CSIR NET Maths previous year question papers

Single choice · Sequences and Series of Functions

1. A

   $(f_{n})_{n}\ge_{1}$ converges uniformly on [0, 1] to a continuous function f.

2. B

   $(f_{n})_{n}\ge_{1}$ converges pointwise on [0, 1] to a discontinuous function f.

3. C

   $(f_{n})_{n}\ge_{1}$ converges pointwise on [0, 1] to a continuous function f.

4. D

   $(f_{n})_{n}\ge_{1}$ does not converge pointwise on [0, 1].

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   If each $f_{n}$ is uniformly continuous and $(f_{n})_{n}\ge_{1}$ converges to f uniformly, then f is uniformly continuous.

2. B

   If each $f_{n}$ is bounded and $(f_{n})_{n}\ge_{1}$ converges to f pointwise, then f is bounded.

3. C

   If each $f_{n}$ is bounded and continuous, $(f_{n})_{n}\ge_{1}$ converges pointwise to a bounded and continuous function f, then the convergence is uniform.

4. D

   If each $f_{n}$ is differentiable and $(f_{n})_{n}\ge_{1}$ converges to f uniformly, then f is differentiable.

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$, and $(f'_{n})_{n}\ge_{1}$ converges uniformly to 0 on the interval (−M, M) for some positive real number M.

2. B

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$, and $(f'_{n})_{n}\ge_{1}$ converges pointwise to 0 on $\mathbb{R}$.

3. C

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$ and $(f'_{n})_{n}\ge_{1}$ does not converge pointwise to 0 on $\mathbb{R}$.

4. D

   $(f_{n})_{n}\ge_{1}$ converges pointwise to 0 on $\mathbb{R}$ but not uniformly on $\mathbb{R}$.

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   [0,1]

2. B

   (0,1]

3. C

   (10⁻$^{2025},1)$

4. D

   (−10⁻$^{2025},10$⁻$^{2025}]$

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   The sequence of functions {$f_{n}$}$_{n}\ge1$ is uniformly bounded.

2. B

   The sequence of functions {$f_{n}$}$_{n}\ge1$ does not converge pointwise.

3. C

   The sequence of functions {$f_{n}$}$_{n}\ge1$ converges uniformly to the constant function 0.

4. D

   The set {$f_{n} : n\in\mathbb{N}$} is compact in C[0,1], where C[0,1] denotes the space of all real valued continuous functions on [0,1] equipped with the supremum norm.

Single choice · Sequences and Series of Functions

1. A

   $\sum(n\ge1) f_{n}$ and $\sum(n\ge1) g_{n}$ are uniformly convergent on the interval [−1, 1].

2. B

   $\sum(n\ge1) f_{n}$ is uniformly convergent on the interval [−1, 1], but $\sum(n\ge1) g_{n}$ is not.

3. C

   $\sum(n\ge1) g_{n}$ is uniformly convergent on the interval [−1, 1], but $\sum(n\ge1) f_{n}$ is not.

4. D

   Neither $\sum(n\ge1) f_{n}$ nor $\sum(n\ge1) g_{n}$ is uniformly convergent on the interval [−1, 1].

Single choice · Sequences and Series of Functions

1. A

   The series is neither absolutely convergent nor uniformly convergent.

2. B

   The series is both absolutely convergent and uniformly convergent.

3. C

   The series is absolutely convergent but not uniformly convergent.

4. D

   The series is uniformly convergent but not absolutely convergent.

Single choice · Sequences and Series of Functions

1. A

   The radius of convergence of the series is 1/e

2. B

   The series converges at x = 5

3. C

   The series converges at x = 3

4. D

   The series converges for all x with |x| < 1/2

Single choice · Sequences and Series of Functions

1. A

   There exists $x \in \mathbb{R}$ such that f(x) is not defined

2. B

   f(x) = 0 for all $x \in \mathbb{R}$

3. C

   f(x) = x for all $x \in \mathbb{R}$

4. D

   f(x) = |x| for all $x \in \mathbb{R}$

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   $(g_{n})_{n}\ge_{1}$ converges uniformly on any compact subset of $\mathbb{R}$ but not on $\mathbb{R}$.

2. B

   $(g_{n})_{n}\ge_{1}$ converges uniformly on $\mathbb{R}$.

3. C

   There exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on $\mathbb{R}$.

4. D

   For every compact subset $K \subseteq \mathbb{R}$, there exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on K.

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   $(x^{n}g(x))_{n}\ge_{1}$ converges uniformly on [0, 1].

2. B

   $(x^{n}g(x))_{n}\ge_{1}$ does not converge uniformly on [0, 1].

3. C

   $(x^{n}h(x))_{n}\ge_{1}$ converges uniformly on [0, 1].

4. D

   $(x^{n}h(x))_{n}\ge_{1}$ does not converge uniformly on [0, 1].

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   $(f_{n})_{n}\ge_{1}$ does not converge pointwise on [0, 1].

2. B

   $(f_{n})_{n}\ge_{1}$ converges pointwise to a continuous function on [0, 1].

3. C

   $(f_{n})_{n}\ge_{1}$ converges pointwise to a discontinuous function on [0, 1].

4. D

   $(f_{n})_{n}\ge_{1}$ does not converge uniformly on [0, 1].

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   f(x) is not uniformly continuous on [0, 1)

2. B

   The sequence $(p_{n}(x))$ converges to f(x) pointwise on [0, 1)

3. C

   The sequence $(p_{n}(x))$ converges to f(x) uniformly on [0, 1)

4. D

   The sequence $(p_{n}(x))$ converges to f(x) uniformly on [0, c] for every 0 < c < 1

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   $(f_{n})$ converges pointwise on [0, 1].

2. B

   $(f_{n})$ converges uniformly on compact subsets of [0, 1) but not on [0, 1).

3. C

   $(f_{n})$ converges uniformly on [0, 1) but not on [0, 1].

4. D

   $(f_{n})$ converges uniformly on [0, 1].