CSIR NET Maths previous year question papers

Single choice · Vector Spaces and Linear Maps

1. A

   The dimension of $X/\ker \varphi$ is 3.

2. B

   $\ker \varphi$ is finite dimensional.

3. C

   The dimension of $X/\ker \varphi$ is 1.

4. D

   X is finite dimensional.

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

1. A

   If f is onto, then so is $f^{n}$ for all n ≥ 1.

2. B

   ker $f^{n+1}$ $= \ker f^{n}$ for some n ≥ 1.

3. C

   If f is onto, then f is one-to-one.

4. D

   If f is one-to-one, then f is onto.

Single choice · Vector Spaces and Linear Maps

1. A

   (A) is true but (B) is false.

2. B

   (B) is true but (A) is false.

3. C

   Both (A) and (B) are true.

4. D

   Both (A) and (B) are false.

Single choice · Vector Spaces and Linear Maps

1. A

   Both V and W are infinite-dimensional

2. B

   dim V = 3 and dim W = 1

3. C

   dim V = 3 and dim W = 2

4. D

   V is infinite-dimensional and dim W = 0

Single choice · Vector Spaces and Linear Maps

1. A

   (x − 2)(x − 1)

2. B

   $x^{2}(x - 2)(x - 1)$

3. C

   $(x - 2)^{2}(x - 1)^{2}$

4. D

   $(x^{2} - 2)(x^{2} - 1)$

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 · 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 · Vector Spaces and Linear Maps

1. A

   There exists an injective $\mathbb{Q}-$linear transformation $T : \mathbb{R} \to \mathbb{Q}[x]$

2. B

   There exists an injective $\mathbb{Q}-$linear transformation $T : \mathbb{Q}[x] \to \mathbb{R}$

3. C

   The $\mathbb{Q}-$vector spaces $\mathbb{Q}[x]$ and $\mathbb{R}$ are isomorphic

4. D

   There do not exist non-zero $\mathbb{Q}-$linear transformations $T : \mathbb{R} \to \mathbb{Q}[x]$

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

1. A

   There exists a polynomial p(x) ∈ W such that $x^{4} - p(x) \in W^\perp$

2. B

   W^⊥ = {0}

3. C

   W and W^⊥ have the same dimension over $\mathbb{R}$

4. D

   W^⊥ is an infinite dimensional vector space over $\mathbb{R}$

Single choice · Lebesgue Measure and Integration

1. A

   S is in bijection with the set of rational numbers.

2. B

   S is in bijection with the set of real numbers.

3. C

   S is in bijection with S × S.

4. D

   S is in bijection with the power set of S.

Single choice · Vector Spaces and Linear Maps

1. A

   If A is a real 3×2 matrix and B a real 2×3 matrix with BA $= I_{2}$, then AB $= I_{3}$.

2. B

   Let A = [[3, 3], [1, 2]]. Then there is a matrix B with integer entries such that AB $= I_{2}$.

3. C

   Let A = [[3, 1], [1, 2]] with entries in $\mathbb{Z}/6\mathbb{Z}$. Then there is a matrix B with entries in $\mathbb{Z}/6\mathbb{Z}$ such that AB $= I_{2}$.

4. D

   If A is a real non-zero 3×3 diagonal matrix, then there is a real matrix B such that AB $= I_{3}$.

Single choice · Rings and Fields

1. A

   K is dense in $\mathbb{C}$.

2. B

   K is an algebraic extension of $\mathbb{Q}$.

3. C

   $\mathbb{C}$ is an algebraic extension of K.

4. D

   The smallest closed subset of $\mathbb{C}$ containing K is NOT a field.

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

1. A

   Let $A_{1} =$ {$x^{2} + y^{2} < 1$}. For each $\gamma \in p(A_{1})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{1})$.

2. B

   Let $A_{2} =$ {$x^{2} + y^{2} \le 1$}. For each $\gamma \in p(A_{2})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{2})$.

3. C

   Let $A_{3} =$ {xy = 0}. For each $\gamma \in p(A_{3})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{3})$.

4. D

   Let $A_{4} =$ {xy = 1}. For each $\gamma \in p(A_{4})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{4})$.

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

1. A

   V is the union of 2 proper subspaces.

2. B

   V is the union of 3 proper subspaces.

3. C

   V has a unique basis.

4. D

   V has precisely two bases.