CSIR NET Maths previous year question papers

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

1. A

   $X_{n}$ converges to X almost surely as $n \to \infty$ implies that $X_{n}$ converges to X in probability as $n \to \infty$.

2. B

   $X_{n}$ converges to X in probability as $n \to \infty$ implies that $X_{n}$ converges to X almost surely as $n \to \infty$.

3. C

   If $\sum(n=1..\infty)$ ℙ[|$X_{n} - X$| $> \delta] < \infty$ for all $\delta > 0$, then $X_{n}$ converges to X almost surely as $n \to \infty$.

4. D

   If $X_{n}$ converges to X in distribution as $n \to \infty$, and X is a constant with probability 1, then $X_{n}$ converges to X in probability as $n \to \infty$.

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 · Limit Theorems and Markov Chains

1. A

   P⁽$^{2}$⁾$_{00} = 3/4$

2. B

   P⁽$^{3}$⁾$_{10} = 39/64$

3. C

   The stationary probability that the chain is in state 2 is 2/7.

4. D

   State 1 is transient.

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

1. A

   e⁻$^{1}$

2. B

   e⁻$^{1}/2$

3. C

   e⁻$^{1}/4$

4. D

   e⁻$^{1}/12$

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

1. A

   $(X_{(1)} - (2/n)\ln 5, X_{(1)})$ is a 97% confidence interval for $\theta$.

2. B

   $(X_{(1)} - (2/n)\ln 5, X_{(1)})$ is a 96% confidence interval for $\theta$.

3. C

   $(X_{(1)} - (1/n)\ln 20, X_{(1)})$ is a 95% confidence interval for $\theta$.

4. D

   $(X_{(1)} - (1/n)\ln 20, X_{(1)})$ is a 96% confidence interval for $\theta$.

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

1. A

   $T_{n}$ is an unbiased estimator of $\theta$.

2. B

   $\lim(n\to\infty) E_\theta(T_{n}) = \theta$ for all $\theta \in \mathbb{R}$.

3. C

   $T_{n}$ is a consistent estimator of $\theta$.

4. D

   max{$X_{1}, X_{2}, \dots, X_{n}$} is a consistent estimator of $\theta$.

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

1. A

   $\delta_{n}$ converges in probability to $\theta$, as $n \to \infty$, for all $\theta \in \mathbb{R}$

2. B

   Z follows normal distribution.

3. C

   $E(Z^{4}) = 48$

4. D

   $E(Z^{2}) = 1$

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   If max{$X_{1}, X_{2}, X_{3}$} is 3.1, then $H_{0}$ is rejected.

2. B

   If max{$X_{1}, X_{2}, X_{3}$} is 1.3, then $H_{0}$ is rejected.

3. C

   If max{$X_{1}, X_{2}, X_{3}$} is 0.1, then $H_{0}$ is rejected.

4. D

   The power of the test at $\theta = 0.3$ is 1.

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   P(T⁺ = 3) = 1/4

2. B

   Var(T⁺) = 7/2

3. C

   P(T⁺ > 3) = 5/8

4. D

   P(T⁺ > 4) = 1/8

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   The sum of squares residuals is strictly positive with probability 1.

2. B

   For every $\beta_{i}, 1 \le i \le n$, there are infinitely many linear unbiased estimators.

3. C

   $Var(\beta$̂$_{1}) = \sigma^{2}$

4. D

   $Y_{3} - Y_{2}$ is the BLUE of $\beta_{3}$.

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   MSE$(\sigma$̂$^{2}($MLE)) < MSE$(\sigma$̂$^{2})$ if r = 2, n = 12

2. B

   $Var(\sigma$̂$^{2}($MLE$)) > Var(\sigma$̂$^{2})$ if 2 ≤ r ≤ n − 2, n ≥ 12

3. C

   $Var(\sigma$̂$^{2}($MLE$)) < Var(\sigma$̂$^{2})$ if 1 ≤ r ≤ n − 2, n ≥ 3

4. D

   MSE$(\sigma$̂$^{2}($MLE)) > MSE$(\sigma$̂$^{2})$ if r = 7, n = 12

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   The distribution of Z is symmetric about 0.

2. B

   $E(Z^{2}) = 2$

3. C

   $Var(Z^{2}) = 1$

4. D

   Z and U are independent random variables.

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 $\mathbb{R}$.

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 · Continuity and Differentiation

1. A

   lim(x→0) f(x) exists.

2. B

   f is continuous at 0.

3. C

   f is differentiable at 0.

4. D

   lim(x→0) f′(x) does not exist.

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 · Functions of Several Variables

1. A

   f is continuous at (0, 0).

2. B

   The partial derivatives $\partial{}f/\partial{}x$ and $\partial{}f/\partial{}y$ exist at (0, 0).

3. C

   f is differentiable at (0, 0).

4. D

   f is not differentiable at (0, 0).

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

1. A

   f is differentiable on $\mathbb{R}^{2}$ \ {(0, 0)}.

2. B

   All the directional derivatives of f exist at (0, 0).

3. C

   f is differentiable on $\mathbb{R}^{2}$.

4. D

   f is not continuous at (0, 0).

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   There exists a subspace W of $\mathbb{C}^{n}$ such that $\mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

2. B

   There exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

3. C

   Suppose that there exists a positive integer k such that Tᵏ is the identity map. Then there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

4. D

   Suppose that there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}. Then there exists a positive integer k such that Tᵏ is the identity map.

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   Both A and B are diagonalizable over $\mathbb{R}$.

2. B

   A is diagonalizable over $\mathbb{C}$ but not over $\mathbb{R}$.

3. C

   Neither A nor B is diagonalizable over $\mathbb{R}$, but both A and B are diagonalizable over $\mathbb{C}$.

4. D

   Neither A nor B is diagonalizable over $\mathbb{C}$.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   S is a basis of W.

2. B

   S is an orthonormal basis of W.

3. C

   There exist f, g ∈ S such that ⟨f, g⟩ = 0.

4. D

   S contains an orthonormal basis of W.

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   f is constant.

2. B

   f has removable singularities at −1 and 1.

3. C

   f is bounded.

4. D

   f has either poles or essential singularities at −1 and 1.

Multi-select · full marks only for the exact set · Zeros and Mappings

1. A

   S_R is an open subset of $\mathbb{C}$.

2. B

   S_R is a bounded subset of $\mathbb{C}$.

3. C

   |P(z)| = R for every z on the boundary of S_R.

4. D

   Every connected component of S_R contains a zero of P(z).