CSIR NET Maths previous year question papers

Single choice

1. A

   0

2. B

   5

3. C

   7

4. D

   9

Single choice · The Real Line

1. A

   A ⊊ C and B ⊊ C

2. B

   A ⊊ C ⊊ B

3. C

   A ⊊ B ⊊ C

4. D

   A ⊊ B = C

Single choice · The Real Line

1. A

   If A ⊊ B and B is finite, then there is a one-to-one map from $S_{2}$ to $S_{1}$.

2. B

   If $B = \mathbb{N}$, then there exists a one-to-one map from $S_{2}$ to B.

3. C

   If $B = \mathbb{N}$ and A is finite, then there exists a one-to-one map from B to $S_{1}$.

4. D

   If A is finite, then $S_{2}$ is finite for any B.

Single choice · Eigenvalues and Canonical Forms

1. A

   101

2. B

   67

3. C

   103

4. D

   113

Single choice · Inner Product Spaces and Forms

1. A

   $f_{4} \in V$

2. B

   {$f_{1}, f_{2}, f_{3}$} is an orthonormal basis of V.

3. C

   The dimension of V is 2.

4. D

   {$f_{1}, f_{2}, f_{3}$} is an orthogonal set but not orthonormal.

Single choice · Inner Product Spaces and Forms

1. A

   Q is a positive definite quadratic form on V.

2. B

   Q takes every positive real value.

3. C

   Q(x) = 2.

4. D

   For all f, g ∈ V, Q(f + g) = Q(f) + Q(g).

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 · Linear Integral Equations

1. A

   y(0) + y(1) = 1/2

2. B

   y(−1) + y(1) = 1

3. C

   y′(0) + y′(1) = 3/2

4. D

   y′(−1) + y′(1) = 3

Single choice · Limit Theorems and Markov Chains

1. A

   $lim_{n}\to\infty P(X_{n} = 0) = 1/2026$

2. B

   $lim_{n}\to\infty P(X_{n} = 0) = 1/2025$

3. C

   $lim_{n}\to\infty P(X_{n} = 0) = 1/2024$

4. D

   $lim_{n}\to\infty P(X_{n} = 0) = 1/2023$

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

   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 · The Real Line

1. A

   There is a bijective map from $\mathbb{N}$ to S.

2. B

   There is a surjective map from S onto (0,1).

3. C

   There is a bijective map from $\mathbb{N}$ to $(0,1)\S$.

4. D

   S is a countable set and $(0,1)\S$ is uncountable.

Single choice · Inner Product Spaces and Forms

1. A

   There exists $(a,b,c) \in \mathbb{R}^{3}${(0,0,0)} such that f(a,b,c) = 0.

2. B

   If there exists $(a,b,c) \in \mathbb{R}^{3}${(0,0,0)} such that f(a,b,c) = 0, then $(a,b,c) \in \mathbb{Q}^{3}$.

3. C

   {$(a,b,c) \in \mathbb{Z}^{3}$ ∣ f(a,b,c) = 0} is a finite set.

4. D

   If there exists $(a,b,c) \in \mathbb{Q}^{3}${(0,0,0)} such that f(a,b,c) = 0, then A has a positive eigenvalue and a negative eigenvalue.

Single choice · Hypothesis Testing

1. A

   8/25

2. B

   13/25

3. C

   17/25

4. D

   22/25

Single choice · The Real Line

1. A

   The set of all finite subsets of $\mathbb{Z}$ is uncountable.

2. B

   Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous and one-one function. If $S \subseteq \mathbb{R}$ is a countably infinite set, then f(S) is a countably infinite set.

3. C

   Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. Then there exists an uncountable subset $T \subseteq \mathbb{R}$ such that $f(T) \subseteq \mathbb{R}$ is uncountable.

4. D

   Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. If R ⊆ image(f) is a countable set, then f⁻$^{1}(R)$ is countable.

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

1. A

   If A is invertible, then AB and BA have the same minimal polynomial.

2. B

   If 0 is an eigenvalue of A, then 0 is an eigenvalue of AB.

3. C

   If 0 is the only eigenvalue of A and of B, then 0 is the only eigenvalue of AB.

4. D

   If AB and BA have the same minimal polynomial, then either A or B is invertible.

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

1. A

   (ST − TS$)^{2} = \lambda{}I$ for some $\lambda \in \mathbb{C}$.

2. B

   The characteristic polynomial of (ST − TS$)^{2}$ is $(x - \lambda)^{2}$ for some $\lambda \in \mathbb{C}$.

3. C

   If ST − TS has only one eigenvalue, then ST − TS $= \lambda{}I$ for some $\lambda \in \mathbb{C}$.

4. D

   If ST − TS has only one eigenvalue, then (ST − TS$)^{2}$ is the zero transformation.

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

1. A

   The number of two dimensional subspaces of V is 13.

2. B

   The number of surjective linear transformations from V to W is 624.

3. C

   The number of one dimensional subspaces of V is 13.

4. D

   The number of linear transformations from V to W is $3^{6}$.

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 · Ordinary Differential Equations

1. A

   $\varphi_{1}(x + 2\pi)$ is also a solution of ODE

2. B

   $\varphi_{2}(x + 4\pi)$ is also a solution of ODE

3. C

   There are NO constants a, b such that $\varphi_{2}(x + 4\pi) = a\varphi_{1}(x) + b\varphi_{2}(x)$

4. D

   There exist $a, b \in \mathbb{R}$ such that $\varphi_{1}(x + 2\pi) = a\varphi_{1}(x) + b\varphi_{2}(x)$

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

1. A

   There exists a q such that cos(x) and e^x cos(x) are solutions of ODE

2. B

   There exists a q such that sin(x) and cos(x) are solutions of ODE

3. C

   There exists a q such that e^x sin(x) and e^x cos(2x) are solutions of ODE

4. D

   There exists a q such that xe^x and x(x − 1)e^x are solutions of ODE

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

1. A

   The probability density function of S is f_S(s) = $(1/2)e^{-s/2}$ if s > 0, and 0 otherwise.

2. B

   The probability density function of T is f_T(t) = 1 if 0 < t < 1, and 0 otherwise.

3. C

   Var(S) = 2.

4. D

   E(T) = 2/3.

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

1. A

   $limsup_{n\to\infty}$ $t_{n} \le$ $limsup_{n\to\infty}$ $s_{n}$

2. B

   $limsup_{n\to\infty}$ $t_{n} \le e$

3. C

   $liminf_{n\to\infty}$ $s_{n} \ge e^{2}$

4. D

   $liminf_{n\to\infty}$ $t_{n} \ge e$