CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Lebesgue Measure and Integration

1. A

   If f is Lebesgue integrable on $\mathbb{R}$, then $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite.

2. B

   If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite, then f is Lebesgue integrable on $\mathbb{R}$.

3. C

   If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite, and $f(x) \ge a_{1}$ for all $x \in \mathbb{R}$, then f is Lebesgue integrable on $\mathbb{R}$.

4. D

   If $\sum_{k\ge0}$ $a_k \mu(A_k)$ is finite and f is bounded, then f is Lebesgue integrable on $\mathbb{R}$.

Multi-select · full marks only for the exact set · Lebesgue Measure and Integration

1. A

   H contains an orthonormal basis of $\ell^{2}$.

2. B

   H is a linear subspace of $\ell^{2}$.

3. C

   H is a bounded subset of $\ell^{2}$.

4. D

   H is a convex subset of $\ell^{2}$.

Multi-select · full marks only for the exact set · Lebesgue Measure and Integration

1. A

   $f \in L^{1}(\mathbb{R})$

2. B

   $f \in L^{2}(\mathbb{R})$

3. C

   $f \in L^{4}(\mathbb{R})$

4. D

   $f \in L^\infty(\mathbb{R})$

Multi-select · full marks only for the exact set · Lebesgue Measure and Integration

1. A

   (‖$f_{n}$‖$_{1})$ is a convergent sequence.

2. B

   (‖$f_{n}$‖$_{2})$ is a convergent sequence.

3. C

   Both (‖$f_{n}$‖$_{1})$ and (‖$f_{n}$‖$_{2})$ are convergent sequences.

4. D

   Neither (‖$f_{n}$‖$_{1})$ nor (‖$f_{n}$‖$_{2})$ is a convergent sequence.

Single choice · Lebesgue Measure and Integration

1. A

   S is in bijection with the set of rational numbers.

2. B

   S is in bijection with the set of real numbers.

3. C

   S is in bijection with S × S.

4. D

   S is in bijection with the power set of S.

Single choice · Lebesgue Measure and Integration

1. A

   Y is an infinite set.

2. B

   Y has $2^{|X|}$ elements.

3. C

   There is a bijective function from X to Y.

4. D

   There is a surjective function from X to Y.

Multi-select · full marks only for the exact set · Lebesgue Measure and Integration

1. A

   $A_{n}$ is finite for every integer n ≥ 1.

2. B

   $A_{n}$ is finite for some integer n ≥ 1.

3. C

   $A_{n}$ is infinite for some integer n ≥ 1.

4. D

   $A_{n}$ is countable (finite or infinite) for every integer n ≥ 1.