CSIR NET Maths previous year question papers

Single choice · Eigenvalues and Canonical Forms

1. A

   $M^{2} = N^{2}$

2. B

   $M^{3} = N^{3}$

3. C

   The characteristic polynomials of M and N are the same.

4. D

   The minimal polynomials of M and N are the same.

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

1. A

   T has at least four linearly independent eigenvectors.

2. B

   dim W ≥ 2.

3. C

   $\ker((T - 2I)^{2025}) = \ker((T - 2I)^{2026})$

4. D

   (T − 2I)(T − 3I) is a nilpotent operator.

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

1. A

   The matrix with rows (0, 0, 0, −1), (1, 0, 0, 0), (0, 1, 0, 2), (0, 0, 1, 0)

2. B

   The matrix with rows (0, 0, 0, 1), (1, 0, 0, 0), (0, 1, 0, −2), (0, 0, 1, 0)

3. C

   The matrix with rows (0, 1, 1, 0), (1, 0, 0, 1), (0, 0, 0, 1), (0, 0, 1, 0)

4. D

   The matrix with rows (0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0)

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

1. A

   There exists a cubic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.

2. B

   There exists a quadratic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.

3. C

   If $f(X)\in\mathbb{Q}[X]$ is such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.

4. D

   If $f(X)\in\mathbb{Q}[X]$ is a cubic polynomial such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.

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

1. A

   There exist $\lambda\in\mathbb{C}$ and a non-zero v∈V such that Av$=\lambda{}v$.

2. B

   There exist $\lambda,\mu\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av$=\lambda{}v$ and Aw$=\mu{}w$.

3. C

   There exist $\lambda,\mu,\delta\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av$=\lambda{}v$ and Aw$=\mu{}v+\delta{}w$.

4. D

   There exists a three-dimensional subspace W such that Aw∈W for all w∈W.

Single choice · Eigenvalues and Canonical Forms

1. A

   A is diagonalizable.

2. B

   A has at least two distinct eigenvalues.

3. C

   The characteristic polynomial of A is $x^{3} - 1$.

4. D

   The minimal polynomial of A cannot have degree 2.

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

   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 · Vector Spaces and Linear Maps

1. A

   The dimension of V is even

2. B

   The trace of T is zero

3. C

   The minimal polynomial of T cannot have two distinct roots

4. D

   The minimal polynomial of T is equal to its characteristic polynomial

Single choice · Eigenvalues and Canonical Forms

1. A

   The eigenspace of A for the eigenvalue 0 is two-dimensional.

2. B

   All the eigenvalues of A are real.

3. C

   $A^{3} = 0$.

4. D

   A is diagonalizable.

Single choice · Eigenvalues and Canonical Forms

1. A

   Both (a) and (b) are true.

2. B

   Both (a) and (b) are false.

3. C

   (a) is true and (b) is false.

4. D

   (a) is false and (b) is true.

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

1. A

   Quadratic forms (a) and (b) are equivalent.

2. B

   Quadratic forms (a) and (c) are equivalent.

3. C

   Quadratic form (b) is positive definite.

4. D

   Quadratic form (c) is positive definite.

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

1. A

   If $T_{1}$ and $T_{2}$ are similar, then $W_{1}$ and $W_{2}$ are isomorphic vector spaces.

2. B

   If $W_{1}$ and $W_{2}$ are isomorphic vector spaces, then $T_{1}$ and $T_{2}$ have the same minimal polynomial.

3. C

   If $W_{1} = W_{2} = V$, then $T_{1}$ and $T_{2}$ are similar.

4. D

   If $W_{1}$ and $W_{2}$ are isomorphic, then $T_{1}$ and $T_{2}$ have the same characteristic polynomial.

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

1. A

   The rank of T is equal to 2.

2. B

   Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$. Then $T^{2}$ must be zero.

3. C

   Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$. Then $T^{3}$ must be zero.

4. D

   (−2, −8, 3, 2, 0) is contained in the null space of T.

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

1. A

   The characteristic polynomial of A is $x^{3}(x - 1)(x - 2\pi{}i)^{3}$.

2. B

   The minimal polynomial of A is $x^{2}(x - 1)(x - 2\pi{}i)^{3}$.

3. C

   The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 3 respectively.

4. D

   The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 2 respectively.