CSIR NET Maths previous year question papers

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

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

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

1. A

   f′(x) ≤ 0 for all x > 0.

2. B

   $lim_{x\to\infty}$ f′(x) = 0.

3. C

   $lim_{x\to\infty}$ x f′(x) need not exist.

4. D

   $lim_{x\to\infty}$ x f′(x) = 0.

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

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   lim(x,y)→(0,0) f(x,y) does not exist.

2. B

   The partial derivatives of f at (0,0) do not exist.

3. C

   lim(x→0) f(x,0) = lim(y→0) f(0,y).

4. D

   f is differentiable at (0,0).

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

   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 · Integration

1. A

   A = 0

2. B

   A = 1

3. C

   A = sin(1)/2

4. D

   A = sin(1/4)

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

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

2. B

   $\lim \sup(n\to\infty) a_{n} = \lim(n\to\infty) b_{n}$

3. C

   $\lim \inf(n\to\infty) a_{n} = \lim(n\to\infty) c_{n}$

4. D

   $\lim(n\to\infty) b_{n} = \lim(n\to\infty) c_{n}$

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

1. A

   $\lim(n\to\infty)$ |$X_{i}(n)$|/n exists for all 1 ≤ i ≤ 8.

2. B

   $\lim \inf(n\to\infty)$ |$X_{i}(n)$|/n ≥ 0 for all 1 ≤ i ≤ 8.

3. C

   $\lim \sup(n\to\infty)$ |$X_{i}(n)$|/n ≥ 1/8 for some 1 ≤ i ≤ 8.

4. D

   $\lim \sup(n\to\infty)$ |$X_{i}(n)$|/n < 1/8 for all 1 ≤ i ≤ 8.

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

1. A

   If $lim_{n}\to\infty b_{n}$ exists in $\mathbb{R}$, then {$a_{n} : n \ge 1$} is bounded

2. B

   If $lim_{n}\to\infty b_{n} = 1$, then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$

3. C

   If $lim_{n}\to\infty b_{n} = 1/2$, then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$

4. D

   If $lim_{n}\to\infty b_{n} = 0$, then $lim_{n}\to\infty a_{n} = 0$

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

1. A

   $A_{n} \to 0$ and $B_{n} \to 0$ as $n \to \infty$

2. B

   If $\sum_{n}$₌$_{1}^\infty a_{n}$ is absolutely convergent, then both $\sum_{n}$₌$_{1}^\infty A_{n}$ and $\sum_{n}$₌$_{1}^\infty B_{n}$ are absolutely convergent

3. C

   Both $\sum_{n}$₌$_{1}^\infty A_{n}$ and $\sum_{n}$₌$_{1}^\infty B_{n}$ are convergent

4. D

   If $\sum_{n}$₌$_{1}^\infty a_{n}$ is not absolutely convergent, then both $\sum_{n}$₌$_{1}^\infty A_{n}$ and $\sum_{n}$₌$_{1}^\infty B_{n}$ are divergent

Single choice · The Real Line

1. A

   $limsup_{n} \exp(\cos((n\pi + (-1)^{n}\cdot2e)/(2n))) > 1$.

2. B

   $lim_{n} \exp(log_{e}((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.

3. C

   $liminf_{n} \exp(\sin((n\pi + (-1)^{n}\cdot2e)/(2n))) < \pi$.

4. D

   $lim_{n} \exp(\tan((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.

Single choice · The Real Line

1. A

   $\limsup a_{n} = \sqrt{3}/2$.

2. B

   $\limsup a_{2n} = 1$.

3. C

   $\limsup a_{2n} = 1/2$.

4. D

   $\limsup a_{3n} = 0$.

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

1. A

   The subsequences {$x_{2n}$₊$_{1}$}, {$x_{2n}$} and {$x_{3n}$} are convergent and have the same limit.

2. B

   The subsequences {$x_{2n}$₊$_{1}$}, {$x_{2n}$} and {$x_{3n}$} are convergent.

3. C

   The subsequences {$x_{kn}$}$_{n}$ are convergent for every k ≥ 2.

4. D

   lim |$x_{n}$₊$_{1} - x_{n}$| = 0.

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

1. A

   If a = 0, then the sequence {$x_{n}$} converges to 1/2.

2. B

   If a = 0, then the sequence {$x_{n}$} converges to −1/2.

3. C

   The sequence {$x_{n}$} converges for every a ∈ (−1/2, 3/2), and it converges to 1/2.

4. D

   If a = 0, then the sequence {$x_{n}$} does not converge.

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

1. A

   There exists an integer n ≥ 1 such that $n^{2} \sin(1/n) \ge x$.

2. B

   There exists an integer n ≥ 1 such that n cos(1/n) ≥ x.

3. C

   There exists an integer n ≥ 1 such that n $e^{-n}$ ≥ x.

4. D

   There exists an integer n ≥ 2 such that n (log $n)^{-1}$ ≥ x.