CSIR NET Maths previous year question papers

Single choice · Zeros and Mappings

1. A

   X is the line segment joining −1 and −i.

2. B

   X = {$e^{i\theta}$ | $\theta \in [\pi, 3\pi/2]$}.

3. C

   X is the line segment joining −1 to 1.

4. D

   X = {$e^{i\theta}$ | $\theta \in [-\pi/2, \pi]$}.

Multi-select · full marks only for the exact set · Zeros and Mappings

1. A

   $lim_{z\to0}$ f(z) does not exist.

2. B

   $lim_{z\to0}$ f(z) exists and has absolute value ≤ 1.

3. C

   $lim_{z\to0}$ f(z) = 0.

4. D

   There exists $\theta \in \mathbb{R}$ such that f(z) = $e^{i\theta}z$ for all z ∈ 𝔻^×.

Multi-select · full marks only for the exact set · Zeros and Mappings

1. A

   |f(−1/5)| ≤ 1/5

2. B

   |f(−1/5)| ≤ 1/3

3. C

   |f′(−1/2)| ≤ 1/2

4. D

   |f′(−1/2)| ≤ 4/3

Single choice · Zeros and Mappings

1. A

   $T_{1}$ and $T_{2}$ conformally map R₊ and R₋ respectively, onto the disc 𝔻

2. B

   $T_{1}$ and $T_{2}$ conformally map R₋ and R₊ respectively, onto the disc 𝔻

3. C

   $T_{1}$ and $T_{2}$ conformally map the disc 𝔻 onto, respectively R₊ and R₋

4. D

   $T_{1}$ and $T_{2}$ conformally map the disc 𝔻 onto, respectively R₋ and R₊

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

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 · Zeros and Mappings

1. A

   S_R is an open subset of $\mathbb{C}$.

2. B

   S_R is a bounded subset of $\mathbb{C}$.

3. C

   |P(z)| = R for every z on the boundary of S_R.

4. D

   Every connected component of S_R contains a zero of P(z).

Single choice · Zeros and Mappings

1. A

   100

2. B

   50

3. C

   30

4. D

   0

Single choice · Zeros and Mappings

1. A

   $(1/2\pi{}i) \int_C ($zf)′/(zf) dz $= n_{0} - n_p + 1$.

2. B

   $(1/2\pi{}i) \int_C ($zf)′/(zf) dz $= n_{0} - n_p - 1$.

3. C

   $(1/2\pi{}i) \int_C ($zf)′/(zf) dz $= n_{0} - n_p$.

4. D

   $(1/2\pi{}i) \int_C ($zf)′/(zf) dz $= n_p - n_{0}$.