CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · The Real Line

1. A

   {$x \in \mathbb{R}$ | [x] = 1}

2. B

   {$x \in \mathbb{R}$ | x − [x] = 1/2}

3. C

   {$x \in \mathbb{R}$ | $x \ge 0, \sqrt{x} \in \mathbb{Q}$}

4. D

   {$x \in \mathbb{R}$ | $x^{2} \in \mathbb{Q}$}

Multi-select · full marks only for the exact set · The Real Line

1. A

   sup A is finite.

2. B

   $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) < \sup A$

3. C

   $\lim \inf(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 2$

4. D

   $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 3$

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   S is empty.

2. B

   S is non-empty.

3. C

   f must be uniformly continuous.

4. D

   f need not be uniformly continuous.

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

1. A

   $(g_{n})_{n}\ge_{1}$ converges uniformly on any compact subset of $\mathbb{R}$ but not on $\mathbb{R}$.

2. B

   $(g_{n})_{n}\ge_{1}$ converges uniformly on $\mathbb{R}$.

3. C

   There exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on $\mathbb{R}$.

4. D

   For every compact subset $K \subseteq \mathbb{R}$, there exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on K.

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 · 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 · 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 · Sequences and Series of Functions

1. A

   $(f_{n})_{n}\ge_{1}$ does not converge pointwise on [0, 1].

2. B

   $(f_{n})_{n}\ge_{1}$ converges pointwise to a continuous function on [0, 1].

3. C

   $(f_{n})_{n}\ge_{1}$ converges pointwise to a discontinuous function on [0, 1].

4. D

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

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

1. A

   f(0) = 0

2. B

   f′(0) exists and f′(0) = 0.

3. C

   f″(0) exists and f″(0) = g(0).

4. D

   f″(0) exists but f″(0) ≠ g(0).

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

1. A

   f is Riemann integrable on [0, 1].

2. B

   The set of discontinuities of f cannot contain a non-empty open set.

3. C

   f is Lebesgue measurable function.

4. D

   f is Borel measurable function.

Multi-select · full marks only for the exact set · Vector Spaces and Linear Maps

1. A

   If f ∈ B, then dim ker f = 3

2. B

   dim B = 3

3. C

   dim A = 4

4. D

   If f ∈ A, then the image of f is a one-dimensional $\mathbb{Q}-$vector space.

Multi-select · full marks only for the exact set · Vector Spaces and Linear Maps

1. A

   DM ≠ MD.

2. B

   D + M is invertible.

3. C

   DM is invertible.

4. D

   rank(DM) = rank(MD).

Multi-select · full marks only for the exact set · Determinants and Matrix Tricks

1. A

   If A and B are upper triangular, then M is diagonalizable over $\mathbb{R}$.

2. B

   If A and B are diagonalizable over $\mathbb{R}$, then M is diagonalizable over $\mathbb{R}$.

3. C

   If A and B are diagonalizable over $\mathbb{R}$, then there exists $\lambda \in \mathbb{R}$ such that $M = \lambda{}I_{2}$.

4. D

   There exists $\lambda \in \mathbb{R}$ such that $M^{2} = \lambda{}I_{2}$.