CSIR NET Maths previous year question papers

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 · Ordinary Differential Equations

1. A

   f is a Lipschitz continuous function on D

2. B

   f is NOT a Lipschitz continuous function on D

3. C

   IVP has at least one solution

4. D

   IVP has NO solution

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 · 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 · 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 · 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 · 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 · Classical Mechanics

1. A

   $G(p,Q) = -Q[1 + log_{e}(p^{2}/(4Q))]$

2. B

   G(p,Q) = −pQ$[1 + log_{e}(Q^{2}/(4p))]$

3. C

   p = 2qe$^P, Q = q^{2}e^P$

4. D

   p = −2qe$^P, Q = -q^{2}e^P$

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 · 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 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 · Ordinary Differential Equations

1. A

   For $\alpha = 1$, all the solutions of the ODE tend to zero as $x \to \infty$

2. B

   For $\alpha = 0$, there exists a solution of the ODE which tends to 1 as $x \to \infty$

3. C

   For $\alpha = -1$, there exists an unbounded solution of the ODE on $\mathbb{R}$

4. D

   For $\alpha = 2$, there exists an unbounded solution of the ODE on $\mathbb{R}$

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

1. A

   $x(\pi) = -e^\pi, y(\pi) = 1/2$

2. B

   $x(-\pi) = e^(-\pi), y(-\pi) = 1/2$

3. C

   x has infinitely many zeros in $\mathbb{R}$

4. D

   y has infinitely many zeros in $\mathbb{R}$