CSIR NET Maths previous year question papers

Single choice · Ordinary Differential Equations

1. A

   a unique solution on $\mathbb{R}$

2. B

   infinitely many solutions on the interval (−2, 2)

3. C

   a unique solution and its maximal interval of existence is $(-\infty, 1)$

4. D

   no solution on the interval (−2, 2)

Single choice · Calculus of Variations

1. A

   5/8

2. B

   3/4

3. C

   3/8

4. D

   5/4

Single choice · Limit Theorems and Markov Chains

1. A

   33/169

2. B

   34/169

3. C

   35/169

4. D

   36/169

Single choice · Topology

1. A

   {m + ne$^(2\pi{}i/3)$ | $m, n \in \mathbb{Z}$} is a dense subset of $\mathbb{C}$.

2. B

   Open connected subsets of $\mathbb{R}^{3}$ need not be path-connected.

3. C

   Let X be a topological space and $p : X \to \mathbb{R} a$ continuous surjective open map. If p⁻$^{1}(${$\alpha$}) is connected for every $\alpha \in \mathbb{R}$, then X must be connected.

4. D

   Compact subsets of any infinite topological space are closed.

Single choice · Inner Product Spaces and Forms

1. A

   A is invertible

2. B

   There exist non-zero vectors v, w such that ⟨v, w⟩ = 0

3. C

   ⟨u, v⟩ ≠ ⟨u, w⟩ for all non-zero vectors u, v, w with v ≠ w

4. D

   Every eigenvalue of $A^{2}$ is positive

Single choice · Cauchy Theory

1. A

   If Re(f), Im(f) are bounded then f is constant

2. B

   If e^(|Re(f)| + |Im(f)|) is bounded, then f is constant

3. C

   If the sum Re(f) + Im(f) and the product Re(f)Im(f) are bounded, then f is constant

4. D

   If sin(Re(f) + Im(f)) is bounded, then f is constant

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   S is empty.

2. B

   S is non-empty.

3. C

   f must be uniformly continuous.

4. D

   f need not be uniformly continuous.

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

1. A

   $f(v_{2}, v_{1}) = -s$.

2. B

   If r = s = t, then either f(v, v) ≥ 0 for all v ∈ V or f(v, v) ≤ 0 for all v ∈ V.

3. C

   If f is positive definite, then r ≠ s, s ≠ t and r ≠ t.

4. D

   If f is positive definite, then $f(v_{1}, v_{2}) = 0$.

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

1. A

   If p divides |G|, then E(p) is a subgroup of G.

2. B

   For all p, E(p) is a subgroup of G.

3. C

   If E(p) is a subgroup of G, then p divides |G|.

4. D

   If G is abelian, then E(p) is a subgroup of G.

Multi-select · full marks only for the exact set · Classical Mechanics

1. A

   |$\theta$̇| $= \sqrt{2/(g\alpha)}$

2. B

   |$\theta$̇| $= \sqrt{1/(g\alpha)}$

3. C

   |$\theta$̇| $= \sqrt{2g\alpha}$

4. D

   |$\theta$̇| $= \sqrt{g\alpha}$

Multi-select · full marks only for the exact set · Limit Theorems and Markov Chains

1. A

   The state space can be partitioned into exactly two equivalence classes.

2. B

   States 0 and 2 are recurrent.

3. C

   States 1 and 3 are recurrent.

4. D

   State 4 is transient.

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

1. A

   $\theta$̂ ∈ (0.0, 0.2)

2. B

   $\theta$̂ ∈ (0.1, 0.3)

3. C

   $\theta$̂ ∈ (0.2, 0.4)

4. D

   $\theta$̂ ∈ (0.3, 0.5)

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   The function h_(a,b) is differentiable on $\mathbb{R}$ for each $(a, b) \in \mathbb{R}^{2}$.

2. B

   There exists $(a, b) \in \mathbb{R}^{2}$ such that h_(a,b) is not differentiable at t = 0.

3. C

   The function f is differentiable at the point (0, 0).

4. D

   If the function f is differentiable at (0, 0) then g is identically zero.

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

1. A

   If K is the splitting field of a non-constant polynomial over $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

2. B

   If K is a normal extension of $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

3. C

   If K is the set of all the roots of the polynomial $x^{121} - x$ in an algebraic closure of 𝔽$_{11}$, then K is Galois over 𝔽$_{11}$, where 𝔽$_{11}$ is the field with 11 elements.

4. D

   If $K = \mathbb{Q}(2^(1/13), \zeta_{13})$, where $\zeta_{13}$ is a primitive 13th root of unity in $\mathbb{C}$, then K is Galois over $\mathbb{Q}$.

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

1. A

   There exists a subspace W of $\mathbb{C}^{n}$ such that $\mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

2. B

   There exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

3. C

   Suppose that there exists a positive integer k such that Tᵏ is the identity map. Then there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.

4. D

   Suppose that there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}. Then there exists a positive integer k such that Tᵏ is the identity map.

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

1. A

   S is a basis of W.

2. B

   S is an orthonormal basis of W.

3. C

   There exist f, g ∈ S such that ⟨f, g⟩ = 0.

4. D

   S contains an orthonormal basis of W.

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

1. A

   f is constant.

2. B

   f has removable singularities at −1 and 1.

3. C

   f is bounded.

4. D

   f has either poles or essential singularities at −1 and 1.

Multi-select · full marks only for the exact set · Linear Programming

1. A

   The optimum value is 3

2. B

   The optimum value is 3/2

3. C

   (0, 2, 1) is an extreme point of the feasible region

4. D

   (1/2, 0, 1) is the optimal solution

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

1. A

   f(z) = 2024 for all $z \in \mathbb{C}$

2. B

   f is a constant function

3. C

   f is an injective function

4. D

   f is a bijective function

Multi-select · full marks only for the exact set · Calculus of Variations

1. A

   $(e/(1 + e^{2}))(e$ˣ + e⁻ˣ$) + x^{2} - 1$

2. B

   $(e/(1 + e^{2}))(e$ˣ + e⁻ˣ$) + 1 - x^{2}$

3. C

   $(e/(1 + e^{2}))(e$ˣ + e⁻ˣ)

4. D

   $(e/(1 + e^{2}))(e$ˣ + e⁻ˣ$) + \sin(2\pi{}x)$

Single choice · The Real Line

1. A

   The series is divergent.

2. B

   The series is convergent.

3. C

   The series is conditionally convergent.

4. D

   The series is absolutely convergent.

Single choice · Calculus of Variations

1. A

   (P) has no stationary function (extremal).

2. B

   y ≡ 0 is the only stationary function (extremal) for (P).

3. C

   (P) has a unique stationary function (extremal) y not identically equal to 0.

4. D

   (P) has infinitely many stationary functions (extremals).

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   There is no continuous real-valued function g defined on any interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

2. B

   There is exactly one continuous real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

3. C

   There is exactly one differentiable real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

4. D

   There are two distinct differentiable real-valued functions g on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

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

1. A

   n⁻$^{1} \sum_{i=1}^{n}$ $X_{i}$

2. B

   n⁻$^{1} \sum_{i=1}^{2n}$ $X_{i}$

3. C

   n⁻$^{1} \sum_{i=1}^{n}$ $X_{2i}$₋$_{1}$

4. D

   n⁻$^{1}(\sum_{i=1}^{n}$ $X_{i} + W)$

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

1. A

   $\int$₋$_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0.

2. B

   $\int$₋$_{1}^{1} f(x)p(x)$ dx = 0 for all real polynomials p(x).

3. C

   $\int$₋$_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 odd.

4. D

   $\int$₋$_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 even.