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

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

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

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

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