CSIR NET Maths previous year question papers

Single choice · Analytic Functions

1. A

   There exists an entire function f such that f⁽$^{n}$⁾$(0) = n!/n^{n}$ for all positive integers n.

2. B

   There exists an entire function f such that f⁽$^{n}$⁾$(0) = n!n^{n}$ for all positive integers n.

3. C

   There exists an entire function f such that f⁽$^{n}$⁾(0) = (n − 1)! for all positive integers n.

4. D

   There exists an entire function f such that f⁽$^{n}$⁾(0) = n!n for all positive integers n.

Single choice · Analytic Functions

1. A

   X is uncountable.

2. B

   X is infinite and countable.

3. C

   Every element of X is an open map.

4. D

   X has exactly one element.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   A is nonempty.

2. B

   A is finite.

3. C

   A is infinite.

4. D

   A is uncountable.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   (P) is true.

2. B

   (P) is FALSE.

3. C

   (Q) is true.

4. D

   (Q) is FALSE.

Single choice · Analytic Functions

1. A

   E is the real axis.

2. B

   E is the imaginary axis.

3. C

   E contains an open subset of $\mathbb{C}$, but $E \ne \mathbb{C}$.

4. D

   $E = \mathbb{C}$

Single choice · Analytic Functions

1. A

   0

2. B

   (e + 1/e)(cos 1)/2

3. C

   (e − 1/e)(cos 1)/2

4. D

   1

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.

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 · Analytic Functions

1. A

   f(z) = f(−z) for all $z \in \mathbb{C}$.

2. B

   f′(0) = f″(0) = f‴(0) = 0

3. C

   There is an entire function $g : \mathbb{C} \to \mathbb{C}$ such that $f(z) = g(z^{4})$ for all $z \in \mathbb{C}$.

4. D

   f is necessarily a constant function.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   The power series converges for every $z \in \mathbb{C}$

2. B

   The power series converges for every $z \in \mathbb{R}$

3. C

   The power series converges for every z ∈ {$2^{n} : n \in \mathbb{N}$}

4. D

   The power series converges for every z ∈ {$1/5^{n} : n \in \mathbb{N}$}

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 · 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 $\mathbb{C}$.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   f(z̄) is an entire function.

2. B

   conj(f(z)) is an entire function.

3. C

   conj(f(z̄)) is an entire function.

4. D

   conj(f(z̄)) + f(z̄) is an entire function.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   The sequence $(p_{n})$ converges to an entire function uniformly on compact subsets of $\mathbb{C}$.

2. B

   For all $n \ge 1, p_{n}$ has a zero in the set {$z \in \mathbb{C}$ : |z| ≤ 2023}.

3. C

   There exists a sequence $(z_{n})$ of complex numbers such that lim |$z_{n}$| $= \infty$ and $p_{n}(z_{n}) = 0$ for all n ≥ 1.

4. D

   Let $S_{n}$ be the set of all zeros of $p_{n}$. If $a_{n} =$ $min_{z\in{}S_{n}}$ |z|, then $a_{n} \to \infty$ as $n \to \infty$.