CSIR NET Maths previous year question papers

Single choice · The Real Line

1. A

   There exist non-negative integers p, q such that both (A) and (B) are true.

2. B

   Both (A) and (B) are false if and only if p = q.

3. C

   For all non-negative integers p and q, (A) or (B) is true.

4. D

   There exists p ≠ q such that both (A) and (B) are false.

Single choice · The Real Line

1. A

   A ⊊ C and B ⊊ C

2. B

   A ⊊ C ⊊ B

3. C

   A ⊊ B ⊊ C

4. D

   A ⊊ B = C

Single choice · The Real Line

1. A

   If A ⊊ B and B is finite, then there is a one-to-one map from $S_{2}$ to $S_{1}$.

2. B

   If $B = \mathbb{N}$, then there exists a one-to-one map from $S_{2}$ to B.

3. C

   If $B = \mathbb{N}$ and A is finite, then there exists a one-to-one map from B to $S_{1}$.

4. D

   If A is finite, then $S_{2}$ is finite for any B.

Single choice · The Real Line

1. A

   If $\ell$ is positive, then $\ell = \sqrt{3}/2$.

2. B

   If $\ell$ is positive, then $\ell = 1/\sqrt{2}$.

3. C

   If $\ell$ is negative, then $\ell = -\sqrt{3}/2$.

4. D

   If $\ell$ is negative, then $\ell = -1/2$.

Single choice · The Real Line

1. A

   $\lim (x\to\infty) f(x) = \infty$

2. B

   $\lim (x\to\infty) f(x) = 0$

3. C

   $\lim (x\to\infty) f(x) = 1$

4. D

   $\lim (x\to\infty) f(x) = 1/2$

Single choice · The Real Line

1. A

   $\limsup x_{n} = 1$

2. B

   $\limsup x_{n} = \sqrt{2}$

3. C

   $\limsup x_{n} = 2$

4. D

   $\limsup x_{n} = 1 + \sqrt{2}$

Single choice · The Real Line

1. A

   The series converges for all $\alpha \in (0,1)$ and for all $\beta \in (1,\infty)$.

2. B

   The series converges for $\alpha = 1$ and $\beta = 1$.

3. C

   The series converges for $\alpha = 1$ and for all $\beta \in (1,\infty)$.

4. D

   The series converges for all $\alpha \in (0,\infty)$ and for all $\beta \in (0,\infty)$.

Single choice · The Real Line

1. A

   There is a bijective map from $\mathbb{N}$ to S.

2. B

   There is a surjective map from S onto (0,1).

3. C

   There is a bijective map from $\mathbb{N}$ to $(0,1)\S$.

4. D

   S is a countable set and $(0,1)\S$ is uncountable.

Single choice · The Real Line

1. A

   The set of all finite subsets of $\mathbb{Z}$ is uncountable.

2. B

   Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous and one-one function. If $S \subseteq \mathbb{R}$ is a countably infinite set, then f(S) is a countably infinite set.

3. C

   Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. Then there exists an uncountable subset $T \subseteq \mathbb{R}$ such that $f(T) \subseteq \mathbb{R}$ is uncountable.

4. D

   Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. If R ⊆ image(f) is a countable set, then f⁻$^{1}(R)$ is countable.

Multi-select · full marks only for the exact set · The Real Line

1. A

   $limsup_{n\to\infty}$ $t_{n} \le$ $limsup_{n\to\infty}$ $s_{n}$

2. B

   $limsup_{n\to\infty}$ $t_{n} \le e$

3. C

   $liminf_{n\to\infty}$ $s_{n} \ge e^{2}$

4. D

   $liminf_{n\to\infty}$ $t_{n} \ge e$

Multi-select · full marks only for the exact set · The Real Line

1. A

   There exists a positive integer N such that for all n≥N, $b_{n,l}$ $= a_{l}$ for all 1≤l≤2023.

2. B

   There exists a positive integer N such that for all n≥N, $b_{n,2025}$ $= a_{2025} - 1$.

3. C

   $\beta_{n}$ is rational for infinitely many n.

4. D

   There exists a positive integer N such that for all n≥N, $b_{n,2024}$ $= a_{2024} - 1$.

Multi-select · full marks only for the exact set · The Real Line

1. A

   A is bounded below but not bounded above.

2. B

   A is bounded above but not bounded below.

3. C

   inf A = −1

4. D

   sup A = 1

Single choice · The Real Line

1. A

   If A is finite, then there exists a bijection between S and U.

2. B

   If A is finite and B = [0, 1], then there is no bijection between S and T.

3. C

   There is no bijection between S and U for any choice of A.

4. D

   If A ≠ B, then there is no bijection between T and U.

Single choice · The Real Line

1. A

   $\lim \inf(n\to\infty) x_{n} \ge 2025$

2. B

   $\lim \sup(n\to\infty) x_{n} \le 2025$

3. C

   $\lim \inf(n\to\infty) x_{n} \le 2025$

4. D

   $\lim \inf(k\to\infty) x_(n_{k}) > 2025$

Single choice · The Real Line

1. A

   $(a_{n})_{n}\ge_{1}$ converges but $(b_{n})_{n}\ge_{1}$ does not converge.

2. B

   $(b_{n})_{n}\ge_{1}$ converges but $(a_{n})_{n}\ge_{1}$ does not converge.

3. C

   Both $(a_{n})_{n}\ge_{1}$ and $(b_{n})_{n}\ge_{1}$ converge.

4. D

   Neither $(a_{n})_{n}\ge_{1}$ nor $(b_{n})_{n}\ge_{1}$ converges.

Single choice · The Real Line

1. A

   A \ B

2. B

   (A \ B) ∪ C

3. C

   A \ (B ∪ C)

4. D

   (A \ B) ∪ (A ∩ C)

Single choice · The Real Line

1. A

   For every $x \in \mathbb{R}$ there exists some n such that $a_{n} > x$

2. B

   For every $x \in \mathbb{R}$ there exists some n such that $a_{n} < x$

3. C

   For every $x \in \mathbb{R}$ there exists some n such that $b_{n} > x$

4. D

   For every $x \in \mathbb{R}$ there exists some n such that $b_{n} < x$

Single choice · The Real Line

1. A

   $\sup A = 2 + 2\sqrt{3}$

2. B

   $\sup A = 3 + 2\sqrt{2}$

3. C

   $\inf A = 2 + 2\sqrt{3}$

4. D

   $\inf A = 3 + 2\sqrt{2}$

Single choice · The Real Line

1. A

   S is empty.

2. B

   There is a bijection between S and $\mathbb{N}$

3. C

   There is a bijection between S and $\mathbb{R}$

4. D

   There is a bijection between S and a non-empty finite set

Single choice · The Real Line

1. A

   There exists a non-empty finite set in C

2. B

   There exists a countably infinite set in C

3. C

   There exists an uncountable set in C

4. D

   C is empty

Single choice · The Real Line

1. A

   if $liminf_{n}\to\infty a_{n} = limsup_{n}\to\infty a_{n}$, then $(a_{n})$ is convergent

2. B

   if inf{$a_{n}$ | n ≥ 1} $= limsup_{n}\to\infty a_{n}$, then $(a_{n})$ is convergent

3. C

   if sup{$a_{n}$ | n ≥ 1} $= liminf_{n}\to\infty a_{n}$, then $(a_{n})$ is constant

4. D

   if sup{$a_{n}$ | n ≥ 1} = inf{$a_{n}$ | n ≥ 1}, then $(a_{n})$ is constant

Multi-select · full marks only for the exact set · The Real Line

1. A

   {$x \in \mathbb{R}$ | [x] = 1}

2. B

   {$x \in \mathbb{R}$ | x − [x] = 1/2}

3. C

   {$x \in \mathbb{R}$ | $x \ge 0, \sqrt{x} \in \mathbb{Q}$}

4. D

   {$x \in \mathbb{R}$ | $x^{2} \in \mathbb{Q}$}

Multi-select · full marks only for the exact set · The Real Line

1. A

   sup A is finite.

2. B

   $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) < \sup A$

3. C

   $\lim \inf(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 2$

4. D

   $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 3$

Multi-select · full marks only for the exact set · The Real Line

1. A

   Let $x, y \in \mathbb{R}$ with x < y. Then there exists $r \in \mathbb{Q}$ such that x < (2^2024/e)r < y.

2. B

   Let $(a_{n})_{n}\ge_{2}$ be a sequence of positive real numbers. If there exists a positive real number L such that $\lim \sup(n\to\infty) a_{n}/\log n = L$, then $\lim \sup(n\to\infty) a_{n} < \infty$.

3. C

   The set of all finite subsets of $\mathbb{Q}$ is countably infinite.

4. D

   The set of continuous functions from $\mathbb{R}$ to the set {0, 1} is infinite.

Multi-select · full marks only for the exact set · The Real Line

1. A

   S is an unbounded subset of $\mathbb{R}$.

2. B

   S is a bounded subset of $\mathbb{R}$ and sup(S) = 1.

3. C

   There exists u ∈ X such that ‖u‖ = 1 and T(u) = 1.

4. D

   S is a closed subset of $\mathbb{R}$.