CSIR NET Maths previous year question papers

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

1. A

   $A_{n}$ is finite for every integer n ≥ 1.

2. B

   $A_{n}$ is finite for some integer n ≥ 1.

3. C

   $A_{n}$ is infinite for some integer n ≥ 1.

4. D

   $A_{n}$ is countable (finite or infinite) for every integer n ≥ 1.

Multi-select · full marks only for the exact set · Vector Spaces and Linear Maps

1. A

   The rank of T is equal to 2.

2. B

   Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$. Then $T^{2}$ must be zero.

3. C

   Suppose that for every vector $v \in \mathbb{R}^{5}$ there exists n such that $T^{n}v = 0$. Then $T^{3}$ must be zero.

4. D

   (−2, −8, 3, 2, 0) is contained in the null space of T.

Multi-select · full marks only for the exact set · Vector Spaces and Linear Maps

1. A

   V is the union of 2 proper subspaces.

2. B

   V is the union of 3 proper subspaces.

3. C

   V has a unique basis.

4. D

   V has precisely two bases.

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   a + b > 8.

2. B

   a + b < 7.

3. C

   a + b is divisible by 2.

4. D

   a > b.

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   X is invertible.

2. B

   X + I is invertible.

3. C

   XY = YX.

4. D

   Y is invertible.

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   A is diagonalizable.

2. B

   If Aᵏ = I for some positive integer k, then $A^{2} = I$.

3. C

   If Aᵏ = 0 for some positive integer k, then $A^{2} = 0$.

4. D

   All eigenvalues of A are real.

Multi-select · full marks only for the exact set · Eigenvalues and Canonical Forms

1. A

   The characteristic polynomial of A is $x^{3}(x - 1)(x - 2\pi{}i)^{3}$.

2. B

   The minimal polynomial of A is $x^{2}(x - 1)(x - 2\pi{}i)^{3}$.

3. C

   The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 3 respectively.

4. D

   The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 2 respectively.

Multi-select · full marks only for the exact set · Inner Product Spaces and Forms

1. A

   A is positive definite.

2. B

   $A^{2}$ is positive definite.

3. C

   There exists a nonzero $v \in \mathbb{R}^{3}$ such that ⟨v, v⟩ = 0.

4. D

   rank A = 2.

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

1. A

   R is isomorphic to a subring of $\mathbb{C}$.

2. B

   For any prime number $p \in \mathbb{Z}$, the ideal generated by $\psi(p)$ is a proper ideal of R.

3. C

   R has infinitely many prime ideals.

4. D

   The ideal generated by $\psi(X)$ is a prime ideal in R.

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

1. A

   n ≡ 1 (mod 3).

2. B

   n ≡ 1 (mod 35).

3. C

   n ≡ 1 (mod 21).

4. D

   n ≡ 1 (mod 5).

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

1. A

   For all prime numbers p, f(X) mod p is irreducible in $(\mathbb{Z}/p\mathbb{Z})[X]$.

2. B

   There exists a prime number p such that g(X) mod p is irreducible in $(\mathbb{Z}/p\mathbb{Z})[X]$.

3. C

   g(X) is irreducible in $\mathbb{Q}[X]$.

4. D

   f(X) is irreducible in $\mathbb{Q}[X]$.

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

1. A

   a is divisible by $3^{4}$.

2. B

   a is divisible by $2^{4}$.

3. C

   a is not divisible by 48.

4. D

   a is divisible by $3^{6}$.

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

1. A

   The Galois group of K over $\mathbb{Q}$ is the symmetric group $S_{3}$.

2. B

   The Galois group of K over $\mathbb{Q}(\omega)$ is the symmetric group $S_{3}$.

3. C

   The Galois group of K over $\mathbb{Q}$ is $\mathbb{Z}/3\mathbb{Z}$.

4. D

   The Galois group of K over $\mathbb{Q}(\omega)$ is $\mathbb{Z}/3\mathbb{Z}$.

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

1. A

   $\mathbb{Q}^{2}$ is connected.

2. B

   If A is a non-empty connected subset of $\mathbb{Q}^{2}$, then A has exactly one element.

3. C

   $\mathbb{Q}^{2}$ is Hausdorff.

4. D

   {$(x, y) \in \mathbb{Q}^{2}$ | $x^{2} + y^{2} = 1$} is compact in the subspace topology.

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

1. A

   The sequence $(p_{n})$ converges to an entire function uniformly on compact subsets of $\mathbb{C}$.

2. B

   For all $n \ge 1, p_{n}$ has a zero in the set {$z \in \mathbb{C}$ : |z| ≤ 2023}.

3. C

   There exists a sequence $(z_{n})$ of complex numbers such that lim |$z_{n}$| $= \infty$ and $p_{n}(z_{n}) = 0$ for all n ≥ 1.

4. D

   Let $S_{n}$ be the set of all zeros of $p_{n}$. If $a_{n} =$ $min_{z\in{}S_{n}}$ |z|, then $a_{n} \to \infty$ as $n \to \infty$.

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

1. A

   f = 0.

2. B

   f is a constant function.

3. C

   There exists a compact subset K of $\mathbb{C}$ such that $f^{-1}(K)$ is not compact.

4. D

   f is a polynomial.

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

1. A

   There exists a holomorphic surjective map $f : \Omega_{1} \to \Omega_{2}$.

2. B

   There exists a holomorphic surjective map $f : \Omega_{2} \to \Omega_{1}$.

3. C

   There exists a holomorphic injective map $f : \Omega_{1} \to \Omega_{2}$.

4. D

   There exists a holomorphic injective map $f : \Omega_{2} \to \Omega_{1}$.

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

1. A

   $I_{k} = 0$ for every integer k.

2. B

   $I_{k} \ne 0$ if k ≥ 1.

3. C

   |$I_{k}$| ≤ |$I_{k}$₊$_{1}$| for every integer k.

4. D

   $lim_{k\to\infty}$ |$I_{k}$| $= \infty$.

Multi-select · full marks only for the exact set · Ordinary Differential Equations

1. A

   There exists an $\alpha \in (0, \infty)$ such that $lim_{x\to\alpha^{-}}$ |y(x)| $= \infty$.

2. B

   y(x) exists on $(0, \infty)$ and it is monotone.

3. C

   y(x) exists on $(0, \infty)$, but not bounded below.

4. D

   y(x) exists on $(0, \infty)$, but not bounded above.

Multi-select · full marks only for the exact set · Ordinary Differential Equations

1. A

   $J = \mathbb{R}$

2. B

   K = (−1, 1)

3. C

   J = (−1, 1)

4. D

   K = [−1, 1]

Multi-select · full marks only for the exact set · Ordinary Differential Equations

1. A

   $\varphi(4) = 16$

2. B

   $\varphi(2) = 2$

3. C

   $\varphi(5) = 25$

4. D

   $\varphi(3) = 3$

Multi-select · full marks only for the exact set · Partial Differential Equations

1. A

   k = 0

2. B

   k = −2

3. C

   k = 4

4. D

   k = 1

Multi-select · full marks only for the exact set · Partial Differential Equations

1. A

   $S_{1} = \emptyset$

2. B

   $S_{2} = \emptyset$

3. C

   $S_{1}$ has exactly one element and $S_{2}$ has exactly two elements.

4. D

   $S_{1}$ and $S_{2}$ are both infinite.

Multi-select · full marks only for the exact set · Numerical Analysis

1. A

   −1/3

2. B

   −1/2

3. C

   5/6

4. D

   17/6

Multi-select · full marks only for the exact set · Numerical Analysis

1. A

   If $\alpha_{0} = 2$, then $\beta_{0} = 2, P = 3/4, Q = 1/4$

2. B

   If $\beta_{0} = 3$, then $\alpha_{0} = 3, P = 5/6, Q = 1/6$

3. C

   If $\alpha_{0} = 2$, then $\beta_{0} = 2, P = 1/4, Q = 3/4$

4. D

   If $\beta_{0} = 3$, then $\alpha_{0} = 3, P = 1/6, Q = 5/6$