CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f′(x) ≤ 0 for all x > 0.

2. B

   $lim_{x\to\infty}$ f′(x) = 0.

3. C

   $lim_{x\to\infty}$ x f′(x) need not exist.

4. D

   $lim_{x\to\infty}$ x f′(x) = 0.

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   If each $f_{n}$ is uniformly continuous and $(f_{n})_{n}\ge_{1}$ converges to f uniformly, then f is uniformly continuous.

2. B

   If each $f_{n}$ is bounded and $(f_{n})_{n}\ge_{1}$ converges to f pointwise, then f is bounded.

3. C

   If each $f_{n}$ is bounded and continuous, $(f_{n})_{n}\ge_{1}$ converges pointwise to a bounded and continuous function f, then the convergence is uniform.

4. D

   If each $f_{n}$ is differentiable and $(f_{n})_{n}\ge_{1}$ converges to f uniformly, then f is differentiable.

Multi-select · full marks only for the exact set · Sequences and Series of Functions

1. A

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$, and $(f'_{n})_{n}\ge_{1}$ converges uniformly to 0 on the interval (−M, M) for some positive real number M.

2. B

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$, and $(f'_{n})_{n}\ge_{1}$ converges pointwise to 0 on $\mathbb{R}$.

3. C

   $(f_{n})_{n}\ge_{1}$ converges uniformly to 0 on $\mathbb{R}$ and $(f'_{n})_{n}\ge_{1}$ does not converge pointwise to 0 on $\mathbb{R}$.

4. D

   $(f_{n})_{n}\ge_{1}$ converges pointwise to 0 on $\mathbb{R}$ but not uniformly on $\mathbb{R}$.

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

1. A

   If A is invertible, then AB and BA have the same minimal polynomial.

2. B

   If 0 is an eigenvalue of A, then 0 is an eigenvalue of AB.

3. C

   If 0 is the only eigenvalue of A and of B, then 0 is the only eigenvalue of AB.

4. D

   If AB and BA have the same minimal polynomial, then either A or B is invertible.

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

1. A

   (ST − TS$)^{2} = \lambda{}I$ for some $\lambda \in \mathbb{C}$.

2. B

   The characteristic polynomial of (ST − TS$)^{2}$ is $(x - \lambda)^{2}$ for some $\lambda \in \mathbb{C}$.

3. C

   If ST − TS has only one eigenvalue, then ST − TS $= \lambda{}I$ for some $\lambda \in \mathbb{C}$.

4. D

   If ST − TS has only one eigenvalue, then (ST − TS$)^{2}$ is the zero transformation.

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

1. A

   {f′(x), f(x) − f(x − 1), 1}

2. B

   {f(x + 1) − f(x), f(x) − f(x − 1), 1}

3. C

   {f(x), f′(x), 1}

4. D

   {f(x + 1), f(x − 1), f(x)}

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

1. A

   The number of two dimensional subspaces of V is 13.

2. B

   The number of surjective linear transformations from V to W is 624.

3. C

   The number of one dimensional subspaces of V is 13.

4. D

   The number of linear transformations from V to W is $3^{6}$.

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 · 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

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