CSIR NET Maths previous year question papers

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

1. A

   B(w, w) = 0

2. B

   There exists an $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$.

3. C

   There are infinitely many $\alpha \in \mathbb{R}$ such that $q(\alpha{}v + w) = 0$.

4. D

   q is equivalent to the quadratic form $Q(x, y) = x^{2} - y^{2}$ for all $(x, y) \in \mathbb{R}^{2}$.

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

1. A

   If f(z) = f(z + 1) for all $z \in \mathbb{C}$ then f is a constant function.

2. B

   If f(z) = f(z + 1) = f(z + i) for all $z \in \mathbb{C}$ then f is a constant function.

3. C

   If f(1/z) has a removable singularity at 0 then f is a constant function.

4. D

   If f is a non-constant function then f(1/z) has a pole at 0.

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

1. A

   $lim_{z\to0}$ f(z) does not exist.

2. B

   $lim_{z\to0}$ f(z) exists and has absolute value ≤ 1.

3. C

   $lim_{z\to0}$ f(z) = 0.

4. D

   There exists $\theta \in \mathbb{R}$ such that f(z) = $e^{i\theta}z$ for all z ∈ 𝔻^×.

Multi-select · full marks only for the exact set · Analytic Functions

1. A

   A is nonempty.

2. B

   A is finite.

3. C

   A is infinite.

4. D

   A is uncountable.

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

1. A

   $\varphi_{1}(x + 2\pi)$ is also a solution of ODE

2. B

   $\varphi_{2}(x + 4\pi)$ is also a solution of ODE

3. C

   There are NO constants a, b such that $\varphi_{2}(x + 4\pi) = a\varphi_{1}(x) + b\varphi_{2}(x)$

4. D

   There exist $a, b \in \mathbb{R}$ such that $\varphi_{1}(x + 2\pi) = a\varphi_{1}(x) + b\varphi_{2}(x)$

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

1. A

   There exists a q such that cos(x) and e^x cos(x) are solutions of ODE

2. B

   There exists a q such that sin(x) and cos(x) are solutions of ODE

3. C

   There exists a q such that e^x sin(x) and e^x cos(2x) are solutions of ODE

4. D

   There exists a q such that xe^x and x(x − 1)e^x are solutions of ODE

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 · Partial Differential Equations

1. A

   The minimum value of u is 1

2. B

   The maximum value of u is 3

3. C

   The minimum value of u is 2

4. D

   The maximum value of u is 3/2

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

1. A

   $(\alpha, \beta) = ((3 - \sqrt{3})/2, \sqrt{3})$ or $(\alpha, \beta) = ((3 + \sqrt{3})/2, -\sqrt{3})$

2. B

   $(\alpha, \beta) = ((3 - \sqrt{2})/2, \sqrt{2})$ or $(\alpha, \beta) = ((3 + \sqrt{2})/2, -\sqrt{2})$

3. C

   $(\alpha, \beta) = ((3 - \sqrt{5})/2, \sqrt{5})$ or $(\alpha, \beta) = ((3 + \sqrt{5})/2, -\sqrt{5})$

4. D

   $(\alpha, \beta) = ((3 - \sqrt{7})/2, \sqrt{7})$ or $(\alpha, \beta) = ((3 + \sqrt{7})/2, -\sqrt{7})$

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

1. A

   $y(x) = (1/15)(x^{4} + 14)$ is an extremal for $J_{3}$

2. B

   $y(x) = (1/3)(x^{2} + 2)$ is an extremal for $J_{1}$

3. C

   y(x) = x is an extremal for $J_{0}$

4. D

   $y(x) = (1/2)(x^{2} - x + 2)$ is an extremal for $J_{1}$

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

1. A

   If (a, b, c) = (−2, 1, −2), then every admissible extremal is a minimizer

2. B

   If (a, b, c) = (1, 0, 2), then every admissible extremal is a minimizer

3. C

   If (a, b, c) = (2, −1, 1), then every admissible extremal is a minimizer

4. D

   If (a, b, c) = (1, −2, 5), then every admissible extremal is a minimizer

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

1. A

   $E(X^{2}) = 4/3$

2. B

   P(|X − 4/5| ≥ 1) = 19/25

3. C

   The upper bound of P(|X − 4/5| ≥ 1), using Chebyshev's inequality, is 52/75

4. D

   The value of the moment generating function, M_X(t), at t = 1 is $(e^{2} - 1)/5$

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

1. A

   $\theta$̂ $= \sum_{i}$₌$_{1}^{n} X_{i}/(n + \tau^{2})$, if the prior is $N(0, 1/\tau^{2}), \tau^{2}$ known and $L(\theta, d) = (\theta - d)^{2}$

2. B

   $\theta$̂ $= \sum_{i}$₌$_{1}^{n} X_{i}/(n + \tau^{2})$, if the prior is $N(0, \tau^{2}), \tau^{2}$ known and $L(\theta, d) =$ |$\theta - d$|

3. C

   $\theta$̂ $= \sum_{i}$₌$_{1}^{n} X_{i}/(n + 1/\tau^{2})$, if the prior is $N(0, 1/\tau^{2}), \tau^{2}$ known and $L(\theta, d) =$ |$\theta - d$|

4. D

   $\theta$̂ $= \sum_{i}$₌$_{1}^{n} X_{i}/n$, if the prior is the Jeffreys prior and $L(\theta, d) = (\theta - d)^{2}$

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

1. A

   Stationary distribution is (0, 0, 0, 0, 1).

2. B

   State 5 is absorbing and recurrent.

3. C

   All states are aperiodic.

4. D

   $lim_{n}\to\infty p_{55}$⁽$^{n}$⁾ = 1.

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

1. A

   $P_{0} = 1/3$

2. B

   $P_{1} = 2/9$

3. C

   The average number of customers in the system is 1

4. D

   The average amount of time that a customer spends in the system is 1/6

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

1. A

   The conditional distribution of X given X + Y = 3 is Binomial(3, 1/3)

2. B

   P(X ≤ 1 | X + Y = 3) = 7/27

3. C

   E(X | X + Y = 3) = 2

4. D

   The value of the characteristic function of X + Y at the point $t = \pi$ is $e^{-12}$

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

1. A

   R(t) = $e^{-2t}$ + $e^{-t}$ − $e^{-3t}$, t > 0

2. B

   The expected lifetime of the given system is 1

3. C

   $h(1) = (2e + e^{2} - 3)/(e + e^{2} - 1)$

4. D

   $h(3) = (4e^{3} + e^{6} - 5)/(e^{3} + e^{6} - 1)$

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

1. A

   The probability density function of S is f_S(s) = $(1/2)e^{-s/2}$ if s > 0, and 0 otherwise.

2. B

   The probability density function of T is f_T(t) = 1 if 0 < t < 1, and 0 otherwise.

3. C

   Var(S) = 2.

4. D

   E(T) = 2/3.

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

1. A

   $(Y + (1/n)\ln(1 - \alpha/2), Y - (1/n)\ln(1 - \alpha/2))$

2. B

   $(Y + (1/2n)\ln(\alpha), Y)$

3. C

   $(Y + (2/n)\ln(\alpha/2), Y + (2/n)\ln(1 - \alpha/2))$

4. D

   $(Y + (1/n)\ln(\alpha/2), Y + (1/n)\ln(1 - \alpha/2))$

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

1. A

   The two fitted lines have the same slope

2. B

   The two fitted models have the same intercept

3. C

   The model with intercept passes through at least one of the observed data points

4. D

   The model without intercept passes through at lesast one of the observed data points

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

1. A

   A possible choice of L is the 5 × 2 matrix with rows (0, 0), (1, 0), (0, 1), (−1, 0), (0, 1)

2. B

   The test statistic for testing $H_{0}$ against H_A follows an F-distribution with (2, n − 4) degrees of freedom under $H_{0}$

3. C

   Sum of squares residuals under the restrictions $L^{T}\beta = 0$ follows $\sigma^{2}\chi^{2}_{n-3}$ distribution

4. D

   Sum of squares residuals without the restrictions $L^{T}\beta = 0$ follows a non-central $\sigma^{2}\chi^{2}_{n-4}$ distribution

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

1. A

   The distribution of $(1/\sqrt{7})\sum_{i}$₌$_{1}^{10}(X_{i} - Y_{i} + 1)$ is N(0, 10)

2. B

   The distribution of $(1/19)\sum_{i}$₌$_{1}^{10}(X_{i} + Y_{i} - 11)^{2}$ is $\chi^{2}-$distribution with degrees of freedom 10

3. C

   The distribution of $2\sqrt{2}(X_{1} - 5)/\surd(\sum_{i}$₌$_{3}^{10}(X_{i} - 5)^{2})$ is t-distribution with degrees of freedom 9

4. D

   The distribution of $2\sum_{i}$₌$_{1}^{3}(Y_{i} - 6)^{2}/\sum_{i}$₌$_{4}^{9}(Y_{i} - 6)^{2}$ is F-distribution with degrees of freedom 3 and 6.

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

1. A

   $\rho$ may be −0.47

2. B

   The first principal component can only explain 32% of the total variation for some $\rho$

3. C

   The second principal component can explain more than 32% of the total variation for any $\rho$

4. D

   The variance of the first principal component is $1 + 2\rho$ for any $\rho$

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

1. A

   $limsup_{n\to\infty}$ $t_{n} \le$ $limsup_{n\to\infty}$ $s_{n}$

2. B

   $limsup_{n\to\infty}$ $t_{n} \le e$

3. C

   $liminf_{n\to\infty}$ $s_{n} \ge e^{2}$

4. D

   $liminf_{n\to\infty}$ $t_{n} \ge e$

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

1. A

   2/e

2. B

   4/e

3. C

   $log_{e}2 - 1$

4. D

   $2 log_{e}2 - 1$