CSIR NET Maths previous year question papers

Single choice · The Real Line

1. A

   $\lim \inf(n\to\infty) x_{n} \ge 2025$

2. B

   $\lim \sup(n\to\infty) x_{n} \le 2025$

3. C

   $\lim \inf(n\to\infty) x_{n} \le 2025$

4. D

   $\lim \inf(k\to\infty) x_(n_{k}) > 2025$

Single choice · The Real Line

1. A

   $(a_{n})_{n}\ge_{1}$ converges but $(b_{n})_{n}\ge_{1}$ does not converge.

2. B

   $(b_{n})_{n}\ge_{1}$ converges but $(a_{n})_{n}\ge_{1}$ does not converge.

3. C

   Both $(a_{n})_{n}\ge_{1}$ and $(b_{n})_{n}\ge_{1}$ converge.

4. D

   Neither $(a_{n})_{n}\ge_{1}$ nor $(b_{n})_{n}\ge_{1}$ converges.

Single choice · Continuity and Differentiation

1. A

   For every bounded subset $A \subseteq \mathbb{R}, f$⁻$^{1}(A)$ is a bounded subset of $\mathbb{R}$.

2. B

   For every Cauchy sequence $(x_{n})_{n}\ge_{1}$ in $\mathbb{R}, (f(x_{n}))_{n}\ge_{1}$ is a Cauchy sequence in $\mathbb{R}$.

3. C

   There exists $x \in \mathbb{R}$ such that f(x) = x.

4. D

   There exists $x \in \mathbb{R}$ such that f(x) = 0.

Single choice · Continuity and Differentiation

1. A

   f is monotone.

2. B

   f is continuous but not strictly monotone.

3. C

   f is not continuous.

4. D

   f is continuous but not uniformly continuous.

Single choice · Sequences and Series of Functions

1. A

   $\sum(n\ge1) f_{n}$ and $\sum(n\ge1) g_{n}$ are uniformly convergent on the interval [−1, 1].

2. B

   $\sum(n\ge1) f_{n}$ is uniformly convergent on the interval [−1, 1], but $\sum(n\ge1) g_{n}$ is not.

3. C

   $\sum(n\ge1) g_{n}$ is uniformly convergent on the interval [−1, 1], but $\sum(n\ge1) f_{n}$ is not.

4. D

   Neither $\sum(n\ge1) f_{n}$ nor $\sum(n\ge1) g_{n}$ is uniformly convergent on the interval [−1, 1].

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 · Eigenvalues and Canonical Forms

1. A

   T is diagonalizable over $\mathbb{R}$.

2. B

   The characteristic polynomial of T is $(x - 2)^{4}$.

3. C

   The characteristic polynomial of T is $(x^{2} + x + 2)(x^{2} - 4)$.

4. D

   For every $\mathbb{R}-$linear transformation $S : \mathbb{R}^{4} \to \mathbb{R}^{4}$, we have ST = TS.

Single choice · Eigenvalues and Canonical Forms

1. A

   A is diagonalizable.

2. B

   A has at least two distinct eigenvalues.

3. C

   The characteristic polynomial of A is $x^{3} - 1$.

4. D

   The minimal polynomial of A cannot have degree 2.

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

   {$(-1/\sqrt{2}, 0, 0, -1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)$}

2. B

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)$}

3. C

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)$}

4. D

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (1/\sqrt{6}, -\sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)$}

Single choice · Inner Product Spaces and Forms

1. A

   B(v, v) = 0 if and only if v = 0.

2. B

   For every $\lambda \in \mathbb{R}$, there exists v such that $B(v, v) = \lambda$.

3. C

   There exists v ≠ 0 such that B(v, w) = 0 for all $w \in \mathbb{R}^{3}$.

4. D

   If B(v, w) = 0 then either v = 0 or w = 0.

Single choice · Metric Spaces

1. A

   A is a closed subset of $\mathbb{R}^{2}$ but B is not a closed subset of $\mathbb{R}^{2}$.

2. B

   B is a closed subset of $\mathbb{R}^{2}$ but A is not a closed subset of $\mathbb{R}^{2}$.

3. C

   Both A and B are closed subsets of $\mathbb{R}^{2}$.

4. D

   Neither A nor B is a closed subset of $\mathbb{R}^{2}$.

Single choice · The Real Line

1. A

   A \ B

2. B

   (A \ B) ∪ C

3. C

   A \ (B ∪ C)

4. D

   (A \ B) ∪ (A ∩ C)

Single choice · Sequences and Series of Functions

1. A

   The series is neither absolutely convergent nor uniformly convergent.

2. B

   The series is both absolutely convergent and uniformly convergent.

3. C

   The series is absolutely convergent but not uniformly convergent.

4. D

   The series is uniformly convergent but not absolutely convergent.

Single choice · The Real Line

1. A

   For every $x \in \mathbb{R}$ there exists some n such that $a_{n} > x$

2. B

   For every $x \in \mathbb{R}$ there exists some n such that $a_{n} < x$

3. C

   For every $x \in \mathbb{R}$ there exists some n such that $b_{n} > x$

4. D

   For every $x \in \mathbb{R}$ there exists some n such that $b_{n} < x$

Single choice · Integration

1. A

   A = 0

2. B

   A = 1

3. C

   A = sin(1)/2

4. D

   A = sin(1/4)

Single choice · Sequences and Series of Functions

1. A

   The radius of convergence of the series is 1/e

2. B

   The series converges at x = 5

3. C

   The series converges at x = 3

4. D

   The series converges for all x with |x| < 1/2

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 · Eigenvalues and Canonical Forms

1. A

   det B = 1

2. B

   det A = 0

3. C

   rank(B) = 2

4. D

   $rank(B^{2}) = 1$

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 · Determinants and Matrix Tricks

1. A

   W = ker(T)

2. B

   W ⊊ ker(T)

3. C

   W ∩ ker(T) ⊊ W

4. D

   W ∩ ker(T) ⊊ ker(T)

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 · The Real Line

1. A

   $\sup A = 2 + 2\sqrt{3}$

2. B

   $\sup A = 3 + 2\sqrt{2}$

3. C

   $\inf A = 2 + 2\sqrt{3}$

4. D

   $\inf A = 3 + 2\sqrt{2}$

Single choice · The Real Line

1. A

   S is empty.

2. B

   There is a bijection between S and $\mathbb{N}$

3. C

   There is a bijection between S and $\mathbb{R}$

4. D

   There is a bijection between S and a non-empty finite set