CSIR NET Maths previous year question papers

Single choice · Cauchy Theory

1. A

   The function zf(1/z) → 0 as |z| $\to \infty$.

2. B

   The function f is constant.

3. C

   c = 0.

4. D

   c > 0.

Single choice · Cauchy Theory

1. A

   |f(z)| $\to \infty$ as |z| $\to \infty$.

2. B

   f(z)/z → 0 as |z| $\to \infty$.

3. C

   zf(z) → 0 as |z| $\to \infty$.

4. D

   f(z) → 0 as |z| $\to \infty$.

Single choice · Zeros and Mappings

1. A

   X is the line segment joining −1 and −i.

2. B

   X = {$e^{i\theta}$ | $\theta \in [\pi, 3\pi/2]$}.

3. C

   X is the line segment joining −1 to 1.

4. D

   X = {$e^{i\theta}$ | $\theta \in [-\pi/2, \pi]$}.

Single choice · Analytic Functions

1. A

   There exists an entire function f such that f⁽$^{n}$⁾$(0) = n!/n^{n}$ for all positive integers n.

2. B

   There exists an entire function f such that f⁽$^{n}$⁾$(0) = n!n^{n}$ for all positive integers n.

3. C

   There exists an entire function f such that f⁽$^{n}$⁾(0) = (n − 1)! for all positive integers n.

4. D

   There exists an entire function f such that f⁽$^{n}$⁾(0) = n!n for all positive integers n.

Single choice · Groups

1. A

   p ∤ 1 + (p − 1)! for some odd prime p.

2. B

   p | $(1234)^{p-1}$ − 1 for all primes p > 700.

3. C

   There exist $a \in \mathbb{Z}$ and a prime p > 11 such that p ∤ aᵖ − a.

4. D

   p ∤ $(p^{2})!/(p!)^{2}$ for some odd prime p.

Single choice · Rings and Fields

1. A

   The set {a ∈ A : 1/a ∉ A} ∪ {0} is an additive subgroup of $\mathbb{Q}$.

2. B

   A has at most one maximal ideal.

3. C

   If $A \ne \mathbb{Q}$, then A has infinitely many prime ideals.

4. D

   For any nonzero a, b ∈ A, a divides b or b divides a in A.

Single choice · Rings and Fields

1. A

   The ideal $2\mathbb{Z}[i]$ is maximal in $\mathbb{Z}[i]$.

2. B

   The ideal $X\mathbb{C}[X, Y]$ is maximal in $\mathbb{C}[X, Y]$.

3. C

   The set of all polynomials in $\mathbb{C}[X]$ whose coefficients add up to 0 is a maximal ideal in $\mathbb{C}[X]$.

4. D

   The ideal $(\sqrt{2} - 1)\mathbb{Z}[\sqrt{2}]$ is maximal in $\mathbb{Z}[\sqrt{2}]$.

Single choice · Topology

1. A

   25

2. B

   120

3. C

   24

4. D

   6

Single choice · Groups

1. A

   The number of pairs $(\sigma, x) \in G \times X$ such that $\sigma(x) = x$ is strictly greater than $\sum_{x}$ |S_G(x)|.

2. B

   The number (1/|G|$)\cdot\sum_{x}$ |S_G(x)| is always an integer.

3. C

   For all x ∈ X, S_G(x) is a normal subgroup of G.

4. D

   For all x, y ∈ X, S_G(x) is isomorphic to S_G(y).

Single choice · Topology

1. A

   If X × Y is Hausdorff, then the statement S is true.

2. B

   If Y is connected, then the statement S is true.

3. C

   If X × Y is regular, then the statement S is true.

4. D

   If Y is compact, then the statement S is true.

Single choice · Analytic Functions

1. A

   X is uncountable.

2. B

   X is infinite and countable.

3. C

   Every element of X is an open map.

4. D

   X has exactly one element.

Single choice · Groups

1. A

   Let G and G′ be finite groups such that S(G) = S(G′). Then G is isomorphic to G′.

2. B

   If S(G) = 4, then |G| = pᵐ for some prime number p and positive integer m.

3. C

   If S(G) = 5, then G is a cyclic group.

4. D

   For every positive integer n, there exists a finite group G such that S(G) = n.

Single choice · Singularities and Residues

1. A

   The function f(1/z) has a removable singularity at 0.

2. B

   The function f(1/z) has an essential singularity at 0.

3. C

   If $\int_{\gamma{}R}$ f(z)dz = 0 for some R > 0, then S is not a singleton set.

4. D

   If S is not a singleton set, then $\int_{\gamma{}R}$ f(z)dz = 0 for some R > 0 such that S ∩ {$z \in \mathbb{C}$ : |z| = R} = ∅.

Single choice · Singularities and Residues

1. A

   For all z with |z| ≥ 1, the sequence {$f_{n}(z)$}$_{n}\ge1$ converges.

2. B

   For all z ∈ 𝔻{0}, the sequence {$f_{n}(z)$}$_{n}\ge1$ converges.

3. C

   The sequence {$f_{n}$}$_{n}\ge1$ converges pointwise to a holomorphic function on 𝔻[0,1).

4. D

   The sequence {$f_{n}$}$_{n}\ge1$ converges pointwise to a holomorphic function on {$z \in \mathbb{C}$ : |z| > 1}.

Single choice · Singularities and Residues

1. A

   $\int_{\gamma1}$ tan(z)dz $= 2\pi{}i$

2. B

   $\int_{\gamma1}$ tan(z)dz $= -2\pi{}i$

3. C

   $\int_{\gamma2}$ tan(z)dz $= 2\pi{}i$

4. D

   $\int_{\gamma2}$ tan(z)dz $= -2\pi{}i$

Single choice · Groups

1. A

   There exists a finite group G such that |H_G| is even.

2. B

   There exists a finite group G such that |H_G| = 4n+1 for some n ≥ 3.

3. C

   For every finite group G, there exists a non-negative integer n such that |H_G| = 4n+1.

4. D

   For every finite group G, there exists a non-negative integer n such that |H_G| = 4n+3.

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 · Groups

1. A

   $10^{7}$

2. B

   10!/7!

3. C

   10!/3!

4. D

   $7^{10}$

Single choice · Analytic Functions

1. A

   0

2. B

   (e + 1/e)(cos 1)/2

3. C

   (e − 1/e)(cos 1)/2

4. D

   1