CSIR NET Maths previous year question papers

Single choice · Groups

1. A

   p ∤ 1 + (p − 1)! for some odd prime p.

2. B

   p | $(1234)^{p-1}$ − 1 for all primes p > 700.

3. C

   There exist $a \in \mathbb{Z}$ and a prime p > 11 such that p ∤ aᵖ − a.

4. D

   p ∤ $(p^{2})!/(p!)^{2}$ for some odd prime p.

Single choice · Rings and Fields

1. A

   The ideal $2\mathbb{Z}[i]$ is maximal in $\mathbb{Z}[i]$.

2. B

   The ideal $X\mathbb{C}[X, Y]$ is maximal in $\mathbb{C}[X, Y]$.

3. C

   The set of all polynomials in $\mathbb{C}[X]$ whose coefficients add up to 0 is a maximal ideal in $\mathbb{C}[X]$.

4. D

   The ideal $(\sqrt{2} - 1)\mathbb{Z}[\sqrt{2}]$ is maximal in $\mathbb{Z}[\sqrt{2}]$.

Single choice · Topology

1. A

   25

2. B

   120

3. C

   24

4. D

   6

Single choice · Topology

1. A

   If X × Y is Hausdorff, then the statement S is true.

2. B

   If Y is connected, then the statement S is true.

3. C

   If X × Y is regular, then the statement S is true.

4. D

   If Y is compact, then the statement S is true.

Single choice · Singularities and Residues

1. A

   $\int_{\gamma1}$ tan(z)dz $= 2\pi{}i$

2. B

   $\int_{\gamma1}$ tan(z)dz $= -2\pi{}i$

3. C

   $\int_{\gamma2}$ tan(z)dz $= 2\pi{}i$

4. D

   $\int_{\gamma2}$ tan(z)dz $= -2\pi{}i$

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

1. A

   Let f(X) ∈ 𝔽_p[X] and $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽$_p(\alpha)$ is the splitting field of f in 𝔽̄_p.

2. B

   Let f, g ∈ 𝔽_p[X] be irreducible polynomials of same degree and $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽$_p(\alpha)$ is the splitting field of g in 𝔽̄_p.

3. C

   𝔽_p[X] has infinitely many irreducible polynomials.

4. D

   The set {a + b | a, b ∈ 𝔽_p} is contained in {$a^{2} + b^{2}$ | a, b ∈ 𝔽_p}.

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

1. A

   X is metrizable.

2. B

   X is compact.

3. C

   X is Hausdorff.

4. D

   X is connected.

Single choice · Topology

1. A

   {m + ne$^(2\pi{}i/3)$ | $m, n \in \mathbb{Z}$} is a dense subset of $\mathbb{C}$.

2. B

   Open connected subsets of $\mathbb{R}^{3}$ need not be path-connected.

3. C

   Let X be a topological space and $p : X \to \mathbb{R} a$ continuous surjective open map. If p⁻$^{1}(${$\alpha$}) is connected for every $\alpha \in \mathbb{R}$, then X must be connected.

4. D

   Compact subsets of any infinite topological space are closed.

Single choice · Cauchy Theory

1. A

   If Re(f), Im(f) are bounded then f is constant

2. B

   If e^(|Re(f)| + |Im(f)|) is bounded, then f is constant

3. C

   If the sum Re(f) + Im(f) and the product Re(f)Im(f) are bounded, then f is constant

4. D

   If sin(Re(f) + Im(f)) is bounded, then f is constant

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

   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 · 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 · 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