CSIR NET Maths previous year question papers

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 · Groups

1. A

   G is an Abelian group.

2. B

   G is a cyclic group.

3. C

   G is a simple group.

4. D

   G is not a simple group.

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

1. A

   f(z̄) is an entire function.

2. B

   conj(f(z)) is an entire function.

3. C

   conj(f(z̄)) is an entire function.

4. D

   conj(f(z̄)) + f(z̄) is an entire function.

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 · Rings and Fields

1. A

   n ≡ 1 (mod 3).

2. B

   n ≡ 1 (mod 35).

3. C

   n ≡ 1 (mod 21).

4. D

   n ≡ 1 (mod 5).

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

1. A

   For all prime numbers p, f(X) mod p is irreducible in $(\mathbb{Z}/p\mathbb{Z})[X]$.

2. B

   There exists a prime number p such that g(X) mod p is irreducible in $(\mathbb{Z}/p\mathbb{Z})[X]$.

3. C

   g(X) is irreducible in $\mathbb{Q}[X]$.

4. D

   f(X) is irreducible in $\mathbb{Q}[X]$.

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

1. A

   a is divisible by $3^{4}$.

2. B

   a is divisible by $2^{4}$.

3. C

   a is not divisible by 48.

4. D

   a is divisible by $3^{6}$.

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

1. A

   The Galois group of K over $\mathbb{Q}$ is the symmetric group $S_{3}$.

2. B

   The Galois group of K over $\mathbb{Q}(\omega)$ is the symmetric group $S_{3}$.

3. C

   The Galois group of K over $\mathbb{Q}$ is $\mathbb{Z}/3\mathbb{Z}$.

4. D

   The Galois group of K over $\mathbb{Q}(\omega)$ is $\mathbb{Z}/3\mathbb{Z}$.

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

1. A

   $\mathbb{Q}^{2}$ is connected.

2. B

   If A is a non-empty connected subset of $\mathbb{Q}^{2}$, then A has exactly one element.

3. C

   $\mathbb{Q}^{2}$ is Hausdorff.

4. D

   {$(x, y) \in \mathbb{Q}^{2}$ | $x^{2} + y^{2} = 1$} is compact in the subspace topology.

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 · Singularities and Residues

1. A

   $I_{k} = 0$ for every integer k.

2. B

   $I_{k} \ne 0$ if k ≥ 1.

3. C

   |$I_{k}$| ≤ |$I_{k}$₊$_{1}$| for every integer k.

4. D

   $lim_{k\to\infty}$ |$I_{k}$| $= \infty$.

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

1. A

   Let $G_{1}$ and $G_{2}$ be finite groups such that |$G_{1}$| and |$G_{2}$| are coprime. Then any homomorphism from $G_{1}$ to $G_{2}$ is trivial.

2. B

   Let G be a finite group and f : G → G a homomorphism that fixes more than half of the elements of G. Then f(x) = x for all x ∈ G.

3. C

   Let G be a finite group having exactly 3 subgroups. Then G is of order $p^{2}$ for some prime p.

4. D

   Any finite abelian group G has at least d(|G|) subgroups, where d(m) is the number of positive divisors of m.