CSIR NET Maths previous year question papers

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

1. A

   If $G_{1}, G_{2}$ are two groups such that Aut$(G_{1})$ is isomorphic to Aut$(G_{2})$, then $G_{1}$ is isomorphic to $G_{2}$.

2. B

   If |G| = 2, then Aut(G × G) is abelian.

3. C

   If G is the group of complex numbers under addition, then Aut(G) is abelian.

4. D

   If G is finite, then Aut(G) is finite.

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

1. A

   If $G_{1}$ is normal in G, then $(G_{2}G_{1})/G_{1} \cong G_{2}/(G_{2} \cap G_{1})$.

2. B

   If $H_{1}$ and $H_{2}$ are normal subgroups of $G_{1}$ and $G_{2}$, respectively, then $(G_{1} \times G_{2})/(H_{1} \times H_{2}) \cong (G_{1}/H_{1}) \times (G_{2}/H_{2})$.

3. C

   If $G_{1}$ is normal in $G_{2}$ and $G_{2}$ is normal in G, then $G_{1}$ is normal in G.

4. D

   Every subgroup of prime index in G is normal.

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

1. A

   If d is a positive integer that divides |G|, then G has a subgroup of order d.

2. B

   The map f : G×G → G given by f(a,b)=ab is not a group homomorphism.

3. C

   Suppose that for every positive integer d that divides |G|, there exists a subgroup of G of order d. Then G has at least three normal subgroups.

4. D

   |G| ≠ 16

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 · 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 · Estimation

1. A

   $(79X + 2024Y, 23Z^{2})$ is a sufficient statistic for $\theta$.

2. B

   $13(X + Y + Z)^{3}$ is a minimal sufficient statistic for $\theta$.

3. C

   2(X + Y + Z) is a complete sufficient statistic for $\theta$.

4. D

   (X − 2Y + Z) is an ancillary statistic.

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 · 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 · 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

   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 · 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 · 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.