CSIR NET Maths previous year question papers

Single choice · Singularities and Residues

1. A

   The function f(1/z) has a removable singularity at 0.

2. B

   The function f(1/z) has an essential singularity at 0.

3. C

   If $\int_{\gamma{}R}$ f(z)dz = 0 for some R > 0, then S is not a singleton set.

4. D

   If S is not a singleton set, then $\int_{\gamma{}R}$ f(z)dz = 0 for some R > 0 such that S ∩ {$z \in \mathbb{C}$ : |z| = R} = ∅.

Single choice · Singularities and Residues

1. A

   For all z with |z| ≥ 1, the sequence {$f_{n}(z)$}$_{n}\ge1$ converges.

2. B

   For all z ∈ 𝔻{0}, the sequence {$f_{n}(z)$}$_{n}\ge1$ converges.

3. C

   The sequence {$f_{n}$}$_{n}\ge1$ converges pointwise to a holomorphic function on 𝔻[0,1).

4. D

   The sequence {$f_{n}$}$_{n}\ge1$ converges pointwise to a holomorphic function on {$z \in \mathbb{C}$ : |z| > 1}.

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 · Singularities and Residues

1. A

   I = 0

2. B

   $(1/2\pi{}i)I \in$ {$4n : n \in \mathbb{Z}, n \ge 1$}

3. C

   $I = 2\pi{}i \sum_{n=0}^\infty$ 1/n!

4. D

   $I = 2\pi{}i \sum_{n=0}^\infty$ 1/(n!(n + 1)!)

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

1. A

   $\int_\gamma1$ dz/(z sin z) = 0

2. B

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

3. C

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

4. D

   $\int_\gamma2$ dz/(z sin z) = 0

Single choice · Singularities and Residues

1. A

   z = 0 is a zero of f of order 3 and a pole of g of order 1.

2. B

   z = 0 is a zero of f of order 2 and a pole of g of order 1.

3. C

   z = 0 is a zero of f of order 3 and a zero of g of order 1.

4. D

   z = 0 is a zero of f of order 2 and a zero of g of order 1.

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 · 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

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 · 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 · Singularities and Residues

1. A

   f has an essential singularity at 0

2. B

   g has an essential singularity at 0

3. C

   f has a removable singularity at 0

4. D

   g has a removable singularity at 0

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

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

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

1. A

   $I_{k} = 0$ for every integer k.

2. B

   $I_{k} \ne 0$ if k ≥ 1.

3. C

   |$I_{k}$| ≤ |$I_{k}$₊$_{1}$| for every integer k.

4. D

   $lim_{k\to\infty}$ |$I_{k}$| $= \infty$.