GATE EC 2022 Question Paper (06-Feb-2022) (Shift 1)

A simple closed path 𝐶 in the complex plane is shown in the figure. If

$\oint \frac{2^z}{z^2-1}dz=-i\pi A$

where i = √-1 then the value of A is _______ (rounded off to two decimal places)

In an electrostatic field, the electric displacement density vector D is given by

$\vec{D}(x,y,z)=(x^3\hat{i}+y^3\hat{j}+x{y}^2\hat{k})$ C/m²,

Where, $\hat{i},\hat{j},\hat{k}$ are the unit vectors along the x-axis, y-axis, and z-axis, respectively. Consider a cubical region 𝑅 centered at the origin with each side of length 1 m, and vertices at (± 0.5 m, ± 0.5 m, ± 0.5 m). The electric charge enclosed within 𝑅 is _________ C (rounded off to two decimal places).

Let x₁(t) = e${}^{-t}$ u(t) and x₂(t) = u(t) - u(t - 2), where u(⋅) denotes the unit step function.

If y(t) denotes the convolution of x₁(t) and x₂(t), then $\underset{t\to \infty }{\lim }y(t)=$ __________ (rounded off to one decimal place).

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, f(x), as shown in the figure.



Consider a 1 bit quantizer that maps positive samples to value 𝛼 and others to value 𝛽. If α and β are the respective choices for α and β that minimize the mean square quantization error, then (α - β) = _________ (rounded off to two decimal places).

An ideal MOS capacitor (p-type semiconductor) is shown in the figure. The MOS capacitor is under strong inversion with V${}_G$ = 2 V. The corresponding inversion charge density (Q${}_{iN}$) is 2.2 μC/cm². Assume oxide capacitance per unit area as C${}_{OX}$ = 1.7 μF/cm². For V${}_G$ = 4 V, the value of Q${}_{IN}$ is _______μC/cm². (rounded off to one decimal place).