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].

Single choice · Continuity and Differentiation

1. A

   f∘g is uniformly continuous, but g∘f is not uniformly continuous.

2. B

   g∘f is uniformly continuous, but f∘g is not uniformly continuous.

3. C

   Neither f∘g nor g∘f is uniformly continuous.

4. D

   Both f∘g and g∘f are uniformly continuous.

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.

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 · Calculus of Variations

1. A

   J[y] ≤ J[0] for every $y \in B_{0}(0, \varepsilon)$

2. B

   J[y] ≤ J[0] for every $y \in B_{1}(0, \varepsilon)$

3. C

   J[y] ≥ J[0] for every $y \in B_{0}(0, \varepsilon)$

4. D

   J[y] ≥ J[0] for every $y \in B_{1}(0, \varepsilon)$

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 · Continuity and Differentiation

1. A

   F is bounded.

2. B

   F is continuous on $\mathbb{R}$.

3. C

   F is uniformly continuous on $\mathbb{R}$.

4. D

   F is not uniformly continuous on $\mathbb{R}$.

Multi-select · full marks only for the exact set · Calculus of Variations

1. A

   $\varphi(x) = 2x + 1$ for every x ∈ [−1, 1]

2. B

   There exists $\varepsilon > 0$ such that $J[y] \ge J[\varphi]$ for every $y \in B_{0}(\varphi, \varepsilon)$

3. C

   There exists $\varepsilon > 0$ such that $J[y] \ge J[\varphi]$ for every $y \in B_{1}(\varphi, \varepsilon)$

4. D

   There exists $\varepsilon > 0$ such that $J[y] \le J[\varphi]$ for every $y \in B_{1}(\varphi, \varepsilon)$

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

Single choice · Continuity and Differentiation

1. A

   f(x) = sin(1/x)

2. B

   f(x) = $e^{-1/x^{2}}$

3. C

   f(x) = eˣ cos(1/x)

4. D

   $f(x) = \cos x \cdot \cos(\pi/x)$

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

1. A

   f is continuous on $[0, \infty)$.

2. B

   f is uniformly continuous on $[0, \infty)$.

3. C

   g is continuous on $[0, \infty)$.

4. D

   g is uniformly continuous on $[0, \infty)$.