CSIR NET Maths previous year question papers

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 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 · Probability

1. A

   Posterior mean = Maximum likelihood estimate of $\theta$, Posterior variance = Var(X|$\theta)$

2. B

   Posterior mean = Maximum likelihood estimate of $\theta$, Posterior variance < Var(X|$\theta)$

3. C

   Posterior mean > Maximum likelihood estimate of $\theta$, Posterior variance = Var(X|$\theta)$

4. D

   Posterior mean > Maximum likelihood estimate of $\theta$, Posterior variance < Var(X|$\theta)$

Single choice · Sampling and Design of Experiments

1. A

   ȳ

2. B

   (2ȳ_S + 3ȳ_B)/5

3. C

   (4ȳ_S + 5ȳ_B)/9

4. D

   (ȳ_S/12 + ȳ_B/15)/(1/12 + 1/15)

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 · Probability

1. A

   (X−1)/|Y|

2. B

   (X+2Y+Z)/(X−2Y+Z)

3. C

   (Z−1)/Y

4. D

   (X−Y)/(X+Y)

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 · Continuity and Differentiation

1. A

   For every bounded subset $A \subseteq \mathbb{R}, f$⁻$^{1}(A)$ is a bounded subset of $\mathbb{R}$.

2. B

   For every Cauchy sequence $(x_{n})_{n}\ge_{1}$ in $\mathbb{R}, (f(x_{n}))_{n}\ge_{1}$ is a Cauchy sequence in $\mathbb{R}$.

3. C

   There exists $x \in \mathbb{R}$ such that f(x) = x.

4. D

   There exists $x \in \mathbb{R}$ such that f(x) = 0.

Single choice · Inner Product Spaces and Forms

1. A

   {$(-1/\sqrt{2}, 0, 0, -1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)$}

2. B

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)$}

3. C

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)$}

4. D

   {$(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (1/\sqrt{6}, -\sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)$}

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

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).