CSIR NET Maths previous year question papers

Single choice · Vector Spaces and Linear Maps

1. A

   125

2. B

   144

3. C

   145

4. D

   480

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.

Single choice · Vector Spaces and Linear Maps

1. A

   1

2. B

   (q^(2n) − q^n)(q^(2n) − q^(n+1))/q^n

3. C

   (q^(2n) − 1)(q^(2n) − q)⋯(q^(2n) − q^(n−1))/q^n

4. D

   $(q^n - 1)(q^n - q)/((q^{2} - 1)(q^{2} - q))$

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

1. A

   {f′(x), f(x) − f(x − 1), 1}

2. B

   {f(x + 1) − f(x), f(x) − f(x − 1), 1}

3. C

   {f(x), f′(x), 1}

4. D

   {f(x + 1), f(x − 1), f(x)}

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

1. A

   The number of two dimensional subspaces of V is 13.

2. B

   The number of surjective linear transformations from V to W is 624.

3. C

   The number of one dimensional subspaces of V is 13.

4. D

   The number of linear transformations from V to W is $3^{6}$.

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

1. A

   If AAᵀ = I, then DF$_A : M_{2}(\mathbb{R}) \to S$ is surjective.

2. B

   If AAᵀ = I, then DF$_A : M_{2}(\mathbb{R}) \to S$ need not be surjective.

3. C

   If A is invertible, then DF$_A : M_{2}(\mathbb{R}) \to S$ is surjective.

4. D

   If A is not invertible, then DF$_A : M_{2}(\mathbb{R}) \to S$ is surjective.

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

1. A

   The dimension of V is 4 or 5.

2. B

   If 2u−3v+5w=0, then {v,w,x,y} is a basis of V.

3. C

   If u−6v+7w=0, then the span of {v,w,x} is a 2-dimensional vector space.

4. D

   If u+w=v+x, then {u,v,w,y} is a basis of V.

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

1. A

   The set {$S\in{}W : S^{2}=S$} contains exactly 2 elements.

2. B

   $dim_\mathbb{R}(W) = 2$.

3. C

   If U∈L(V) is such that $U^{2}=U$ and $(T+U)^{2}=T+U$, then TU=0.

4. D

   If U∈L(V) is such that $U^{2}=U$ and $(T-U)^{2}=T-U$, then (TU$)^{2}=$TU.

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

   5

2. B

   10

3. C

   15

4. D

   20

Single choice · Vector Spaces and Linear Maps

1. A

   $(T^{3} - 3T^{2} + 3T)^2025(x) = x$

2. B

   $(T^{3} - 3T^{2} + 3T)^2025(x) = x + 1$

3. C

   $(T^{3} - 3T^{2} + 3T)^2025(x) = 2025! x$

4. D

   $(T^{3} - 3T^{2} + 3T)^2025(x) = 2025! x + 1$

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

   There exists a vector $v \in \mathbb{R}^{10}$ such that Av ≠ 0 and $A^{2}v = 0$

2. B

   There exists a vector $v \in \mathbb{R}^{10}$ such that $A^{2}v \ne 0$

3. C

   A must have a non-zero eigenvalue

4. D

   $A^{7} = 0$

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 · Vector Spaces and Linear Maps

1. A

   If f ∈ B, then dim ker f = 3

2. B

   dim B = 3

3. C

   dim A = 4

4. D

   If f ∈ A, then the image of f is a one-dimensional $\mathbb{Q}-$vector space.

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

1. A

   DM ≠ MD.

2. B

   D + M is invertible.

3. C

   DM is invertible.

4. D

   rank(DM) = rank(MD).

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

1. A

   (1, 1, 1, 1, 1)

2. B

   (0, 0, 1, 2, 4)

3. C

   (1, 0, 1, 0, 1)

4. D

   (1, 0, 1, 0, 2)

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 · 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 · Vector Spaces and Linear Maps

1. A

   A does not have any non-zero invariant subspace in $\mathbb{C}^{n}$.

2. B

   A has an invariant subspace in $\mathbb{C}^{n}$ of dimension n − 3.

3. C

   All eigenvalues of A are real.

4. D

   $A^{2}$ does not have any invariant subspace in $\mathbb{C}^{n}$ of dimension n − 1.

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 · 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 $= I_{3}$.

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 · 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.