CSIR NET Maths previous year question papers

Single choice · Ordinary Differential Equations

1. A

   (a + $be^{-2x})x$, $a, b \in \mathbb{R}$

2. B

   (a + $be^{2x})x$, $a, b \in \mathbb{R}$

3. C

   ae^x + bx, $a, b \in \mathbb{R}$

4. D

   (a + be$^x)x, a, b \in \mathbb{R}$

Single choice · Ordinary Differential Equations

1. A

   $\pi^{2}/12$

2. B

   $2\pi^{2}/3$

3. C

   $\pi^{2}/4$

4. D

   $\pi^{2}/16$

Single choice · Ordinary Differential Equations

1. A

   Both (P) and (Q) are true.

2. B

   (P) is true, (Q) is FALSE.

3. C

   (P) is FALSE, (Q) is true.

4. D

   Both (P) and (Q) are FALSE.

Single choice · Ordinary Differential Equations

1. A

   2

2. B

   3

3. C

   4

4. D

   −3

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 · 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 · 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 · 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.

Single choice · Ordinary Differential Equations

1. A

   S is an empty set.

2. B

   S is a non-empty finite set.

3. C

   S is a countably infinite set.

4. D

   S is an uncountable set.

Single choice · Ordinary Differential Equations

1. A

   a unique solution on $\mathbb{R}$

2. B

   infinitely many solutions on the interval (−2, 2)

3. C

   a unique solution and its maximal interval of existence is $(-\infty, 1)$

4. D

   no solution on the interval (−2, 2)

Single choice · Ordinary Differential Equations

1. A

   $d^{2}u/$dx$^{2} + \lambda{}u = 0, u(0) = 0 = u(1)$

2. B

   $d^{2}u/$dx$^{2} - \lambda{}u = 0, u(0) = 0 = u(1)$

3. C

   du/dx $+ \lambda{}u = 0, u(0) = 0 = u(1)$

4. D

   du/dx $- \lambda{}u = 0, u(0) = 0 = u(1)$

Single choice · Ordinary Differential Equations

1. A

   e⁻ˣ(P(x) + 1) is a constant function on $\mathbb{R}$

2. B

   e^(−2x)P(x) is a constant function on $\mathbb{R}$

3. C

   $s(x) = e^(2x)/2, x \in \mathbb{R}$

4. D

   P(x) → 1 as $x \to \infty$

Single choice · Ordinary Differential Equations

1. A

   $y_{2}$ is a strictly increasing function on $(0, \infty)$

2. B

   $e^(-2x)y_{2}(x) \to 1$ as $x \to \infty$

3. C

   $y_{2}$ is a strictly decreasing function on $(0, \infty)$

4. D

   $e^(-2x)y_{2}(x) \to 0$ as $x \to \infty$

Single choice · Ordinary Differential Equations

1. A

   K(x, t) = (1 + x)(1 − t) for 0 ≤ x ≤ t ≤ 1, and (1 + t)(1 − x) for 0 ≤ t < x ≤ 1

2. B

   K(x, t) = −1 − x for 0 ≤ x ≤ t ≤ 1, and −1 − t for 0 ≤ t < x ≤ 1

3. C

   $K(x, t) = 1 - x^{2}$ for 0 ≤ x ≤ t ≤ 1, and $1 - t^{2}$ for 0 ≤ t < x ≤ 1

4. D

   K(x, t) = (1 + x)(t − 1) for 0 ≤ x ≤ t ≤ 1, and (1 + t)(x − 1) for 0 ≤ t < x ≤ 1

Single choice · Ordinary Differential Equations

1. A

   both $S_{1}$ and $S_{2}$ are true

2. B

   $S_{1}$ is true but $S_{2}$ is false

3. C

   $S_{1}$ is false but $S_{2}$ is true

4. D

   both $S_{1}$ and $S_{2}$ are false

Single choice · Ordinary Differential Equations

1. A

   −e/2

2. B

   −e/3

3. C

   e/3

4. D

   e

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 $\mathbb{R}$

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 · Ordinary Differential Equations

1. A

   $\lim(t\to\infty) t$⁻$^{2}x(t)y(t) = 0$

2. B

   x(1) = 0, y(1/2) = 0

3. C

   x(1/2) = 0, y(1) = 0

4. D

   $\lim(t\to\infty) t$⁻$^{2}x(t)y(t) = 2$

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 · Ordinary Differential Equations

1. A

   r(t) → 0 as $t \to +\infty$

2. B

   r(ln 2) = e⁻$^{1}$

3. C

   r(ln 2) = 2e⁻$^{1}$

4. D

   r(t)eᵗ → 0 as $t \to +\infty$

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 $x \in \mathbb{R}$