CSIR NET Maths previous year question papers

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 · Limit Theorems and Markov Chains

1. A

   E(|$X_{1}$|$) = \infty$

2. B

   $(1/n)\sum(i=1..n)X_{i} \to 0$ in probability as $n \to \infty$

3. C

   E(1/|$X_{1}$|$) < \infty$

4. D

   $E(X_{1}^{2}) = \infty$

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

1. A

   The state space can be partitioned into exactly two equivalence classes.

2. B

   States 0 and 2 are recurrent.

3. C

   States 1 and 3 are recurrent.

4. D

   State 4 is transient.

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

1. A

   P(XY = 0) = (4 + 5e⁻$^{1})/9$

2. B

   $E[(3X - Y)^{2}] = 6$

3. C

   Var(X + Y) = 2

4. D

   Cov(2X + Y, X − 2Y) = −10/9

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

1. A

   For M = 0, the maximum likelihood estimate of $\theta$ is equal to 0.

2. B

   For M = 10, the maximum likelihood estimate of $\theta$ does not exist.

3. C

   The maximum likelihood estimator of $\theta$ exists for M ∈ {1, 2, …, 9} and is equal to ln(10/M − 1).

4. D

   The method of moments estimator of $\theta$ exists and is equal to M/10.

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

1. A

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

2. B

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

3. C

   $T_{n} - 1$ is a consistent estimator of $\theta$.

4. D

   $\lim(n\to\infty) E_\theta(T_{n}) = \theta + 1, \forall\theta \in \mathbb{R}$

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

1. A

   $\theta$̂ ∈ (0.0, 0.2)

2. B

   $\theta$̂ ∈ (0.1, 0.3)

3. C

   $\theta$̂ ∈ (0.2, 0.4)

4. D

   $\theta$̂ ∈ (0.3, 0.5)

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

1. A

   The size of the test is 0.01.

2. B

   The power of the test is 0.5.

3. C

   The test is unbiased at level $\alpha = 0.05$.

4. D

   If all of these three components survive for less than five hours, then the p-value of the test is 0.001.

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

1. A

   Every linear function of $\beta_{i}, 1 \le i \le n$, is not estimable.

2. B

   Every $\beta_{i}, 1 \le i \le n$, has infinitely many linear unbiased estimators, but a unique best linear unbiased estimator.

3. C

   Each $\beta_{i}, 1 \le i \le n$, has only one linear unbiased estimator.

4. D

   $Y_{2} - Y_{1}$ is the best linear unbiased estimator of $\beta_{2}$.

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

1. A

   MSE$(\sigma$̂$^{2}($MLE)) < MSE$(\sigma$̂$^{2})$ if r = 3, n = 12

2. B

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

3. C

   $Var(\sigma$̂$^{2}($MLE$)) > Var(\sigma$̂$^{2})$ if 1 ≤ r ≤ 6, n ≥ 12

4. D

   MSE$(\sigma$̂$^{2}($MLE)) > MSE$(\sigma$̂$^{2})$ if r = 6, n = 12

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

1. A

   The distribution of the second principal component is normal with mean 0 and variance 2.

2. B

   The variance of the first principal component is 4.

3. C

   The first principal component explains more than 70% of the total variance.

4. D

   The correlation coefficient between the first and the second principal components is 0.

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

1. A

   F is Riemann integrable.

2. B

   If F(x) = 0 for all x ∈ [0, 1], then f(x) = 0 for all x ∈ [0, 1].

3. C

   F is uniformly continuous on [0, 1].

4. D

   F is differentiable on (0, 1).

Multi-select · full marks only for the exact set · Lebesgue Measure and Integration

1. A

   $f \in L^{1}(\mathbb{R})$

2. B

   $f \in L^{2}(\mathbb{R})$

3. C

   $f \in L^{4}(\mathbb{R})$

4. D

   $f \in L^\infty(\mathbb{R})$

Multi-select · full marks only for the exact set · Lebesgue Measure and Integration

1. A

   (‖$f_{n}$‖$_{1})$ is a convergent sequence.

2. B

   (‖$f_{n}$‖$_{2})$ is a convergent sequence.

3. C

   Both (‖$f_{n}$‖$_{1})$ and (‖$f_{n}$‖$_{2})$ are convergent sequences.

4. D

   Neither (‖$f_{n}$‖$_{1})$ nor (‖$f_{n}$‖$_{2})$ is a convergent sequence.

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

1. A

   f is differentiable at 0 when $\alpha > 1$.

2. B

   f is differentiable at 0 when $\alpha = 1/2$.

3. C

   f is continuous at 0 when $\alpha = 0$.

4. D

   f is continuous at 0 when $\alpha > 0$.

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

1. A

   The function h_(a,b) is differentiable on $\mathbb{R}$ for each $(a, b) \in \mathbb{R}^{2}$.

2. B

   There exists $(a, b) \in \mathbb{R}^{2}$ such that h_(a,b) is not differentiable at t = 0.

3. C

   The function f is differentiable at the point (0, 0).

4. D

   If the function f is differentiable at (0, 0) then g is identically zero.

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

1. A

   There exists a 5 × 5 real matrix whose minimal polynomial is $(x^{2} + x + 1)x^{3}$.

2. B

   There exists a 5 × 5 real matrix whose minimal polynomial is $(x^{2} + x + 1)^{2}$.

3. C

   There exists a 5 × 5 complex matrix whose minimal polynomial is $(x^{2} + x + 1)^{2}x$.

4. D

   There exists a 5 × 5 complex matrix whose minimal polynomial is $(x^{2} + x + 1)^{2}$.

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

1. A

   n = 3 and the characteristic polynomials of A and B are the same.

2. B

   n = 4 and A has two distinct eigenvalues.

3. C

   n = 5 and A has some eigenvalue for which the dimensions of the eigenspaces of A and B are the same.

4. D

   n = 6 and A has only one eigenvalue and for this eigenvalue the dimensions of the eigenspaces of A and B are the same.

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

1. A

   f has a pole at z = 0.

2. B

   There is an entire function g such that f = g on $\mathbb{C}$ \ {0}.

3. C

   f is a polynomial of degree at most 2.

4. D

   f has an essential singularity at z = 0.

Multi-select · full marks only for the exact set · Singularities and Residues

1. A

   f has a removable singularity at z = 0.

2. B

   f has a simple pole at z = 0.

3. C

   $\int_C (p(z)/z)$dz = 0, where C is any counterclockwise oriented smooth closed curve in $\mathbb{C}$.

4. D

   $\int_\gamma f(z)$dz $= 2\pi$ia$_{1}$, where $\gamma =$ {$z \in \mathbb{C}$ : |z| = 2025} is a counterclockwise oriented circle.

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

1. A

   If an abelian group G has property (P), every subgroup of G has property (P).

2. B

   If an abelian group G has property (P), then every element of G has finite order.

3. C

   The group $\mathbb{Z}$ has property (P).

4. D

   If an abelian group G has property (P), then no element has order $p^{2}$, where p is a prime number.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   The ring of polynomials in 3 variables over $\mathbb{R}$.

2. B

   The ring of complex analytic functions on the open unit disc in $\mathbb{C}$.

3. C

   The ring of 2 × 2 real matrices.

4. D

   The ring of continuous functions from [0, 1] to $\mathbb{R}$.

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

1. A

   Any group with p(p + 2) elements, where p and p + 2 are both prime numbers, has property (J).

2. B

   Any finite abelian group has property (J).

3. C

   The symmetric group $S_{3}$ has property (J).

4. D

   Any group with 77 elements has property (J).

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   If K is the splitting field of a non-constant polynomial over $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

2. B

   If K is a normal extension of $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

3. C

   If K is the set of all the roots of the polynomial $x^{121} - x$ in an algebraic closure of 𝔽$_{11}$, then K is Galois over 𝔽$_{11}$, where 𝔽$_{11}$ is the field with 11 elements.

4. D

   If $K = \mathbb{Q}(2^(1/13), \zeta_{13})$, where $\zeta_{13}$ is a primitive 13th root of unity in $\mathbb{C}$, then K is Galois over $\mathbb{Q}$.

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

1. A

   If X and Y are compact, Y is Hausdorff and f is onto, then X is also Hausdorff.

2. B

   If X is an infinite compact set and f is a homeomorphism from X to f(X) (where f(X) is given the subspace topology), then Y is compact.

3. C

   If X is Hausdorff and f is onto, then Y is Hausdorff.

4. D

   If f is a homeomorphism, then X is second-countable if and only if Y is second-countable.