CSIR NET Maths previous year question papers

Single choice · The Real Line

1. A

   If A is finite, then there exists a bijection between S and U.

2. B

   If A is finite and B = [0, 1], then there is no bijection between S and T.

3. C

   There is no bijection between S and U for any choice of A.

4. D

   If A ≠ B, then there is no bijection between T and U.

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 · Analytic Functions

1. A

   E is the real axis.

2. B

   E is the imaginary axis.

3. C

   E contains an open subset of $\mathbb{C}$, but $E \ne \mathbb{C}$.

4. D

   $E = \mathbb{C}$

Single choice · Zeros and Mappings

1. A

   $T_{1}$ and $T_{2}$ conformally map R₊ and R₋ respectively, onto the disc 𝔻

2. B

   $T_{1}$ and $T_{2}$ conformally map R₋ and R₊ respectively, onto the disc 𝔻

3. C

   $T_{1}$ and $T_{2}$ conformally map the disc 𝔻 onto, respectively R₊ and R₋

4. D

   $T_{1}$ and $T_{2}$ conformally map the disc 𝔻 onto, respectively R₋ and R₊

Single choice · Cauchy Theory

1. A

   f(1)

2. B

   f(2)

3. C

   f′(1)

4. D

   f′(2)

Single choice · Singularities and Residues

1. A

   z = 0 is a zero of f of order 3 and a pole of g of order 1.

2. B

   z = 0 is a zero of f of order 2 and a pole of g of order 1.

3. C

   z = 0 is a zero of f of order 3 and a zero of g of order 1.

4. D

   z = 0 is a zero of f of order 2 and a zero of g of order 1.

Single choice · Groups

1. A

   For each $p < 100, \sum(n=1..\infty) (S_p/\varphi(p))^{n}$ converges.

2. B

   For each $100 \le p \le 200, \sum(n=1..\infty) (S_p/\varphi(p))^{n}$ diverges.

3. C

   For each $p > 200, \sum(n=1..\infty) (S_p/\varphi(p))^{n}$ converges.

4. D

   The element 4 mod 101 in $(\mathbb{Z}/101\mathbb{Z})$ˣ is a primitive root.

Single choice · Groups

1. A

   If G is abelian, then |X| = n.

2. B

   If |X| divides n, then G is abelian.

3. C

   |X| < n

4. D

   |X| divides n.

Single choice · Rings and Fields

1. A

   $\psi$ is surjective.

2. B

   If $p_{1}(x)y + p_{2}(x) \in \ker \psi$ for polynomials $p_{1}(x), p_{2}(x) \in \mathbb{C}[x]$, then both $p_{1}$ and $p_{2}$ are the zero polynomial.

3. C

   There exist non-zero polynomials $p_{1}(x), p_{2}(x) \in \mathbb{C}[x]$ such that $\psi(p_{1}(x)y + p_{2}(x)) = 0$.

4. D

   There exists $f \in \mathbb{C}[x, y]$ such that $\psi(f(x, y)) = 0$ and $\psi(f^{2}(x, y)) \ne 0$.

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 · Ordinary Differential Equations

1. A

   S is an empty set.

2. B

   S is a non-empty finite set.

3. C

   S is a countably infinite set.

4. D

   S is an uncountable set.

Single choice · Ordinary Differential Equations

1. A

   a unique solution on $\mathbb{R}$

2. B

   infinitely many solutions on the interval (−2, 2)

3. C

   a unique solution and its maximal interval of existence is $(-\infty, 1)$

4. D

   no solution on the interval (−2, 2)

Single choice · Partial Differential Equations

1. A

   no solution

2. B

   exactly one solution

3. C

   exactly two solutions

4. D

   infinitely many solutions

Single choice · Partial Differential Equations

1. A

   a = 1 and b = 0

2. B

   a = 1 and b = −1

3. C

   a = 2 and b = 1

4. D

   a = 1 and b = 2

Single choice · Numerical Analysis

1. A

   0

2. B

   1

3. C

   2

4. D

   3