CSIR NET Maths previous year question papers

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 · Estimation

1. A

   $S_{n}(X_{1}:_{n})$⁻$^{1}$ is a consistent estimator of $\theta$

2. B

   $({\bar{X}}_{n} - X_{1}:_{n})(X_{1}:_{n})$⁻$^{1}$ is a consistent estimator of $\theta$

3. C

   $({\bar{X}}_{n} - X_{1}:_{n})({\bar{X}}_{n} - S_{n})$⁻$^{1}$ is a consistent estimator of $\theta$

4. D

   $S_{n}({\bar{X}}_{n})$⁻$^{1}$ is a consistent estimator of $\theta$

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).

Multi-select · full marks only for the exact set · Partial Differential Equations

1. A

   For every $x \in \mathbb{R}, \lim(t\to\infty) u(x, t) = 0$

2. B

   For every $x \in \mathbb{R}, \lim(t\to\infty) u(x, t) = \infty$

3. C

   For every $t \in (0, \infty), \lim(x\to\infty) u(x, t) = 0$

4. D

   For every $t \in (0, \infty), \lim(x\to-\infty) u(x, t) = \infty$

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   $G_{n}$ converges in r-th mean to $X_{1}$ as $n \to \infty$, for any r > 0

2. B

   $G_{n}$ converges in probability to $X_{1}$ as $n \to \infty$

3. C

   $G_{n}$ converges in distribution to $X_{2}$ as $n \to \infty$

4. D

   $G_{n}$ converges in r-th mean to $X_{2}$ as $n \to \infty$, for any r > 0

Multi-select · full marks only for the exact set · Probability

1. A

   $Y_{3}$ follows gamma distribution with $E(Y_{3}) = 3$.

2. B

   E(Y(2,n)) → 2 as $n \to \infty$

3. C

   $Y_{2}$ follows beta distribution with $E(Y_{2}) = 1/2$.

4. D

   Y(3,n) follows beta distribution with parameters 3 and n − 1.

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   $N_{n}/(3n)$ converges to e⁻$^{2}$ with probability one.

2. B

   $N_{n}$ converges to e⁻$^{2}$ in probability.

3. C

   $N_{n}/n$ converges to 3e⁻$^{2}$ in distribution.

4. D

   $(N_{n} - 3$ne⁻$^{2})/\sqrt{3n}$ converges in distribution to a normal random variable with mean zero and variance e⁻$^{2}(1 - 3e$⁻$^{2})$.

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 · Probability

1. A

   $Y_{2}$ follows gamma distribution with $E(Y_{2}) = 2$.

2. B

   E(Y(3,n)) → 3 as $n \to \infty$

3. C

   $Y_{3}$ follows beta distribution with $E(Y_{3}) = 1/3$.

4. D

   Y(2,n) follows beta distribution with parameters 2 and n − 1.

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

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   $\lim(n\to\infty) P((\sqrt{n}\cdot\sum_{i}$₌$_{1}^{n}X_{i})/(\sum_{i}$₌$_{1}^{n}X_{i}^{2}) \le 0) = 1/2$

2. B

   $(\sum_{i}$₌$_{1}^{n}X_{i})/(\sum_{i}$₌$_{1}^{n}X_{i}^{2})$ converges in probability to 0 as $n \to \infty$

3. C

   $(1/n)\sum_{i}$₌$_{1}^{n}X_{i}^{2}$ converges in probability to 1 as $n \to \infty$

4. D

   $\lim(n\to\infty) P((\sum_{i}$₌$_{1}^{n}X_{i})/\sqrt{n} \le 0) = 1/2$

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.