CSIR NET Maths previous year question papers

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

1. A

   Let f(X) ∈ 𝔽_p[X] and $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽$_p(\alpha)$ is the splitting field of f in 𝔽̄_p.

2. B

   Let f, g ∈ 𝔽_p[X] be irreducible polynomials of same degree and $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽$_p(\alpha)$ is the splitting field of g in 𝔽̄_p.

3. C

   𝔽_p[X] has infinitely many irreducible polynomials.

4. D

   The set {a + b | a, b ∈ 𝔽_p} is contained in {$a^{2} + b^{2}$ | a, b ∈ 𝔽_p}.

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

1. A

   $x^{2} +$ xy$^{2} - x - y + 1$ is irreducible in $\mathbb{Q}[x,y]$.

2. B

   $x^{3} + 100x^{2} + 25x + 10$ is irreducible in $\mathbb{Z}[x]$.

3. C

   $2x^{2} + 3$xy $+ y^{2} + 3x + 2y + 1$ is irreducible in $\mathbb{Q}[x,y]$.

4. D

   $x^{4} + 4x^{2} + 3$ is irreducible in $\mathbb{Z}[x]$.

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

1. A

   If there are no elements of order 4 in G, then G is abelian.

2. B

   If there are exactly two elements of order 4 in G, then G is abelian.

3. C

   If there are exactly six elements of order 4 in G, then G is abelian.

4. D

   There is an element of order 4 in G.

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

1. A

   X is metrizable.

2. B

   X is compact.

3. C

   X is Hausdorff.

4. D

   X is connected.

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

1. A

   $(\mathbb{R}, \tau_E)$ is a normal space.

2. B

   Let $\tau$ be a topology on $\mathbb{R}$. If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau_E)$ to $(\mathbb{R},\tau)$, then $(\mathbb{R},\tau)$ is a regular space.

3. C

   Let $\tau$ be a topology on $\mathbb{R}$. If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau)$ to $(\mathbb{R},\tau_E)$, then $(\mathbb{R},\tau)$ is a Hausdorff space.

4. D

   $\mathbb{R}$ is a regular space in the finite-complement topology.

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

1. A

   R has a unique maximal ideal.

2. B

   Every finitely generated ideal of R is principal.

3. C

   R is an integral domain.

4. D

   R is a euclidean domain.

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

1. A

   (P) is true.

2. B

   (P) is FALSE.

3. C

   (Q) is true.

4. D

   (Q) is FALSE.

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

1. A

   The function g has a pole at 0.

2. B

   If $g(\alpha)=0$, then |$\alpha$|≠1.

3. C

   The function g has only finitely many zeros.

4. D

   $max_{|z|=1}$ |g(z)| ≥ 1

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

1. A

   |f(−1/5)| ≤ 1/5

2. B

   |f(−1/5)| ≤ 1/3

3. C

   |f′(−1/2)| ≤ 1/2

4. D

   |f′(−1/2)| ≤ 4/3

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

1. A

   $\int_\gamma1$ dz/(z sin z) = 0

2. B

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

3. C

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

4. D

   $\int_\gamma2$ dz/(z sin z) = 0

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 · Groups

1. A

   If d is a positive integer that divides |G|, then G has a subgroup of order d.

2. B

   The map f : G×G → G given by f(a,b)=ab is not a group homomorphism.

3. C

   Suppose that for every positive integer d that divides |G|, there exists a subgroup of G of order d. Then G has at least three normal subgroups.

4. D

   |G| ≠ 16

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.

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

Single choice · Analytic Functions

1. A

   f can be extended to $\mathbb{C}$ as an entire function.

2. B

   f must have infinitely many zeros in 𝔻.

3. C

   f is not a polynomial.

4. D

   exp(f) cannot take every complex value.

Single choice · Cauchy Theory

1. A

   $\lim(y\to+\infty) f'($iy) = 0

2. B

   $\lim(y\to+\infty) f'($iy) is a complex number with absolute value 1.

3. C

   $\lim(y\to+\infty)$ |f′(iy)| $= +\infty$

4. D

   $\lim(y\to+\infty) f'($iy) is not a real number.

Single choice · Singularities and Residues

1. A

   I_(m,n) = 1 if m = n

2. B

   I_(m,n) = 1 if m + 1 = n

3. C

   I_(m,n) = 1 if m = n + 1

4. D

   I_(m,n) = 1 if m = n + 2