CSIR NET Maths previous year question papers

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

   Both A and B are diagonalizable over $\mathbb{R}$.

2. B

   A is diagonalizable over $\mathbb{C}$ but not over $\mathbb{R}$.

3. C

   Neither A nor B is diagonalizable over $\mathbb{R}$, but both A and B are diagonalizable over $\mathbb{C}$.

4. D

   Neither A nor B is diagonalizable over $\mathbb{C}$.

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

1. A

   S is a cyclic group.

2. B

   S has $\varphi(2025)$ elements, where $\varphi$ denotes the Euler $\varphi-$function.

3. C

   S has $\varphi(2025)$ generators, where $\varphi$ denotes the Euler $\varphi-$function.

4. D

   S has 81 elements.

Multi-select · full marks only for the exact set · Linear Integral Equations

1. A

   f(x) = cos x

2. B

   f(x) = cos 5x

3. C

   f(x) = sin x

4. D

   f(x) = sin 5x

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

1. A

   1

2. B

   2

3. C

   3

4. D

   5

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

1. A

   {$v \in \mathbb{Q}^{5}$ | Bv = 0} is a three-dimensional vector space over $\mathbb{Q}$.

2. B

   The linear transformation $T : \mathbb{Q}^{3} \to \mathbb{Q}^{5}$ given by T(v) = Bᵗv is injective.

3. C

   The column span of B is two-dimensional.

4. D

   The linear transformation $T : \mathbb{Q}^{3} \to \mathbb{Q}^{3}$ given by T(v) = BBᵗv is injective.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   f(z̄) is an entire function.

2. B

   conj(f(z)) is an entire function.

3. C

   conj(f(z̄)) is an entire function.

4. D

   conj(f(z̄)) + f(z̄) is an entire function.

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

1. A

   For all prime numbers p, f(X) mod p is irreducible in $(\mathbb{Z}/p\mathbb{Z})[X]$.

2. B

   There exists a prime number p such that g(X) mod p is irreducible in $(\mathbb{Z}/p\mathbb{Z})[X]$.

3. C

   g(X) is irreducible in $\mathbb{Q}[X]$.

4. D

   f(X) is irreducible in $\mathbb{Q}[X]$.