected component of S_R contains a zero of P(z).

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

1. A

   f(0) = 0

2. B

   f(z) = 0 for all z ∈ 𝔻.

3. C

   There exists a nonzero constant c such that f(z) = ce^(−1/z) for all z ∈ 𝔻 \ {0}.

4. D

   There exists a nonzero constant c and a positive integer n such that f(z) = cz$^{n}e^(-1/z)$ for all z ∈ 𝔻 \ {0}.

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

1. A

   f(z) = f(−z) for all $z \in \mathbb{C}$.

2. B

   f′(0) = f″(0) = f‴(0) = 0

3. C

   There is an entire function $g : \mathbb{C} \to \mathbb{C}$ such that $f(z) = g(z^{4})$ for all $z \in \mathbb{C}$.

4. D

   f is necessarily a constant function.

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

1. A

   The value of the Euler $\varphi-$function is even for all integers n ≥ 3.

2. B

   Let G be a finite group and S a subset of G with |S| > |G|/2. Then {ab : a, b ∈ S} = G.

3. C

   The polynomial ring $\mathbb{R}[x_{1}, \dots, x_{n}]$ is a Euclidean domain for all integers n ≥ 1.

4. D

   The subset {f ∈ C([0, 1]) : f(1/2) = 0} of the ring C([0, 1]) of continuous functions from [0, 1] to $\mathbb{R}$ is a prime ideal.

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

1. A

   $\mathbb{Q}$ with ordinary addition

2. B

   $\mathbb{C}$ \ {0} with ordinary multiplication

3. C

   The cyclic group of order 5

4. D

   The symmetric group $S_{5}$

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

1. A

   S is a cyclic group.

2. B

   S has $\varphi(2025)$ elements, where $\varphi$ denotes the Euler $\varphi-$function.

3. C

   S has $\varphi(2025)$ generators, where $\varphi$ denotes the Euler $\varphi-$function.

4. D

   S has 81 elements.

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

1. A

   The set of all irrational numbers is dense in $(\mathbb{R}, \tau)$.

2. B

   For each prime number p, the set {$0, \sqrt{p}$} is dense in $(\mathbb{R}, \tau)$.

3. C

   [0, 1] is compact in $(\mathbb{R}, \tau)$.

4. D

   $(\mathbb{R}, \tau)$ is Hausdorff.

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

1. A

   The spaces B, the closure of W, and C are homeomorphic.

2. B

   The spaces B, W, C are homeomorphic.

3. C

   If C is compact, then A, B, C are homeomorphic.

4. D

   If A is connected, then B and C are connected.

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

1. A

   f(z) = 2024 for all $z \in \mathbb{C}$

2. B

   f is a constant function

3. C

   f is an injective function

4. D

   f is a bijective 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

   If f(a) is a unit in S for every non-zero element a ∈ R, then S is a field

2. B

   If f(a) is a unit in S for every non-zero element a ∈ R, then f(R) is a field

3. C

   If R is a field, then f(a) is a unit in S for every non-zero element a ∈ R

4. D

   If a is a unit in R, then f(a) is a unit in S

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 · 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

Single choice · Groups

1. A

   If |G| $= p^{6}$, then G has a subgroup of index $p^{2}$.

2. B

   If |G| $= p^{6}$, then G has at least five normal subgroups.

3. C

   Center of G can be infinite.

4. D

   There exists G with |G| $= p^{6}$ such that G has exactly six normal subgroups.

Single choice · Rings and Fields

1. A

   0

2. B

   2

3. C

   4

4. D

   8

Single choice · Rings and Fields

1. A

   0

2. B

   1

3. C

   2

4. D

   3

Single choice · Singularities and Residues

1. A

   $\sum_{l=0}^\infty$ 1/(l+1)!

2. B

   $\sum_{l=0}^\infty$ 1/(l!(l+1))

3. C

   $\sum_{l=0}^\infty$ 1/(l!(l+1)!)

4. D

   $\sum_{l=0}^\infty$ $1/((l^{2}+l)!)$

Single choice · Analytic Functions

1. A

   f is an entire function.

2. B

   f has a simple pole at z = 0.

3. C

   f has a Taylor series expansion $f(z) = \sum a_{n}z^{n}$, where $a_{0} = 1, a_{1} = 0$ and $a_{n}$₊$_{2} = a_{n} + a_{n}$₊$_{1}$ for n ≥ 0.

4. D

   f has a Taylor series expansion $f(z) = \sum a_{n}z^{n}$, where $a_{0} = 1, a_{1} = 1$ and $a_{n}$₊$_{2} = a_{n} + a_{n}$₊$_{1}$ for n ≥ 0.