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.

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

1. A

   If f(z) = f(z + 1) for all $z \in \mathbb{C}$ then f is a constant function.

2. B

   If f(z) = f(z + 1) = f(z + i) for all $z \in \mathbb{C}$ then f is a constant function.

3. C

   If f(1/z) has a removable singularity at 0 then f is a constant function.

4. D

   If f is a non-constant function then f(1/z) has a pole at 0.

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

1. A

   $lim_{z\to0}$ f(z) does not exist.

2. B

   $lim_{z\to0}$ f(z) exists and has absolute value ≤ 1.

3. C

   $lim_{z\to0}$ f(z) = 0.

4. D

   There exists $\theta \in \mathbb{R}$ such that f(z) = $e^{i\theta}z$ for all z ∈ 𝔻^×.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   A is nonempty.

2. B

   A is finite.

3. C

   A is infinite.

4. D

   A is uncountable.

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

1. A

   I = 0

2. B

   $(1/2\pi{}i)I \in$ {$4n : n \in \mathbb{Z}, n \ge 1$}

3. C

   $I = 2\pi{}i \sum_{n=0}^\infty$ 1/n!

4. D

   $I = 2\pi{}i \sum_{n=0}^\infty$ 1/(n!(n + 1)!)

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

1. A

   G = GL$_{2}($𝔽_p) and H is a subgroup of order p.

2. B

   G = SL$_{2}($𝔽_p) and H is a subgroup of order p.

3. C

   p ≡ 3 (mod 4), G = GL$_{2}($𝔽_p)/SL$_{2}($𝔽_p) and H is a subgroup of order 2.

4. D

   p ≡ 1 (mod 4), G = GL$_{2}($𝔽_p)/SL$_{2}($𝔽_p) and H is a subgroup of order 2.

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

1. A

   If $G_{1}, G_{2}$ are two groups such that Aut$(G_{1})$ is isomorphic to Aut$(G_{2})$, then $G_{1}$ is isomorphic to $G_{2}$.

2. B

   If |G| = 2, then Aut(G × G) is abelian.

3. C

   If G is the group of complex numbers under addition, then Aut(G) is abelian.

4. D

   If G is finite, then Aut(G) is finite.

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

1. A

   If $G_{1}$ is normal in G, then $(G_{2}G_{1})/G_{1} \cong G_{2}/(G_{2} \cap G_{1})$.

2. B

   If $H_{1}$ and $H_{2}$ are normal subgroups of $G_{1}$ and $G_{2}$, respectively, then $(G_{1} \times G_{2})/(H_{1} \times H_{2}) \cong (G_{1}/H_{1}) \times (G_{2}/H_{2})$.

3. C

   If $G_{1}$ is normal in $G_{2}$ and $G_{2}$ is normal in G, then $G_{1}$ is normal in G.

4. D

   Every subgroup of prime index in G is normal.

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

1. A

   If f is onto, then so is $f^{n}$ for all n ≥ 1.

2. B

   ker $f^{n+1}$ $= \ker f^{n}$ for some n ≥ 1.

3. C

   If f is onto, then f is one-to-one.

4. D

   If f is one-to-one, then f is onto.

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

1. A

   f(X) is irreducible in $\mathbb{Q}[X]$, but g(X) is not.

2. B

   g(X) is irreducible in $\mathbb{Q}[X]$, but f(X) is not.

3. C

   Both f(X) and g(X) are irreducible in $\mathbb{Q}[X]$.

4. D

   Neither f(X) nor g(X) is irreducible in $\mathbb{Q}[X]$.