CSIR NET Maths previous year question papers

Single choice

1. A

   Suresh

2. B

   Ramesh

3. C

   Ramesh's son

4. D

   Ramesh's cousin

Single choice

1. A

   Rahman-Violet, George-Red, Vedant-Green

2. B

   Rahman-Green, George-Violet, Vedant-Red

3. C

   Rahman-Green, George-Red, Vedant-Violet

4. D

   Rahman-Red, George-Violet, Vedant-Green

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 $= -2\pi{}i$

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

1. A

   If trace of TT* is zero, then T = 0.

2. B

   Let v ∈ V be such that T*T(v) = 0. Then T(v) = 0.

3. C

   Suppose T = T* and N > 1 be an integer. Let v ∈ V be such that $T^{2^N}(v)$ = 0. Then T(v) = 0.

4. D

   Suppose that TT* = T*T and N > 1 be an integer. Let v ∈ V be such that T^N(v) = 0. Then T(v) = 0.

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

1. A

   There is NO neighbourhood of the origin on which (CP) has a solution

2. B

   (CP) has a unique solution defined on some neighbourhood of the origin

3. C

   (CP) has a unique solution defined on some neighbourhood of the point (0, 1) in the xy-plane

4. D

   (CP) has an infinite number of solutions, each of which is defined on some neighbourhood of the origin

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

1. A

   If AAᵀ = I, then DF$_A : M_{2}(\mathbb{R}) \to S$ is surjective.

2. B

   If AAᵀ = I, then DF$_A : M_{2}(\mathbb{R}) \to S$ need not be surjective.

3. C

   If A is invertible, then DF$_A : M_{2}(\mathbb{R}) \to S$ is surjective.

4. D

   If A is not invertible, then DF$_A : M_{2}(\mathbb{R}) \to S$ is surjective.

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 · Integration

1. A

   If f∈F is a continuous function and f(n)=0 for all positive integers n, then $\lim(x\to\infty)\int_{0}$ˣf(t)dt exists in $\mathbb{R}$.

2. B

   If f∈F is uniformly continuous and $\lim(x\to\infty)\int_{0}$ˣf(t)dt exists in $\mathbb{R}$, then $\lim(x\to\infty)f(x)=0$.

3. C

   There exists a function f∈F such that $\lim(x\to\infty)\int_{0}$ˣf(t)dt exists in $\mathbb{R}$ but $\lim(x\to\infty)f(x)\ne0$.

4. D

   If f∈F is Lebesgue measurable and $\lim(x\to\infty)\int_{0}$ˣf(t)dt=0, then f=0 almost everywhere on $[0,\infty)$.

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 · Limit Theorems and Markov Chains

1. A

   $p_{0}+p_{3} = 17/29$

2. B

   $p_{0} > p_{3}$

3. C

   $p_{3}/p_{0} = 2$

4. D

   $p_{3}-p_{0} = 15/29$

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

1. A

   $E(t) \ge \pi$ for all t>0

2. B

   $E(t) \le \pi$ for all t>0

3. C

   $E(t) \ge \pi/2$ for all t>0

4. D

   $E(t) \le \pi/2$ for all t>0

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

1. A

   There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $g:\Omega_\varepsilon\to(-\delta,\delta)$ such that g(0,0)=0 and F(x,y,g(x,y))=−1 for all $(x,y)\in\Omega_\varepsilon$.

2. B

   There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $h:\Omega_\varepsilon\to(-\delta,\delta)$ such that h(0,0)=0 and F(x,h(x,z),z)=−1 for all $(x,z)\in\Omega_\varepsilon$.

3. C

   There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $k_{1},k_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $k_{1}(0)=k_{2}(0)=0$ and $F(x,k_{1}(x),k_{2}(x))=-1$ for all $x\in(-\varepsilon,\varepsilon)$.

4. D

   There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $j_{1},j_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $j_{1}(0)=j_{2}(0)=0$ and $F(j_{1}(z),j_{2}(z),z)=-1$ for all $z\in(-\varepsilon,\varepsilon)$.

Multi-select · full marks only for the exact set · Linear Models and Multivariate

1. A

   YᵀAY$/\sigma^{2}$ follows a central $\chi^{2}$ distribution with (n−1) degrees of freedom.

2. B

   YᵀBY$/\sigma^{2}$ follows a central $\chi^{2}$ distribution with p degrees of freedom if $\beta_{1}=\beta_{2}=\cdots=\beta_{p}=0$.

3. C

   YᵀBY/YᵀCY follows a central F distribution with (p,n−p) degrees of freedom.

4. D

   YᵀBY and YᵀCY are independently distributed if and only if $\beta_{1}=\beta_{2}=\cdots=\beta_{p}=0$.

Single choice

1. A

   1

2. B

   2

3. C

   3

4. D

   9

Single choice

1. A

   Their volumes are equal.

2. B

   Volume of the rectangular container is larger than that of the square container.

3. C

   Volume of the triangular container is smaller than that of the square container.

4. D

   Volume of the square container is larger than that of the circular container.

Single choice

1. A

   not calculable due to insufficient data

2. B

   60

3. C

   68

4. D

   225

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

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