CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   The transposition (1 2) belongs to G.

2. B

   |G| is divisible by p.

3. C

   A p-cycle belongs to G.

4. D

   $G = S_{p}$

Multi-select · full marks only for the exact set · Cauchy Theory

1. A

   f is a real valued function.

2. B

   f(z) = 1 for all $z\in\mathbb{C}$.

3. C

   f is not an entire function.

4. D

   f has finitely many zeros on the imaginary axis.

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

1. A

   {$x \in \mathbb{R}$ | [x] = 1}

2. B

   {$x \in \mathbb{R}$ | x − [x] = 1/2}

3. C

   {$x \in \mathbb{R}$ | $x \ge 0, \sqrt{x} \in \mathbb{Q}$}

4. D

   {$x \in \mathbb{R}$ | $x^{2} \in \mathbb{Q}$}

Multi-select · full marks only for the exact set · Zeros and Mappings

1. A

   The series $\sum(n\ge1) f_{n}$ converges only at z = 0.

2. B

   The series $\sum(n\ge1) f_{n}$ converges pointwise only on a countable set E ⊆ 𝔻 but not on 𝔻 \ E.

3. C

   The series $\sum(n\ge1) f_{n}$ converges pointwise at all points of 𝔻 but not uniformly on some compact subsets of 𝔻.

4. D

   The series $\sum(n\ge1) f_{n}$ converges uniformly on all compact subsets of 𝔻 to a holomorphic function on 𝔻.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   $\mathbb{Q}[x]/(x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$

2. B

   $\mathbb{Q}[x]/(x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$

3. C

   $\mathbb{Z}[x]/(x - 101)$

4. D

   $\mathbb{Q}[x]/(x^{43} - 41x + 41)$

Multi-select · full marks only for the exact set · Metric Spaces

1. A

   d_X is a metric on X which induces the topology $\tau$.

2. B

   The topological space $(X, \tau)$ is connected.

3. C

   $(P_{n})_{n}\ge_{1}$ converges to $P_{0}$ in the topological space $(X, \tau)$.

4. D

   The topological space $(X, \tau)$ is compact.

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

1. A

   P(XY = 0) = (4 + 5e⁻$^{1})/9$

2. B

   $E[(3X - Y)^{2}] = 6$

3. C

   Var(X + Y) = 2

4. D

   Cov(2X + Y, X − 2Y) = −10/9

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   The ring of polynomials in 3 variables over $\mathbb{R}$.

2. B

   The ring of complex analytic functions on the open unit disc in $\mathbb{C}$.

3. C

   The ring of 2 × 2 real matrices.

4. D

   The ring of continuous functions from [0, 1] to $\mathbb{R}$.

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

1. A

   Any group with p(p + 2) elements, where p and p + 2 are both prime numbers, has property (J).

2. B

   Any finite abelian group has property (J).

3. C

   The symmetric group $S_{3}$ has property (J).

4. D

   Any group with 77 elements has property (J).

Multi-select · full marks only for the exact set · Linear Integral Equations

1. A

   R(x, t) = 1

2. B

   R(x, t) = e^(t−x)

3. C

   u(ln 8) = 10

4. D

   u(ln 7) = 9

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

1. A

   f(0) = 0

2. B

   f′(0) exists and f′(0) = 0.

3. C

   f″(0) exists and f″(0) = g(0).

4. D

   f″(0) exists but f″(0) ≠ g(0).

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

1. A

   f is Riemann integrable on [0, 1].

2. B

   The set of discontinuities of f cannot contain a non-empty open set.

3. C

   f is Lebesgue measurable function.

4. D

   f is Borel measurable function.

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 · Determinants and Matrix Tricks

1. A

   If A and B are upper triangular, then M is diagonalizable over $\mathbb{R}$.

2. B

   If A and B are diagonalizable over $\mathbb{R}$, then M is diagonalizable over $\mathbb{R}$.

3. C

   If A and B are diagonalizable over $\mathbb{R}$, then there exists $\lambda \in \mathbb{R}$ such that $M = \lambda{}I_{2}$.

4. D

   There exists $\lambda \in \mathbb{R}$ such that $M^{2} = \lambda{}I_{2}$.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   R is never an integral domain.

2. B

   r = −r for all r ∈ R.

3. C

   Every nonzero prime ideal of R is maximal.

4. D

   R must be a commutative ring.

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   P⁽$^{2}$⁾$_{00} = 3/4$

2. B

   P⁽$^{3}$⁾$_{10} = 39/64$

3. C

   The stationary probability that the chain is in state 2 is 2/7.

4. D

   State 1 is transient.

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

1. A

   $(X_{(1)} - (2/n)\ln 5, X_{(1)})$ is a 97% confidence interval for $\theta$.

2. B

   $(X_{(1)} - (2/n)\ln 5, X_{(1)})$ is a 96% confidence interval for $\theta$.

3. C

   $(X_{(1)} - (1/n)\ln 20, X_{(1)})$ is a 95% confidence interval for $\theta$.

4. D

   $(X_{(1)} - (1/n)\ln 20, X_{(1)})$ is a 96% confidence interval for $\theta$.

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

1. A

   Let $x, y \in \mathbb{R}$ with x < y. Then there exists $r \in \mathbb{Q}$ such that x < (2^2024/e)r < y.

2. B

   Let $(a_{n})_{n}\ge_{2}$ be a sequence of positive real numbers. If there exists a positive real number L such that $\lim \sup(n\to\infty) a_{n}/\log n = L$, then $\lim \sup(n\to\infty) a_{n} < \infty$.

3. C

   The set of all finite subsets of $\mathbb{Q}$ is countably infinite.

4. D

   The set of continuous functions from $\mathbb{R}$ to the set {0, 1} is infinite.

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

1. A

   lim(x→0) f(x) exists.

2. B

   f is continuous at 0.

3. C

   f is differentiable at 0.

4. D

   lim(x→0) f′(x) does not exist.

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

1. A

   f is continuous at (0, 0).

2. B

   The partial derivatives $\partial{}f/\partial{}x$ and $\partial{}f/\partial{}y$ exist at (0, 0).

3. C

   f is differentiable 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 differentiable on $\mathbb{R}^{2}$ \ {(0, 0)}.

2. B

   All the directional derivatives of f exist at (0, 0).

3. C

   f is differentiable on $\mathbb{R}^{2}$.

4. D

   f is not continuous at (0, 0).

Multi-select · full marks only for the exact set · Zeros and Mappings

1. A

   S_R is an open subset of $\mathbb{C}$.

2. B

   S_R is a bounded subset of $\mathbb{C}$.

3. C

   |P(z)| = R for every z on the boundary of S_R.

4. D

   Every connected component of S_R contains a zero of P(z).

Multi-select · full marks only for the exact set · Singularities and Residues

1. A

   f(0) = 0

2. B

   f(z) = 0 for all z ∈ 𝔻.

3. C

   There exists a nonzero constant c such that f(z) = ce^(−1/z) for all z ∈ 𝔻 \ {0}.

4. D

   There exists a nonzero constant c and a positive integer n such that f(z) = cz$^{n}e^(-1/z)$ for all z ∈ 𝔻 \ {0}.

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

1. A

   $\mathbb{Q}$ with ordinary addition

2. B

   $\mathbb{C}$ \ {0} with ordinary multiplication

3. C

   The cyclic group of order 5

4. D

   The symmetric group $S_{5}$

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

1. A

   The set of all irrational numbers is dense in $(\mathbb{R}, \tau)$.

2. B

   For each prime number p, the set {$0, \sqrt{p}$} is dense in $(\mathbb{R}, \tau)$.

3. C

   [0, 1] is compact in $(\mathbb{R}, \tau)$.

4. D

   $(\mathbb{R}, \tau)$ is Hausdorff.