CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   $(x^{n}g(x))_{n}\ge_{1}$ converges uniformly on [0, 1].

2. B

   $(x^{n}g(x))_{n}\ge_{1}$ does not converge uniformly on [0, 1].

3. C

   $(x^{n}h(x))_{n}\ge_{1}$ converges uniformly on [0, 1].

4. D

   $(x^{n}h(x))_{n}\ge_{1}$ does not converge uniformly on [0, 1].

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

1. A

   The set {$I, A, \dots, A^{n}$} is linearly independent but the set {$I, A, \dots, A^(n^{2})$} is not linearly independent.

2. B

   If A is a singular matrix, then the set {I, A, …, A^k} spans a (k + 1)-dimensional subspace of V for all k ≤ rank(A).

3. C

   The sets {$I, A, \dots, A^{n}$} and {$I, A, \dots, A^(n^{2})$} both span the same subspace of V.

4. D

   The set {$I, A, \dots, A^{n}$} is linearly dependent.

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

1. A

   $\varphi$ is one-to-one but not onto.

2. B

   $\varphi$ is onto but not one-to-one.

3. C

   There exists $f \in \mathbb{R}[x]$ such that deg f ≤ n and $\ker \varphi =$ {fg | $g \in \mathbb{R}[x]$}.

4. D

   deg h ≥ n for every nonzero $h \in \ker \varphi$.

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

1. A

   The characteristic polynomial of T can be $x^{2} + x + 1$.

2. B

   The characteristic polynomial of T can be x(x − 1).

3. C

   The minimal polynomial of T can be $x^{2}$.

4. D

   The characteristic polynomial of T can be $x(x - e^(2\pi{}i/3))$.

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

1. A

   There exists a nonzero $v \in \mathbb{R}^{4}$ such that $q_{1}(T)v = 0$.

2. B

   There exists a nonzero $v \in \mathbb{R}^{4}$ such that $q_{1}(T)v = 0$ and $q_{2}(T)v = 0$.

3. C

   For all nonzero $v \in \mathbb{R}^{4}, v$ and Tv are linearly independent.

4. D

   If $q_{1}(T)v = 0$ for some nonzero $v \in \mathbb{R}^{4}$, then {v, Tv, $T^{2}v, T^{3}v$} is a basis of $\mathbb{R}^{4}$.

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

1. A

   The set {$v_{1}, v_{2}$} is linearly independent.

2. B

   If $c_{1} \ne c_{2}$, then ⟨$v_{1}, v_{2}$⟩ = 0.

3. C

   If $c_{1} = c_{2}$, then $c_{3} = c_{2}$.

4. D

   If $c_{1} \ne c_{2}$, then $c_{3} = c_{2}$.

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

1. A

   $f(v_{2}, v_{1}) = -s$.

2. B

   If r = s = t, then either f(v, v) ≥ 0 for all v ∈ V or f(v, v) ≤ 0 for all v ∈ V.

3. C

   If f is positive definite, then r ≠ s, s ≠ t and r ≠ t.

4. D

   If f is positive definite, then $f(v_{1}, v_{2}) = 0$.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   f is a constant function.

2. B

   If $f(0) \in \mathbb{R}$, then f(0) + f(1) = 1.

3. C

   $f(\mathbb{C})$ is connected.

4. D

   If $f(2) \in \mathbb{R}$, then f(i) + f(1) = 2.

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

1. A

   The series $\sum(n\ge1) f_{n}$ converges only at z = 0.

2. B

   The series $\sum(n\ge1) f_{n}$ converges pointwise only on a countable set E ⊆ 𝔻 but not on 𝔻 \ E.

3. C

   The series $\sum(n\ge1) f_{n}$ converges pointwise at all points of 𝔻 but not uniformly on some compact subsets of 𝔻.

4. D

   The series $\sum(n\ge1) f_{n}$ converges uniformly on all compact subsets of 𝔻 to a holomorphic function on 𝔻.

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

1. A

   If p divides |G|, then E(p) is a subgroup of G.

2. B

   For all p, E(p) is a subgroup of G.

3. C

   If E(p) is a subgroup of G, then p divides |G|.

4. D

   If G is abelian, then E(p) is a subgroup of G.

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

1. A

   $\mathbb{Q}[x]/(x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$

2. B

   $\mathbb{Q}[x]/(x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$

3. C

   $\mathbb{Z}[x]/(x - 101)$

4. D

   $\mathbb{Q}[x]/(x^{43} - 41x + 41)$

Multi-select · full marks only for the exact set · Metric Spaces

1. A

   d_X is a metric on X which induces the topology $\tau$.

2. B

   The topological space $(X, \tau)$ is connected.

3. C

   $(P_{n})_{n}\ge_{1}$ converges to $P_{0}$ in the topological space $(X, \tau)$.

4. D

   The topological space $(X, \tau)$ is compact.

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

1. A

   For $\alpha = 1$, all the solutions of the ODE tend to zero as $x \to \infty$

2. B

   For $\alpha = 0$, there exists a solution of the ODE which tends to 1 as $x \to \infty$

3. C

   For $\alpha = -1$, there exists an unbounded solution of the ODE on $\mathbb{R}$

4. D

   For $\alpha = 2$, there exists an unbounded solution of the ODE on $\mathbb{R}$

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

1. A

   $x(\pi) = -e^\pi, y(\pi) = 1/2$

2. B

   $x(-\pi) = e^(-\pi), y(-\pi) = 1/2$

3. C

   x has infinitely many zeros in $\mathbb{R}$

4. D

   y has infinitely many zeros in $\mathbb{R}$

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 · Calculus of Variations

1. A

   J[y] ≤ J[0] for every $y \in B_{0}(0, \varepsilon)$

2. B

   J[y] ≤ J[0] for every $y \in B_{1}(0, \varepsilon)$

3. C

   J[y] ≥ J[0] for every $y \in B_{0}(0, \varepsilon)$

4. D

   J[y] ≥ J[0] for every $y \in B_{1}(0, \varepsilon)$

Multi-select · full marks only for the exact set · Calculus of Variations

1. A

   There are infinitely many extremals

2. B

   There are more than one but only finitely many extremals

3. C

   There is no extremal

4. D

   There is a unique extremal

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

1. A

   The BVP has a non-zero solution if $\lambda = 2$

2. B

   The BVP has a non-zero solution if $\lambda = 3$

3. C

   The BVP has a non-zero solution y = y(x) satisfying $y(k\pi/2) = 0$ for every integer k if $\lambda = 6$

4. D

   The BVP has a non-zero solution y = y(x) satisfying $y(k\pi/3) = 0$ for every integer k if $\lambda = 6$

Multi-select · full marks only for the exact set · Classical Mechanics

1. A

   |$\theta$̇| $= \sqrt{2/(g\alpha)}$

2. B

   |$\theta$̇| $= \sqrt{1/(g\alpha)}$

3. C

   |$\theta$̇| $= \sqrt{2g\alpha}$

4. D

   |$\theta$̇| $= \sqrt{g\alpha}$

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

1. A

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

2. B

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

3. C

   $X_{n} - X$ converges to Z in distribution as $n \to \infty$, where Z follows the distribution N(0, 2).

4. D

   X/|$X_{n}$| converges to M in distribution as $n \to \infty$, where M follows standard Cauchy distribution.

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.