CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Metric Spaces

1. A

   There exists $\alpha \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖T(f) − T(g)‖ $\le \alpha$‖f − g‖.

2. B

   There exists $\alpha \in (0, 1)$ such that for all f, g ∈ C[0, 1], ‖$T^{2}(f) - T^{2}(g)$‖ $\le \alpha$‖f − g‖.

3. C

   The set {f ∈ C[0, 1] : T(f) = f} is a singleton set.

4. D

   ‖$T^{n}$‖ $\to \infty$ as $n \to \infty$.

Multi-select · full marks only for the exact set · Metric Spaces

1. A

   The preimage of a compact set under p is a compact set.

2. B

   The preimage of a connected set under p is a connected set.

3. C

   Every point $x \in \mathbb{R}$ has an open neighbourhood U_x such that the restriction $p|_{U_x}$ is a homeomorphism onto an open set in $\mathbb{R}$.

4. D

   The image of a bounded set under p is a bounded set.

Multi-select · full marks only for the exact set · Metric Spaces

1. A

   S is dense in $\mathbb{R}$.

2. B

   S \ $\mathbb{Q}$ is dense in $\mathbb{R}$.

3. C

   S \ $\mathbb{Q}$ is discrete with subspace topology on S.

4. D

   S is connected.

Multi-select · full marks only for the exact set · Metric Spaces

1. A

   For all x,y∈X and every non-empty subset A of X, we have d(x,A)−d(y,A) ≤ d(x,y).

2. B

   For every non-empty subset A of X, the function x↦d(x,A) is uniformly continuous on X.

3. C

   A non-empty subset A of X is closed if and only if d(x,A)>0 for all x in X outside A.

4. D

   If X is compact and $C_{1},\dots,C_{k}$ are non-empty closed sets of X such that $\cap{}C_{i}=\emptyset$, then the minimum value of $x\mapsto\sum{}d(x,C_{i})$ on X is 0.

Single choice · Metric Spaces

1. A

   A is a closed subset of $\mathbb{R}^{2}$ but B is not a closed subset of $\mathbb{R}^{2}$.

2. B

   B is a closed subset of $\mathbb{R}^{2}$ but A is not a closed subset of $\mathbb{R}^{2}$.

3. C

   Both A and B are closed subsets of $\mathbb{R}^{2}$.

4. D

   Neither A nor B is a closed subset of $\mathbb{R}^{2}$.

Multi-select · full marks only for the exact set · Metric Spaces

1. A

   d_X is a metric on X which induces the topology $\tau$.

2. B

   The topological space $(X, \tau)$ is connected.

3. C

   $(P_{n})_{n}\ge_{1}$ converges to $P_{0}$ in the topological space $(X, \tau)$.

4. D

   The topological space $(X, \tau)$ is compact.