CSIR NET Maths previous year question papers

Single choice · Partial Differential Equations

1. A

   u(1, 1) = 2

2. B

   u(2, 2) = 4

3. C

   u(3, 3) = 3/2

4. D

   u(4, 4) = 2/3

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 · 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 · 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 $(x,y)\in\Omega\cup\partial\Omega$.

Single choice · Partial Differential Equations

1. A

   u(x, t) = 0, whenever x = t

2. B

   u(x, t) = 0, whenever x = −t

3. C

   u(−x, t) = u(x, t), whenever x > 0, t > 0

4. D

   u(−x, t) = −u(x, t), whenever 0 < x ≤ t

Single choice · Calculus of Variations

1. A

   {$-\sqrt{2}, \sqrt{2}$}

2. B

   {$\sqrt{2}/k : k \in \mathbb{Z}, k \ne 0$}

3. C

   {$\sqrt{2/k} : k \in \mathbb{N}$}

4. D

   {$-\sqrt{2}, 0, \sqrt{2}$}

Single choice · Calculus of Variations

1. A

   $\Delta{}u = -$ue$^(u^{2})$ in $B(0,1), \partial{}u/\partial\nu = f$ on $\partial{}B(0,1)$

2. B

   $\Delta{}u =$ ue$^(u^{2}) + f$ in B(0,1), u = 0 on $\partial{}B(0,1)$

3. C

   $\Delta{}u =$ ue$^(u^{2})$ in $B(0,1), \partial{}u/\partial\nu = -f$ on $\partial{}B(0,1)$

4. D

   $\Delta{}u =$ ue$^(u^{2})$ in $B(0,1), \partial{}u/\partial\nu + u = f$ on $\partial{}B(0,1)$

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 · Calculus of Variations

1. A

   There are infinitely many extremals

2. B

   There are more than one but only finitely many extremals

3. C

   There is no extremal

4. D

   There is a unique extremal

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

1. A

   The BVP has a non-zero solution if $\lambda = 2$

2. B

   The BVP has a non-zero solution if $\lambda = 3$

3. C

   The BVP has a non-zero solution y = y(x) satisfying $y(k\pi/2) = 0$ for every integer k if $\lambda = 6$

4. D

   The BVP has a non-zero solution y = y(x) satisfying $y(k\pi/3) = 0$ for every integer k if $\lambda = 6$

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

1. A

   Every solution of the ODE is bounded on $(0, \infty)$

2. B

   There exists a solution of the ODE which is unbounded on $(0, \infty)$

3. C

   Every solution of the ODE is unbounded on $(0, \infty)$

4. D

   Every solution of the ODE tends to zero as $x \to \infty$

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

1. A

   $u(x_{0}, y_{0}) = \sqrt{2} e^\pi$

2. B

   $u(x_{0}, y_{0}) = e^\pi$

3. C

   $u(x_{0}, y_{0}) = -1$

4. D

   $u(x_{0}, y_{0}) = -e^\pi$

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

1. A

   There exists an $\alpha \in (0, \infty)$ such that for every $b \in \mathbb{R}$ with $b > \alpha$, the solution y_b is bounded above on I_b

2. B

   There exists an $\alpha \in (0, \infty)$ such that for every $b \in \mathbb{R}$ with $b > \alpha$, the solution y_b is bounded below on I_b

3. C

   There exists an $\alpha \in (-\infty, 0)$ such that for every $b \in \mathbb{R}$ with $b < \alpha$, the solution y_b is bounded above on I_b

4. D

   There exists an $\alpha \in (-\infty, 0)$ such that for every $b \in \mathbb{R}$ with $b < \alpha$, the solution y_b is bounded below on I_b

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

1. A

   S(2) has exactly two elements

2. B

   S(1/2) has exactly one element

3. C

   S(2) is empty

4. D

   S(1/2) has exactly two elements

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

1. A

   $19\sqrt{2}/8$

2. B

   $19\sqrt{2}$

3. C

   19/8

4. D

   $19/(2\sqrt{2})$

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

1. A

   There is a positive $y_{0}$ such that the solution of the IVP is unbounded

2. B

   There is a negative $y_{0}$ such that the solution of the IVP is bounded

3. C

   For every $y_{0} \in \mathbb{R}$, every solution of the IVP is bounded

4. D

   For every $y_{0} \in \mathbb{R}$, there is a solution to the IVP for all $x \in \mathbb{R}$

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

1. A

   $R(x, t, \lambda) = 3$xt$/(3 - 8\lambda) - 5x^{2}t^{2}/(5 - 32\lambda)$

2. B

   $R(x, t, \lambda) = 3$xt$/(3 - 8\lambda) + 5x^{2}t^{2}/(5 - 32\lambda)$

3. C

   $u(1) = -5/(5 - 32\lambda)$

4. D

   $u(1) = 3/(3 - 8\lambda)$