CSIR NET Maths previous year question papers

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

1. A

   f is a constant function.

2. B

   If $f(0) \in \mathbb{R}$, then f(0) + f(1) = 1.

3. C

   $f(\mathbb{C})$ is connected.

4. D

   If $f(2) \in \mathbb{R}$, then f(i) + f(1) = 2.

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

1. A

   The series $\sum(n\ge1) f_{n}$ converges only at z = 0.

2. B

   The series $\sum(n\ge1) f_{n}$ converges pointwise only on a countable set E ⊆ 𝔻 but not on 𝔻 \ E.

3. C

   The series $\sum(n\ge1) f_{n}$ converges pointwise at all points of 𝔻 but not uniformly on some compact subsets of 𝔻.

4. D

   The series $\sum(n\ge1) f_{n}$ converges uniformly on all compact subsets of 𝔻 to a holomorphic function on 𝔻.

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

1. A

   If p divides |G|, then E(p) is a subgroup of G.

2. B

   For all p, E(p) is a subgroup of G.

3. C

   If E(p) is a subgroup of G, then p divides |G|.

4. D

   If G is abelian, then E(p) is a subgroup of G.

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

1. A

   $\mathbb{Q}[x]/(x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$

2. B

   $\mathbb{Q}[x]/(x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$

3. C

   $\mathbb{Z}[x]/(x - 101)$

4. D

   $\mathbb{Q}[x]/(x^{43} - 41x + 41)$

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

1. A

   f has a pole at z = 0.

2. B

   There is an entire function g such that f = g on $\mathbb{C}$ \ {0}.

3. C

   f is a polynomial of degree at most 2.

4. D

   f has an essential singularity at z = 0.

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

1. A

   f has a removable singularity at z = 0.

2. B

   f has a simple pole at z = 0.

3. C

   $\int_C (p(z)/z)$dz = 0, where C is any counterclockwise oriented smooth closed curve in $\mathbb{C}$.

4. D

   $\int_\gamma f(z)$dz $= 2\pi$ia$_{1}$, where $\gamma =$ {$z \in \mathbb{C}$ : |z| = 2025} is a counterclockwise oriented circle.

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

1. A

   If an abelian group G has property (P), every subgroup of G has property (P).

2. B

   If an abelian group G has property (P), then every element of G has finite order.

3. C

   The group $\mathbb{Z}$ has property (P).

4. D

   If an abelian group G has property (P), then no element has order $p^{2}$, where p is a prime number.

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

1. A

   The ring of polynomials in 3 variables over $\mathbb{R}$.

2. B

   The ring of complex analytic functions on the open unit disc in $\mathbb{C}$.

3. C

   The ring of 2 × 2 real matrices.

4. D

   The ring of continuous functions from [0, 1] to $\mathbb{R}$.

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

1. A

   Any group with p(p + 2) elements, where p and p + 2 are both prime numbers, has property (J).

2. B

   Any finite abelian group has property (J).

3. C

   The symmetric group $S_{3}$ has property (J).

4. D

   Any group with 77 elements has property (J).

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

1. A

   If K is the splitting field of a non-constant polynomial over $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

2. B

   If K is a normal extension of $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

3. C

   If K is the set of all the roots of the polynomial $x^{121} - x$ in an algebraic closure of 𝔽$_{11}$, then K is Galois over 𝔽$_{11}$, where 𝔽$_{11}$ is the field with 11 elements.

4. D

   If $K = \mathbb{Q}(2^(1/13), \zeta_{13})$, where $\zeta_{13}$ is a primitive 13th root of unity in $\mathbb{C}$, then K is Galois over $\mathbb{Q}$.

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

1. A

   If X and Y are compact, Y is Hausdorff and f is onto, then X is also Hausdorff.

2. B

   If X is an infinite compact set and f is a homeomorphism from X to f(X) (where f(X) is given the subspace topology), then Y is compact.

3. C

   If X is Hausdorff and f is onto, then Y is Hausdorff.

4. D

   If f is a homeomorphism, then X is second-countable if and only if Y is second-countable.

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

1. A

   R is never an integral domain.

2. B

   r = −r for all r ∈ R.

3. C

   Every nonzero prime ideal of R is maximal.

4. D

   R must be a commutative ring.

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

1. A

   $\mathbb{R}[x]/(f)$ is a field.

2. B

   $\mathbb{R}[x]/(f)$ is a finite dimensional $\mathbb{R}-$vector space.

3. C

   $\mathbb{R}[x]/(f)$ is a direct sum of fields, each of which is isomorphic to $\mathbb{R}$ or $\mathbb{C}$.

4. D

   There are no non-zero elements $u \in \mathbb{R}[x]/(f)$ such that u^m = 0 for some m ≥ 1.

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

1. A

   f is constant.

2. B

   f has removable singularities at −1 and 1.

3. C

   f is bounded.

4. D

   f has either poles or essential singularities at −1 and 1.

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