CSIR NET Maths previous year question papers

Single choice · Cauchy Theory

1. A

   The function zf(1/z) → 0 as |z| $\to \infty$.

2. B

   The function f is constant.

3. C

   c = 0.

4. D

   c > 0.

Single choice · Cauchy Theory

1. A

   |f(z)| $\to \infty$ as |z| $\to \infty$.

2. B

   f(z)/z → 0 as |z| $\to \infty$.

3. C

   zf(z) → 0 as |z| $\to \infty$.

4. D

   f(z) → 0 as |z| $\to \infty$.

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   If f(z) = f(z + 1) for all $z \in \mathbb{C}$ then f is a constant function.

2. B

   If f(z) = f(z + 1) = f(z + i) for all $z \in \mathbb{C}$ then f is a constant function.

3. C

   If f(1/z) has a removable singularity at 0 then f is a constant function.

4. D

   If f is a non-constant function then f(1/z) has a pole at 0.

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   The function g has a pole at 0.

2. B

   If $g(\alpha)=0$, then |$\alpha$|≠1.

3. C

   The function g has only finitely many zeros.

4. D

   $max_{|z|=1}$ |g(z)| ≥ 1

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   f is a real valued function.

2. B

   f(z) = 1 for all $z\in\mathbb{C}$.

3. C

   f is not an entire function.

4. D

   f has finitely many zeros on the imaginary axis.

Single choice · Cauchy Theory

1. A

   f(1)

2. B

   f(2)

3. C

   f′(1)

4. D

   f′(2)

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 · 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

   $2\pi{}i$

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 · 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.

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   f(z) = 2024 for all $z \in \mathbb{C}$

2. B

   f is a constant function

3. C

   f is an injective function

4. D

   f is a bijective function

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   There exists an infinite set X such that f(z) = 0 for all z ∈ X

2. B

   There exists a non-empty closed set X such that f(z) = 0 for all z ∈ X

3. C

   The set X_k is unbounded for each k ≥ 1

4. D

   If there exists a bounded sequence (z_k)_(k≥1) such that z_k ∈ X_k for each k ≥ 1, then f has a zero

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   f = 0.

2. B

   f is a constant function.

3. C

   There exists a compact subset K of $\mathbb{C}$ such that $f^{-1}(K)$ is not compact.

4. D

   f is a polynomial.

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   There exists a holomorphic surjective map $f : \Omega_{1} \to \Omega_{2}$.

2. B

   There exists a holomorphic surjective map $f : \Omega_{2} \to \Omega_{1}$.

3. C

   There exists a holomorphic injective map $f : \Omega_{1} \to \Omega_{2}$.

4. D

   There exists a holomorphic injective map $f : \Omega_{2} \to \Omega_{1}$.