Differential Equations - General Aptitude - General Aptitude (GA) Previous Year Questions

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2015 · General Aptitude (GA) · General Aptitude · Differential Equations

Electronics & Communication Engineering (EC) 2015 [Session 1]

The solution of the differential equation \(\frac{d^2 y}{dt^2} + 2\frac{dy}{dt} + y = 0\) with \(y(0) = y'(0) = 1\) is

A

\((2-t)e^t\)

B

\((1+2t)e^{-t}\)

C

\((2+t)e^{-t}\)

D

\((1-2t)e^t\)

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2

2015 · General Aptitude (GA) · General Aptitude · Differential Equations

Electronics & Communication Engineering (EC) 2015 [Session 2]

The general solution of the differential equation \( \frac{dy}{dx} = \frac{1+\cos 2y}{1-\cos 2x} \) is

A

\(\tan y - \cot x = c\) (c is a constant)

B

\(\tan x - \cot y = c\) (c is a constant)

C

\(\tan y + \cot x = c\) (c is a constant)

D

\(\tan x + \cot y = c\) (c is a constant)

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3

2015 · General Aptitude (GA) · General Aptitude · Differential Equations

Electronics & Communication Engineering (EC) 2015 [Session 2]

Consider the differential equation \[ \frac{dx}{dt} = 10 - 0.2x \] with initial condition \(x(0) = 1\). The response \(x(t)\) for \(t > 0\) is

A

\(2 - e^{-0.2t}\)

B

\(2 - e^{0.2t}\)

C

\(50 - 49e^{-0.2t}\)

D

\(50 - 49e^{0.2t}\)

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4

2015 · General Aptitude (GA) · General Aptitude · Differential Equations

Electronics & Communication Engineering (EC) 2015 [Session 2]

Input \(x(t)\) and output \(y(t)\) of an LTI system are related by the differential equation \(y''(t) - y'(t) - 6y(t) = x(t)\). If the system is neither causal nor stable, the impulse response \(h(t)\) of the system is

A

\(\frac{1}{5}e^{3t}u(-t) + \frac{1}{5}e^{-2t}u(-t)\)

B

\(-\frac{1}{5}e^{3t}u(-t) + \frac{1}{5}e^{-2t}u(-t)\)

C

\(\frac{1}{5}e^{3t}u(-t) - \frac{1}{5}e^{-2t}u(t)\)

D

\(-\frac{1}{5}e^{3t}u(-t) - \frac{1}{5}e^{-2t}u(t)\)

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5

2015 · General Aptitude (GA) · General Aptitude · Differential Equations

Electronics & Communication Engineering (EC) 2015 [Session 2]

The state variable representation of a system is given as \[ \dot{x} = \begin{bmatrix} 0 & 1 \\ 0 & -1 \end{bmatrix} x, \quad x(0) = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \] \[ y = \begin{bmatrix} 0 & 1 \end{bmatrix} x \] The response y(t) is

A

\sin(t)

B

1 - e^t

C

1 - \cos(t)

D

0

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6

2015 · General Aptitude (GA) · General Aptitude · Differential Equations

Electronics & Communication Engineering (EC) 2015 [Session 2]

The output of a standard second-order system for a unit step input is given as \[ y(t) = 1 - \frac{2}{\sqrt{3}} e^{-t} \cos\left( \sqrt{3}t - \frac{\pi}{6} \right) \]. The transfer function of the system is

A

\(\frac{2}{(s+2)(s+\sqrt{3})}\)

B

\(\frac{1}{s^2 + 2s + 1}\)

C

\(\frac{3}{s^2 + 2s + 3}\)

D

\(\frac{4}{s^2 + 2s + 4}\)

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7

2015 · General Aptitude (GA) · General Aptitude · Differential Equations

Electronics & Communication Engineering (EC) 2015 [Session 3]

Consider the differential equation
d^2x(t)/dt^2 + 3 dx(t)/dt + 2x(t) = 0.
Given x(0) = 20 and x(1) = 10/e, where e = 2.718, the value of x(2) is ______.

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8

2015 · General Aptitude (GA) · General Aptitude · Differential Equations

Electronics & Communication Engineering (EC) 2015 [Session 3]

A network is described by the state model as

\[\begin{aligned} \dot{x}_1 &= 2x_1 - x_2 + 3u \\ \dot{x}_2 &= -4x_2 - u \\ y &= 3x_1 - 2x_2 \end{aligned}\]

The transfer function \(H(s) = \frac{Y(s)}{U(s)}\) is

A

\(\frac{11s+35}{(s-2)(s+4)}\)

B

\(\frac{11s-35}{(s-2)(s+4)}\)

C

\(\frac{11s+38}{(s-2)(s+4)}\)

D

\(\frac{11s-38}{(s-2)(s+4)}\)

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9

2016 · General Aptitude (GA) · General Aptitude · Differential Equations

Electronics & Communication Engineering (EC) 2016 [Session 1]

Which one of the following is a property of the solutions to the Laplace equation: \( \nabla^2 f = 0 \)?

A

The solutions have neither maxima nor minima anywhere except at the boundaries.

B

The solutions are not separable in the coordinates.

C

The solutions are not continuous.

D

The solutions are not dependent on the boundary conditions.

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