CSIR NET Maths previous year question papers

Single choice · Analytic Functions

1. A

   f can be extended to $\mathbb{C}$ as an entire function.

2. B

   f must have infinitely many zeros in 𝔻.

3. C

   f is not a polynomial.

4. D

   exp(f) cannot take every complex value.

Single choice · Cauchy Theory

1. A

   $\lim(y\to+\infty) f'($iy) = 0

2. B

   $\lim(y\to+\infty) f'($iy) is a complex number with absolute value 1.

3. C

   $\lim(y\to+\infty)$ |f′(iy)| $= +\infty$

4. D

   $\lim(y\to+\infty) f'($iy) is not a real number.

Single choice · Singularities and Residues

1. A

   I_(m,n) = 1 if m = n

2. B

   I_(m,n) = 1 if m + 1 = n

3. C

   I_(m,n) = 1 if m = n + 1

4. D

   I_(m,n) = 1 if m = n + 2

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

Single choice · Groups

1. A

   If |G| $= p^{6}$, then G has a subgroup of index $p^{2}$.

2. B

   If |G| $= p^{6}$, then G has at least five normal subgroups.

3. C

   Center of G can be infinite.

4. D

   There exists G with |G| $= p^{6}$ such that G has exactly six normal subgroups.

Single choice · Rings and Fields

1. A

   0

2. B

   2

3. C

   4

4. D

   8

Single choice · Rings and Fields

1. A

   0

2. B

   1

3. C

   2

4. D

   3

Single choice · Singularities and Residues

1. A

   $\sum_{l=0}^\infty$ 1/(l+1)!

2. B

   $\sum_{l=0}^\infty$ 1/(l!(l+1))

3. C

   $\sum_{l=0}^\infty$ 1/(l!(l+1)!)

4. D

   $\sum_{l=0}^\infty$ $1/((l^{2}+l)!)$

Single choice · Analytic Functions

1. A

   f is an entire function.

2. B

   f has a simple pole at z = 0.

3. C

   f has a Taylor series expansion $f(z) = \sum a_{n}z^{n}$, where $a_{0} = 1, a_{1} = 0$ and $a_{n}$₊$_{2} = a_{n} + a_{n}$₊$_{1}$ for n ≥ 0.

4. D

   f has a Taylor series expansion $f(z) = \sum a_{n}z^{n}$, where $a_{0} = 1, a_{1} = 1$ and $a_{n}$₊$_{2} = a_{n} + a_{n}$₊$_{1}$ for n ≥ 0.

Single choice · Groups

1. A

   G is a union of proper subgroups.

2. B

   G is a union of proper subgroups if |G| has at least two distinct prime divisors.

3. C

   If G is abelian, then G is a union of proper subgroups.

4. D

   G is a union of proper subgroups if and only if G is not cyclic.

Single choice · Rings and Fields

1. A

   88

2. B

   −88

3. C

   −2

4. D

   0

Single choice · Rings and Fields

1. A

   K is dense in $\mathbb{C}$.

2. B

   K is an algebraic extension of $\mathbb{Q}$.

3. C

   $\mathbb{C}$ is an algebraic extension of K.

4. D

   The smallest closed subset of $\mathbb{C}$ containing K is NOT a field.

Single choice · Analytic Functions

1. A

   For $c_{1}, c_{2} \in \mathbb{C}$, the level curves $u = c_{1}$ and $v = c_{2}$ are orthogonal wherever they intersect.

2. B

   $\nabla{}u \cdot \nabla{}v = 0$ at every point.

3. C

   If f is an entire function, then $\nabla{}u \cdot \nabla{}v = 0$ at every point.

4. D

   If $\nabla{}u \cdot \nabla{}v = 0$ at every point, then f is an entire function.

Single choice · Analytic Functions

1. A

   f(ℍ$) = \mathbb{C}$∖{0}.

2. B

   f(ℍ) ∩ ℍ is countable.

3. C

   f(ℍ) is bounded.

4. D

   f(ℍ) is a convex subset of $\mathbb{C}$.

Single choice · Zeros and Mappings

1. A

   100

2. B

   50

3. C

   30

4. D

   0