1
GATE ME 2023
MCQ (Single Correct Answer)
+2
-0.66
Which one of the options given is the inverse Laplace transform of $\rm \frac{1}{s^3-s}$? 𝑒(𝑑) denotes the unit-step function.
A
$\rm \left(-1+\frac{1}{2}e^{-t}+\frac{1}{2}e^t\right)u(t)$
B
$\rm \left(\frac{1}{3}e^{-t}-e^t\right)u(t)$
C
$\rm \left(-1+\frac{1}{2}e^{-(t-1)}+\frac{1}{2}e^{(t-1)}\right)u(t-1)$
D
$\rm \left(-1-\frac{1}{2}e^{-(t-1)}-\frac{1}{2}e^{(t-1)}\right)u(t-1)$
2
GATE ME 2023
Numerical
+2
-0

The smallest perimeter that a rectangle with area of 4 square units can have is ______ units.

(Answer in integer)

Your input ____
3
GATE ME 2023
Numerical
+2
-0

Consider the second-order linear ordinary differential equation

$\rm x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y=0, x\ge1$

with the initial conditions 

$\rm y(x=1)=6, \left.\frac{dy}{dx}\right|_{x=1}=2$

The value of 𝑦 at π‘₯ = 2 equals ________.

(Answer in integer)

Your input ____
4
GATE ME 2023
Numerical
+2
-0

The initial value problem

$\rm \frac{dy}{dt}+2y=0, y(0)=1$

is solved numerically using the forward Euler’s method with a constant and positive time step of Ξ”t. 

Let 𝑦𝑛 represent the numerical solution obtained after 𝑛 steps. The condition |𝑦n+1| ≀ |𝑦n| is satisfied if and only if Ξ”t does not exceed _____________.

(Answer in integer)

Your input ____
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