CSIR NET Maths previous year question papers

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

1. A

   The maximum possible value of T is 115

2. B

   Right-tailed test based on T is appropriate for testing $H_{0}$ against $H_{1}$

3. C

   Under $H_{0}, E(T) = 76$

4. D

   Under $H_{0}, P(R_{1} = 1, R_{9} = 16) = 1/240$

Multi-select · full marks only for the exact set · Sampling and Design of Experiments

1. A

   The design is incomplete

2. B

   The design is connected

3. C

   The design is balanced

4. D

   The design is orthogonal

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

1. A

   If the observed value of X̄ is 0.6, then $\varphi$ does not reject $H_{0}$

2. B

   If the observed value of X̄ is 1.6, then $\varphi$ rejects $H_{0}$

3. C

   If the observed value of X̄ is 1.3, then $\varphi$ rejects $H_{0}$

4. D

   If the observed value of X̄ is 1.2, then $\varphi$ does not reject $H_{0}$

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

1. A

   $x^{2} +$ xy$^{2} - x - y + 1$ is irreducible in $\mathbb{Q}[x,y]$.

2. B

   $x^{3} + 100x^{2} + 25x + 10$ is irreducible in $\mathbb{Z}[x]$.

3. C

   $2x^{2} + 3$xy $+ y^{2} + 3x + 2y + 1$ is irreducible in $\mathbb{Q}[x,y]$.

4. D

   $x^{4} + 4x^{2} + 3$ is irreducible in $\mathbb{Z}[x]$.

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

1. A

   $S_{n}(X_{1}:_{n})$⁻$^{1}$ is a consistent estimator of $\theta$

2. B

   $({\bar{X}}_{n} - X_{1}:_{n})(X_{1}:_{n})$⁻$^{1}$ is a consistent estimator of $\theta$

3. C

   $({\bar{X}}_{n} - X_{1}:_{n})({\bar{X}}_{n} - S_{n})$⁻$^{1}$ is a consistent estimator of $\theta$

4. D

   $S_{n}({\bar{X}}_{n})$⁻$^{1}$ is a consistent estimator of $\theta$

Multi-select · full marks only for the exact set · Sampling and Design of Experiments

1. A

   Mean of the observations in dataset B is 0

2. B

   Median of the observations in dataset B is 0

3. C

   Variance of the observations in dataset B ≥ Variance of the observations in dataset A

4. D

   Range of the observations in dataset B ≥ Range of the observations in dataset A

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

1. A

   If there are no elements of order 4 in G, then G is abelian.

2. B

   If there are exactly two elements of order 4 in G, then G is abelian.

3. C

   If there are exactly six elements of order 4 in G, then G is abelian.

4. D

   There is an element of order 4 in G.

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

1. A

   There exists a positive integer N such that for all n≥N, $b_{n,l}$ $= a_{l}$ for all 1≤l≤2023.

2. B

   There exists a positive integer N such that for all n≥N, $b_{n,2025}$ $= a_{2025} - 1$.

3. C

   $\beta_{n}$ is rational for infinitely many n.

4. D

   There exists a positive integer N such that for all n≥N, $b_{n,2024}$ $= a_{2024} - 1$.

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 · Ordinary Differential Equations

1. A

   There exists a non-trivial solution $\varphi$ of ODE, and $\alpha,\beta\in\mathbb{R}, \alpha<\beta$ such that $\varphi$ has infinitely many zeros in $[\alpha,\beta]$.

2. B

   There exists a non-trivial solution $\varphi$ of ODE, and a sequence {$x_{n}$} such that $x_{n}\to\infty$ and $\varphi(x_{n})=0$ for all $n\in\mathbb{N}$.

3. C

   If $\varphi_{1},\varphi_{2}$ are two linearly independent solutions of ODE, then their Wronskian is a constant function on $\mathbb{R}$.

4. D

   There exists a non-trivial solution of ODE which vanishes at most at one point in $(0,\infty)$.

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 · Estimation

1. A

   The method of moments estimator of $\theta$ is −2X̄

2. B

   The method of moments estimator of $\theta$ is 2X̄

3. C

   The maximum likelihood estimator of $\theta$ is min{$-2X_{(1)}/3, 2X_{(n)}$}

4. D

   The maximum likelihood estimator of $\theta$ is max{$-2X_{(1)}/3, 2X_{(n)}$}

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

1. A

   $(\mathbb{R}, \tau_E)$ is a normal space.

2. B

   Let $\tau$ be a topology on $\mathbb{R}$. If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau_E)$ to $(\mathbb{R},\tau)$, then $(\mathbb{R},\tau)$ is a regular space.

3. C

   Let $\tau$ be a topology on $\mathbb{R}$. If the identity function of $\mathbb{R}$ is a continuous map from $(\mathbb{R},\tau)$ to $(\mathbb{R},\tau_E)$, then $(\mathbb{R},\tau)$ is a Hausdorff space.

4. D

   $\mathbb{R}$ is a regular space in the finite-complement topology.

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

1. A

   R(0,1,1/3) = 3/2

2. B

   R(1/2,1/2,1/3) = 2/3

3. C

   y(1) = 3e+1

4. D

   y(1) = e+3

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

1. A

   Posterior mean of $\theta$ given X=2 is less than the prior mean

2. B

   Posterior mean of $\theta$ given X=3 is greater than the prior mean

3. C

   Posterior mean of $\theta$ given X=4 is not equal to the prior mean

4. D

   Posterior mean of $\theta$ given X=5 is equal to the prior mean

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

1. A

   The set {$B \in M_{4}(\mathbb{C})$ ∣ AB+BA=0} is an 8-dimensional $\mathbb{C}-$vector space.

2. B

   The set {$B \in M_{4}(\mathbb{C})$ ∣ AB−BA=0} is an 8-dimensional $\mathbb{C}-$vector space.

3. C

   There exists $B \in M_{4}(\mathbb{C})$ such that AB+BA$=0, B^{2}=I$ and Trace(B)=0.

4. D

   If $B \in M_{4}(\mathbb{C})$ is such that AB+BA=0, then B is diagonalisable.

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 · Inner Product Spaces and Forms

1. A

   $W_{0} =$ {0}

2. B

   $W_{1} =$ {0}

3. C

   $W_{2} =$ {0}

4. D

   $W_{n} =$ {0}

Multi-select · full marks only for the exact set

1. A

   In every calendar month of 2024, at least one person was troublesome.

2. B

   In any two consecutive calendar months of 2024, at least one person was troublesome in at least one of the two calendar months.

3. C

   At least one person was troublesome in two calendar months of 2024.

4. D

   At least one person was troublesome in January 2024.

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

1. A

   The dimension of V is 4 or 5.

2. B

   If 2u−3v+5w=0, then {v,w,x,y} is a basis of V.

3. C

   If u−6v+7w=0, then the span of {v,w,x} is a 2-dimensional vector space.

4. D

   If u+w=v+x, then {u,v,w,y} is a basis of V.

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

1. A

   y′(1/2) = 3/4

2. B

   $y'(1/\sqrt{3}) = 1$

3. C

   y′(1/3) = 1/3

4. D

   y′(1/4) = 1/2

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

1. A

   19(X̄ᵀSX$)({\bar{X}}^{T}\sum{\bar{X}})$⁻$^{1}$ ~ $\chi^{2}_{19}$

2. B

   E(S⁻$^{1}) = (19/6)\sum$⁻$^{1}$

3. C

   S ~ $W_{12}(19, 19\sum)$

4. D

   trace$(19\sum$⁻$^{1}S)$ ~ $\chi^{2}_{228}$

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 · Numerical Analysis

1. A

   (4,1;2,3) ∈ S

2. B

   (1,2;3,4) ∈ S

3. C

   (1,5;1,10) ∈ S

4. D

   (−5,2;1,−4) ∈ S

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

1. A

   R has a unique maximal ideal.

2. B

   Every finitely generated ideal of R is principal.

3. C

   R is an integral domain.

4. D

   R is a euclidean domain.