CSIR NET Maths previous year question papers

Single choice · Inner Product Spaces and Forms

1. A

   If v and w are eigenvectors of B for distinct eigenvalues, then ⟨v,w⟩ = 0.

2. B

   If v is an eigenvector of B and w ≠ 0 is such that ⟨w,v⟩ = 0, then w is an eigenvector of B.

3. C

   For every subspace $W \subseteq \mathbb{R}^{4}, W + W\perp = \mathbb{R}^{4}$, where W⊥ = {$v \in \mathbb{R}^{4}$ ∣ ⟨v,w⟩ = 0 for all w ∈ W}.

4. D

   If $W \subseteq \mathbb{R}^{4}$ is an eigenspace of B, then W has an orthonormal basis.

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 · Inner Product Spaces and Forms

1. A

   det A = 0

2. B

   det A = −1

3. C

   B(x, x) ≠ 0 for all nonzero $x \in \mathbb{R}^{4}$.

4. D

   If $x \in \mathbb{R}^{4}$ is nonzero, then there exists $y \in \mathbb{R}^{4}$ such that B(x, y) ≠ 0.

Single choice · Vector Spaces and Linear Maps

1. A

   If A is one-to-one but not onto, then m > n

2. B

   If A is onto but not one-to-one, then m < n

3. C

   If A is bijective, then m = n

4. D

   If A is one-to-one, then m = n

Single choice · Vector Spaces and Linear Maps

1. A

   The product of two 2×2 real matrices of rank 2 is of rank 2.

2. B

   The product of two 3×3 real matrices of rank 2 is of rank at most 2.

3. C

   The product of two 3×3 real matrices of rank 2 is of rank at least 2.

4. D

   The product of two 2×2 real matrices of rank 1 can be the zero matrix.

Single choice · Inner Product Spaces and Forms

1. A

   $A^{2} = 0$.

2. B

   rank A = 1.

3. C

   rank A = 0.

4. D

   There exists $u \in \mathbb{R}^{2}, u \ne 0$ such that (u, u) = 0.