CSIR NET Maths previous year question papers

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

1. A

   The sequence $(y_{n})$ does NOT converge

2. B

   $y_{n} \to 0$ as $n \to \infty$

3. C

   $0 \le y_{n} \le 1$ for n = 0, 1, 2, …

4. D

   |y(nh$) - y_{n}$| → 0 as $n \to \infty$

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

1. A

   u(0) = 0

2. B

   $u(2\pi/\sqrt{3}) =$ $(1/6)(e^{4\pi/\sqrt{3}}$ − $e^{-2\pi/\sqrt{3}})$

3. C

   $u(\pi/(2\sqrt{3})) =$ $(1/6)(e^{\pi/\sqrt{3}}$ $- \sqrt{3}$ $e^{-\pi/(2\sqrt{3})})$

4. D

   $u(\pi/(2\sqrt{3})) =$ $(1/6)(e^{\pi/\sqrt{3}}$ $+ \sqrt{3}$ $e^{-\pi/\sqrt{3}})$

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

1. A

   $f'(\pi) = 3$

2. B

   $f(\pi) + f'(\pi) = 4 - \pi$

3. C

   $f(\pi) + f'(\pi) = 2 - \pi$

4. D

   f(0) + f′(0) = −4

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

1. A

   $\partial/\partial{}t [f, g] = [\partial{}f/\partial{}t, g] + [f, \partial{}g/\partial{}t]$

2. B

   If f is a constant of motion, and f is independent of t, then [H, f] is a constant of motion

3. C

   [[H, f], g] + [[g, H], f] + [[f, g], H] = 0

4. D

   If f and g are constants of motion, then [f, g] is a constant of motion

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

1. A

   $S_{n}/\sqrt{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

2. B

   $(T_{n} - n)/\sqrt{3n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

3. C

   $\sqrt{n} S_{n}/T_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

4. D

   $(T_{n} - n)/\sqrt{V}_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)

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

1. A

   X̄ is an unbiased estimator of $\theta_{1}$

2. B

   $({\bar{X}} - X_{(1)})$ is an unbiased estimator of $\theta_{2}$

3. C

   $(X_{(1)} + X_{(n)})/2$ is the uniformly minimum variance unbiased estimator of $\theta_{1}$

4. D

   $(n + 1)(X_{(n)} - X_{(1)})/(2(n - 1))$ is the uniformly minimum variance unbiased estimator of $\theta_{2}$

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

1. A

   $U_{n}$ is an unbiased estimator of $\theta$

2. B

   $S_{n}^{2}$ is an unbiased estimator of $\theta^{2}$

3. C

   $(1/3)V_{n}$ is a consistent estimator of $\theta^{2}$

4. D

   The statistic $(U_{n}, V_{n})$ is complete

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

1. A

   The likelihood function is maximized at $\theta = 1/(x - 1)$

2. B

   The maximum value of the likelihood function is $e^{-1}/(x$ − 1)

3. C

   The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if (x − $1)e^{-x}$ < k, for some k > 0

4. D

   The likelihood ratio test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ if x > c, for some c > 1

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

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