CSIR NET Maths previous year question papers

Single choice · Groups

1. A

   If G(n) = 1, then n is prime.

2. B

   G(8) = 2

3. C

   If $\gcd(n, \varphi(n)) > 1$, then G(n) > 1. (Here $\varphi$ denotes the Euler $\varphi-$function.)

4. D

   $\lim \sup(n\to\infty) G(n) = 2$

Single choice · Groups

1. A

   The cyclic group of order 6.

2. B

   The symmetric group $S_{5}$.

3. C

   The alternating group $A_{5}$.

4. D

   The dihedral group with ten elements.

Single choice · Rings and Fields

1. A

   I = (x, y)

2. B

   I = (x + y)

3. C

   $I = (x^{2} + y^{2})$

4. D

   I = (xy − 1)

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

Single choice · Cauchy Theory

1. A

   0

2. B

   $\pi{}i$

3. C

   $-\pi{}i$

4. D

   $2\pi{}i$

Single choice · Singularities and Residues

1. A

   $I_{r}$ ≠ 0 if r > max{|a|, b}

2. B

   $I_{r}$ ≠ 0 if r < max{|a|, b}

3. C

   $I_{r}$ = 0 if r > max{|a|, b} and |a| = b

4. D

   $I_{r}$ = 0 if |a| < r < b

Single choice · Zeros and Mappings

1. A

   If |z| < 1, then |1 − āz| < |z − a|

2. B

   If |z − a| = |1 − āz|, then |z| = 1

3. C

   If |z| = 1, then |z − a| < |1 − āz|

4. D

   If |1 − āz| < |z − a|, then |z| < 1

Single choice · Groups

1. A

   525

2. B

   35

3. C

   840

4. D

   315

Single choice · Groups

1. A

   30

2. B

   60

3. C

   45

4. D

   10

Single choice · Rings and Fields

1. A

   R is not commutative

2. B

   The ideal (X − 1) is a maximal ideal in R

3. C

   The ideal (X − 1, 2) is a prime ideal in R

4. D

   The ideal (X, 5) is a maximal ideal in R

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.