CSIR NET Maths previous year question papers

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 · Inner Product Spaces and Forms

1. A

   S is a basis of W.

2. B

   S is an orthonormal basis of W.

3. C

   There exist f, g ∈ S such that ⟨f, g⟩ = 0.

4. D

   S contains an orthonormal basis of W.

Multi-select · full marks only for the exact set · The Real Line

1. A

   If $lim_{n}\to\infty b_{n}$ exists in $\mathbb{R}$, then {$a_{n} : n \ge 1$} is bounded

2. B

   If $lim_{n}\to\infty b_{n} = 1$, then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$

3. C

   If $lim_{n}\to\infty b_{n} = 1/2$, then $lim_{n}\to\infty a_{n}$ exists in $\mathbb{R}$

4. D

   If $lim_{n}\to\infty b_{n} = 0$, then $lim_{n}\to\infty a_{n} = 0$

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f is strictly increasing

2. B

   f is strictly decreasing

3. C

   f is either strictly increasing or strictly decreasing

4. D

   f is onto

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f need not be surjective

2. B

   f must be surjective if K = [0, 1]

3. C

   f is injective and f⁻$^{1} : f(K) \to K$ is continuous

4. D

   f is injective, but f⁻$^{1} : f(K) \to K$ need not be continuous

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   f(x) is not uniformly continuous on [0, 1)

2. B

   The sequence $(p_{n}(x))$ converges to f(x) pointwise on [0, 1)

3. C

   The sequence $(p_{n}(x))$ converges to f(x) uniformly on [0, 1)

4. D

   The sequence $(p_{n}(x))$ converges to f(x) uniformly on [0, c] for every 0 < c < 1

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 · Eigenvalues and Canonical Forms

1. A

   There exists a non-zero vector $v_{1} \in \mathbb{R}^{4}$ such that Tv$_{1} = 2v_{1}$

2. B

   There exists a non-zero vector $v_{2} \in \mathbb{R}^{4}$ such that Tv$_{2} = v_{2}$

3. C

   For every non-zero vector $v \in \mathbb{R}^{4}$, the set {2v, 3Tv} is linearly independent

4. D

   T is a one-one function

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

1. A

   The minimal polynomial of B is $x^{2} + x + 1$

2. B

   The minimal polynomial of B is $x^{2} - x + 1$

3. C

   $B^{3} = I_{4}$

4. D

   $B^{3} + I_{4} = 0$

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

   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

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

1. A

   q is a quadratic form in $x_{1}, x_{2}, y_{1}, y_{2}$

2. B

   There exists $(t_{1}, t_{2}) \in \mathbb{R}^{2}$ such that $q_{1}(t_{1}, t_{2}) = 5$

3. C

   There does not exist $(s_{1}, s_{2}) \in \mathbb{R}^{2}$ such that $q_{2}(s_{1}, s_{2}) = -5$

4. D

   Given $\alpha \in \mathbb{R}$, there exists a vector $w \in \mathbb{R}^{4}$ such that $q(w) = \alpha$

Multi-select · full marks only for the exact set · The Real Line

1. A

   $A_{n} \to 0$ and $B_{n} \to 0$ as $n \to \infty$

2. B

   If $\sum_{n}$₌$_{1}^\infty a_{n}$ is absolutely convergent, then both $\sum_{n}$₌$_{1}^\infty A_{n}$ and $\sum_{n}$₌$_{1}^\infty B_{n}$ are absolutely convergent

3. C

   Both $\sum_{n}$₌$_{1}^\infty A_{n}$ and $\sum_{n}$₌$_{1}^\infty B_{n}$ are convergent

4. D

   If $\sum_{n}$₌$_{1}^\infty a_{n}$ is not absolutely convergent, then both $\sum_{n}$₌$_{1}^\infty A_{n}$ and $\sum_{n}$₌$_{1}^\infty B_{n}$ are divergent

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f is continuous on $\mathbb{R}$

2. B

   f is differentiable on $\mathbb{R}$

3. C

   f is differentiable only at 0

4. D

   f is not differentiable at 0

Multi-select · full marks only for the exact set · The Real Line

1. A

   S is the empty set

2. B

   S has exactly one element

3. C

   S has at least two elements

4. D

   S has to be a finite set

Multi-select · full marks only for the exact set · Integration

1. A

   The integral I is convergent

2. B

   The integral I is not convergent

3. C

   The integral $I_{a}$ converges for a = 1/2 but not for a = 0

4. D

   The integral $I_{a}$ converges for all a ≥ 0

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   $\partial{}f/\partial{}x(0, 0)$ exists

2. B

   $\partial{}f/\partial{}y(0, 0)$ exists

3. C

   f is not continuous at (0, 0)

4. D

   f is not differentiable at (0, 0)

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   f is injective in a neighbourhood of (0, 0)

2. B

   There exists an open neighbourhood U of (0, 0) in $\mathbb{R}^{2}$ such that $f_{3}$ is a function of $f_{1}$ and $f_{2}$

3. C

   f maps an open neighbourhood of (0, 0) in $\mathbb{R}^{2}$ onto an open subset of $\mathbb{R}^{3}$

4. D

   (0, 0) is an isolated point of f⁻$^{1}(${f(0,0)})

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

1. A

   There exists a polynomial p(x) ∈ W such that $x^{4} - p(x) \in W^\perp$

2. B

   W^⊥ = {0}

3. C

   W and W^⊥ have the same dimension over $\mathbb{R}$

4. D

   W^⊥ is an infinite dimensional vector space over $\mathbb{R}$

Multi-select · full marks only for the exact set · The Real Line

1. A

   The subsequences {$x_{2n}$₊$_{1}$}, {$x_{2n}$} and {$x_{3n}$} are convergent and have the same limit.

2. B

   The subsequences {$x_{2n}$₊$_{1}$}, {$x_{2n}$} and {$x_{3n}$} are convergent.

3. C

   The subsequences {$x_{kn}$}$_{n}$ are convergent for every k ≥ 2.

4. D

   lim |$x_{n}$₊$_{1} - x_{n}$| = 0.

Multi-select · full marks only for the exact set · The Real Line

1. A

   If a = 0, then the sequence {$x_{n}$} converges to 1/2.

2. B

   If a = 0, then the sequence {$x_{n}$} converges to −1/2.

3. C

   The sequence {$x_{n}$} converges for every a ∈ (−1/2, 3/2), and it converges to 1/2.

4. D

   If a = 0, then the sequence {$x_{n}$} does not converge.

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   There is no continuous real-valued function g defined on any interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

2. B

   There is exactly one continuous real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

3. C

   There is exactly one differentiable real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

4. D

   There are two distinct differentiable real-valued functions g on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

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 · 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.