CSIR NET Maths previous year question papers

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 $\mathbb{R}$

Single choice · The Real Line

1. A

   The series is divergent.

2. B

   The series is convergent.

3. C

   The series is conditionally convergent.

4. D

   The series is absolutely convergent.

Single choice · The Real Line

1. A

   $limsup_{n} \exp(\cos((n\pi + (-1)^{n}\cdot2e)/(2n))) > 1$.

2. B

   $lim_{n} \exp(log_{e}((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.

3. C

   $liminf_{n} \exp(\sin((n\pi + (-1)^{n}\cdot2e)/(2n))) < \pi$.

4. D

   $lim_{n} \exp(\tan((n\pi^{2} + (-1)^{n}e^{2})/(7n)))$ does not exist.

Single choice · Continuity and Differentiation

1. A

   f(x) = sin(1/x)

2. B

   f(x) = $e^{-1/x^{2}}$

3. C

   f(x) = eˣ cos(1/x)

4. D

   $f(x) = \cos x \cdot \cos(\pi/x)$

Single choice · Eigenvalues and Canonical Forms

1. A

   The eigenspace of A for the eigenvalue 0 is two-dimensional.

2. B

   All the eigenvalues of A are real.

3. C

   $A^{3} = 0$.

4. D

   A is diagonalizable.

Single choice · Eigenvalues and Canonical Forms

1. A

   Both (a) and (b) are true.

2. B

   Both (a) and (b) are false.

3. C

   (a) is true and (b) is false.

4. D

   (a) is false and (b) is true.

Single choice · Inner Product Spaces and Forms

1. A

   ⟨x, y⟩ $= \sum_{i,j=1}^n$ $x_{i}y_{j}$

2. B

   ⟨x, y⟩ $= \sum_{i,j=1}^n$ $(x_{i}^{2} + y_{j}^{2})$

3. C

   ⟨x, y⟩ $= \sum_{j=1}^n$ $j^{3} x_{j}y_{j}$

4. D

   ⟨x, y⟩ $= \sum_{j=1}^n$ $x_{j} y_{n}$₋$_{j}$₊$_{1}$

Single choice · Lebesgue Measure and Integration

1. A

   S is in bijection with the set of rational numbers.

2. B

   S is in bijection with the set of real numbers.

3. C

   S is in bijection with S × S.

4. D

   S is in bijection with the power set of S.

Single choice · The Real Line

1. A

   inf U = 5.

2. B

   inf U = 4.

3. C

   inf U = 3.

4. D

   inf U = 2.

Single choice · The Real Line

1. A

   $\limsup a_{n} = \sqrt{3}/2$.

2. B

   $\limsup a_{2n} = 1$.

3. C

   $\limsup a_{2n} = 1/2$.

4. D

   $\limsup a_{3n} = 0$.

Single choice · The Real Line

1. A

   (a) is convergent, but (b) is not convergent.

2. B

   (a) is not convergent, but (b) is convergent.

3. C

   Both (a) and (b) are convergent.

4. D

   Neither (a) nor (b) is convergent.

Single choice · Continuity and Differentiation

1. A

   F(x) has exactly one real root.

2. B

   F(x) has exactly four real roots.

3. C

   F(x) has at most two real roots.

4. D

   F(x) has at most one real root.

Single choice · Continuity and Differentiation

1. A

   f never vanishes in [0, 1].

2. B

   f has at most finitely many zeros in [0, 1].

3. C

   f has infinitely many zeros in [0, 1].

4. D

   f(1/2) = 0.

Single choice · Continuity and Differentiation

1. A

   S = ∅

2. B

   S ≠ ∅ and S is a proper subset of $(1, \infty)$

3. C

   $(2, \infty)$ is a proper subset of S

4. D

   S ∩ (0, 1) ≠ ∅

Single choice · Lebesgue Measure and Integration

1. A

   Y is an infinite set.

2. B

   Y has $2^{|X|}$ elements.

3. C

   There is a bijective function from X to Y.

4. D

   There is a surjective function from X to Y.

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 $= I_{3}$.

Single choice · Eigenvalues and Canonical Forms

1. A

   1

2. B

   n

3. C

   n(n+1)/2

4. D

   n(n+1)(2n+1)/6

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.