CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Zeros and Mappings

1. A

   |f(−1/5)| ≤ 1/5

2. B

   |f(−1/5)| ≤ 1/3

3. C

   |f′(−1/2)| ≤ 1/2

4. D

   |f′(−1/2)| ≤ 4/3

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

1. A

   $\int_\gamma1$ dz/(z sin z) = 0

2. B

   $\int_\gamma1$ dz$/(z \sin z) = 2\pi{}i$

3. C

   $\int_\gamma2$ dz$/(z \sin z) = 2\pi{}i$

4. D

   $\int_\gamma2$ dz/(z sin z) = 0

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

1. A

   If T is not diagonalizable over $\mathbb{R}$, then $T^{2}$ has an eigenvalue ≤0.

2. B

   If $T^{2}$ has only negative eigenvalues, then dim V is an even integer.

3. C

   If $T^{2}$ has only non-negative eigenvalues, then T is diagonalizable over $\mathbb{R}$.

4. D

   For each non-zero v∈V, {v,Tv,$T^{2}v$} is linearly dependent.

Multi-select · full marks only for the exact set · Hypothesis Testing

1. A

   $H_{0}$ is rejected

2. B

   The p-value of the test is greater than 0.01

3. C

   Under $H_{0}, E(K) = 3.5$

4. D

   If $X_{(i)}$ denotes the i-th smallest observation of the sample, then $[X_{(2)},X_{(6)}]$ is a confidence interval for M with confidence coefficient at least 0.95

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

1. A

   $\lim(x\to0+) y_{2}(x)/(x^{2}y_{1}(x))$ exists.

2. B

   $\lim(x\to0+) y_{2}(x)/($xy$_{1}(x))$ exists.

3. C

   lim(x→0+) xy$_{1}(x)/y_{2}(x)$ does NOT exist.

4. D

   $\lim(x\to0+) x^{2}y_{1}(x)/y_{2}(x)$ exists.

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

1. A

   P(X−Y=0 ∣ X+Y=2) = P(X−Y=0 ∣ X+Y=12)

2. B

   E((X−Y)/(X+Y)) = 0

3. C

   Cov(X+Y, X−Y) = 0

4. D

   (X+Y) and (X−Y) are independent

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

1. A

   There exist $\lambda\in\mathbb{C}$ and a non-zero v∈V such that Av$=\lambda{}v$.

2. B

   There exist $\lambda,\mu\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av$=\lambda{}v$ and Aw$=\mu{}w$.

3. C

   There exist $\lambda,\mu,\delta\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av$=\lambda{}v$ and Aw$=\mu{}v+\delta{}w$.

4. D

   There exists a three-dimensional subspace W such that Aw∈W for all w∈W.

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

1. A

   The transposition (1 2) belongs to G.

2. B

   |G| is divisible by p.

3. C

   A p-cycle belongs to G.

4. D

   $G = S_{p}$

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

1. A

   The set {$S\in{}W : S^{2}=S$} contains exactly 2 elements.

2. B

   $dim_\mathbb{R}(W) = 2$.

3. C

   If U∈L(V) is such that $U^{2}=U$ and $(T+U)^{2}=T+U$, then TU=0.

4. D

   If U∈L(V) is such that $U^{2}=U$ and $(T-U)^{2}=T-U$, then (TU$)^{2}=$TU.

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

1. A

   If d is a positive integer that divides |G|, then G has a subgroup of order d.

2. B

   The map f : G×G → G given by f(a,b)=ab is not a group homomorphism.

3. C

   Suppose that for every positive integer d that divides |G|, there exists a subgroup of G of order d. Then G has at least three normal subgroups.

4. D

   |G| ≠ 16

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

1. A

   Cumulative hazard function of each component is H(t)=2ln(1+t), t>0

2. B

   $S(t)=1/(1+t)^{2}, t>0$

3. C

   P(Y<1/2)={S(1/2)}$^{4}$

4. D

   P(Y<1/2)=65/81

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

1. A

   lim(x,y)→(0,0) f(x,y) does not exist.

2. B

   The partial derivatives of f at (0,0) do not exist.

3. C

   lim(x→0) f(x,0) = lim(y→0) f(0,y).

4. D

   f is differentiable at (0,0).

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

1. A

   $(X_{1},X_{2},\dots,X_{n})$ is a sufficient statistic for $(\alpha,\sigma)$

2. B

   $(\sum{}t X_{t}, \sum{}t$ tX$_{t}, \sum{}t t^{2}X_{t}^{2})$ is a jointly minimal sufficient statistic for $(\alpha,\sigma)$

3. C

   $X_{4}-X_{3}-X_{2}+X_{1}$ is an ancillary statistic

4. D

   $(X_{4}+X_{1}-X_{2}-X_{3})/(X_{5}+X_{2}-X_{3}-X_{4})$ is an ancillary statistic

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

1. A

   y(1/2) = 1

2. B

   y′(0) = 1

3. C

   y(1/4) = 2

4. D

   y′(1/2) = 1

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 $\beta_{1}=\beta_{2}=\cdots=\beta_{p}=0$.

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

1. A

   y(0)+y′(0) = 3/2

2. B

   y(1/2)+y′(1/2) = 7/2

3. C

   y(−1)+y′(−1) = −1

4. D

   y(1/3)+y′(1/3) = 14/9

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

1. A

   p = 1

2. B

   $\lim(n\to\infty)$ |$e_{n}$₊$_{1}$|/|$e_{n}$|^p = 1/2

3. C

   p = 2

4. D

   $\lim(n\to\infty)$ |$e_{n}$₊$_{1}$|/|$e_{n}$|^p = 1

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

1. A

   There exists a unique solution to BVP.

2. B

   The BVP does NOT have a solution.

3. C

   There exists a solution u to BVP such that u(x,y)=(1+e)/2 for some $(x,y)\in\Omega\cup\partial\Omega$.

4. D

   There exists a solution u to BVP such that $u(x,y)=(1+e^{3})/2$ for some $(x,y)\in\Omega\cup\partial\Omega$.

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

1. A

   $Var(\beta$̃$_{1}) > Var(\beta$̃$_{2}), E(\beta$̃$_{1}-\beta)^{2} > E(\beta$̃$_{2}-\beta)^{2}$

2. B

   $Var(\beta$̃$_{1}) < Var(\beta$̃$_{2}), E(\beta$̃$_{1}-\beta)^{2} < E(\beta$̃$_{2}-\beta)^{2}$

3. C

   $Var(\beta$̃$_{1}) > Var(\beta$̃$_{2}), E(\beta$̃$_{1}) < E(\beta$̃$_{2})$

4. D

   $Var(\beta$̃$_{1}) < Var(\beta$̃$_{2}), E(\beta$̃$_{1}) > E(\beta$̃$_{2})$

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

1. A

   f is a real valued function.

2. B

   f(z) = 1 for all $z\in\mathbb{C}$.

3. C

   f is not an entire function.

4. D

   f has finitely many zeros on the imaginary axis.

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

1. A

   (5, 0, 1) is a corner point.

2. B

   (4, 1, 1) is an optimal point.

3. C

   The optimal solution is 9.

4. D

   The optimal solution is 10.

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

1. A

   {$x \in \mathbb{R}$ | [x] = 1}

2. B

   {$x \in \mathbb{R}$ | x − [x] = 1/2}

3. C

   {$x \in \mathbb{R}$ | $x \ge 0, \sqrt{x} \in \mathbb{Q}$}

4. D

   {$x \in \mathbb{R}$ | $x^{2} \in \mathbb{Q}$}

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

1. A

   sup A is finite.

2. B

   $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) < \sup A$

3. C

   $\lim \inf(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 2$

4. D

   $\lim \sup(n\to\infty) 1/(\sqrt{n + \sqrt{n}} - \sqrt{n}) = 3$

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 · Sequences and Series of Functions

1. A

   $(g_{n})_{n}\ge_{1}$ converges uniformly on any compact subset of $\mathbb{R}$ but not on $\mathbb{R}$.

2. B

   $(g_{n})_{n}\ge_{1}$ converges uniformly on $\mathbb{R}$.

3. C

   There exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on $\mathbb{R}$.

4. D

   For every compact subset $K \subseteq \mathbb{R}$, there exists a subsequence of $(h_{n})_{n}\ge_{1}$ that converges uniformly on K.