CSIR NET Maths previous year question papers

Single choice · Eigenvalues and Canonical Forms

1. A

   $t^{2} + 7t + 10$

2. B

   $t^{2} - 7t + 29$

3. C

   $t^{2} - 7t - 10$

4. D

   $t^{2} - 7t + 10$

Single choice · Eigenvalues and Canonical Forms

1. A

   101

2. B

   67

3. C

   103

4. D

   113

Single choice · Eigenvalues and Canonical Forms

1. A

   There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = f(B) = f(C) = 0 and f(D) = I.

2. B

   There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = f(B) = 0, f(C) = I and f(D) = I.

3. C

   There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = 0, f(B) = f(C) = I and f(D) ≠ 0.

4. D

   There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = f(B) = 0, f(C) = I and f(D) = 0.

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.

Single choice · Eigenvalues and Canonical Forms

1. A

   The characteristic polynomial of A has repeated roots.

2. B

   Every eigenvalue of A has absolute value 1.

3. C

   A does not have any eigenvalue on the imaginary axis.

4. D

   $A^{2}$ is the identity matrix.

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

1. A

   If A is invertible, then AB and BA have the same minimal polynomial.

2. B

   If 0 is an eigenvalue of A, then 0 is an eigenvalue of AB.

3. C

   If 0 is the only eigenvalue of A and of B, then 0 is the only eigenvalue of AB.

4. D

   If AB and BA have the same minimal polynomial, then either A or B is invertible.

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

1. A

   (ST − TS$)^{2} = \lambda{}I$ for some $\lambda \in \mathbb{C}$.

2. B

   The characteristic polynomial of (ST − TS$)^{2}$ is $(x - \lambda)^{2}$ for some $\lambda \in \mathbb{C}$.

3. C

   If ST − TS has only one eigenvalue, then ST − TS $= \lambda{}I$ for some $\lambda \in \mathbb{C}$.

4. D

   If ST − TS has only one eigenvalue, then (ST − TS$)^{2}$ is the zero transformation.

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

   The set {$B \in M_{4}(\mathbb{C})$ ∣ AB+BA=0} is an 8-dimensional $\mathbb{C}-$vector space.

2. B

   The set {$B \in M_{4}(\mathbb{C})$ ∣ AB−BA=0} is an 8-dimensional $\mathbb{C}-$vector space.

3. C

   There exists $B \in M_{4}(\mathbb{C})$ such that AB+BA$=0, B^{2}=I$ and Trace(B)=0.

4. D

   If $B \in M_{4}(\mathbb{C})$ is such that AB+BA=0, then B is diagonalisable.

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

   If T is not diagonalizable over $\mathbb{R}$, then $T^{2}$ has an eigenvalue ≤0.

2. B

   If $T^{2}$ has only negative eigenvalues, then dim V is an even integer.

3. C

   If $T^{2}$ has only non-negative eigenvalues, then T is diagonalizable over $\mathbb{R}$.

4. D

   For each non-zero v∈V, {v,Tv,$T^{2}v$} is linearly dependent.

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

   T is diagonalizable over $\mathbb{R}$.

2. B

   The characteristic polynomial of T is $(x - 2)^{4}$.

3. C

   The characteristic polynomial of T is $(x^{2} + x + 2)(x^{2} - 4)$.

4. D

   For every $\mathbb{R}-$linear transformation $S : \mathbb{R}^{4} \to \mathbb{R}^{4}$, we have ST = TS.

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.

Single choice · Eigenvalues and Canonical Forms

1. A

   There exists a real invertible matrix P such that PAP⁻$^{1}$ is a diagonal matrix and Pᵗ$P = I_{3}$.

2. B

   There exists a real invertible matrix P such that PAP⁻$^{1}$ is a diagonal matrix and Pᵗ$P \ne I_{3}$.

3. C

   One of the eigenvalues of A is not real.

4. D

   A has only real eigenvalues and it is not diagonalizable over $\mathbb{R}$.

Single choice · Eigenvalues and Canonical Forms

1. A

   det B = 1

2. B

   det A = 0

3. C

   rank(B) = 2

4. D

   $rank(B^{2}) = 1$

Single choice · Eigenvalues and Canonical Forms

1. A

   24 − ab = 4c

2. B

   a + b = 8

3. C

   c = 6

4. D

   ab = 0

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 · 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 · Eigenvalues and Canonical Forms

1. A

   There exists a subspace W of $\mathbb{C}^{n}$ such that $\mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

2. B

   There exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

3. C

   Suppose that there exists a positive integer k such that Tᵏ is the identity map. Then there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

4. D

   Suppose that there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}. Then there exists a positive integer k such that Tᵏ is the identity map.