CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   $q = 2^{18}$

2. B

   $q = 2^{32}$

3. C

   $q = 2^{12}$

4. D

   $q = 2^{243}$

Multi-select · full marks only for the exact set · Topology

1. A

   X × [0, 1] is closed in X × Y with respect to the product topology

2. B

   X × [0, 1] is compact with respect to the product topology

3. C

   X is compact

4. D

   X × Y is compact with respect to the product topology

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   R is isomorphic to a subring of $\mathbb{C}$.

2. B

   For any prime number $p \in \mathbb{Z}$, the ideal generated by $\psi(p)$ is a proper ideal of R.

3. C

   R has infinitely many prime ideals.

4. D

   The ideal generated by $\psi(X)$ is a prime ideal in R.

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 $n \to \infty$.

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 $f : \Omega_{2} \to \Omega_{1}$.

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f(x) = [x] sin(1/x) for x ≠ 0, f(0) = 0 (where [x] is the integer part) has a discontinuity at 0 which is removable.

2. B

   $f : [0,\infty) \to \mathbb{R}, f(x) = \sin(\log x)$ for x ≠ 0, f(0) = 0, has a discontinuity at 0 which is NOT removable.

3. C

   f(x) = $e^{1/x}$ for x < 0, f(x) = $e^{1/(x+1)}$ for x ≥ 0, has a jump discontinuity at 0.

4. D

   If $f, g : [0,1] \to \mathbb{R}$ are of bounded variation, then fg has at most countably many discontinuities.