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

Two linear time-invariant systems with transfer functions

${\mathrm{G}}_1(\mathrm{s})=\frac{10}{{\mathrm{s}}^2+\mathrm{s}+1}$ and ${\mathrm{G}}_2(\mathrm{s})=\frac{10}{{\mathrm{s}}^2+\mathrm{s}\sqrt{10}+10}$

have unit step responses y1(t) and y2(t), respectively. Which of the following statements is/are true?

A waveguide consists of two infinite parallel plates (perfect conductors) at a separation of 10${}^{-4cm,withairasthedielectric.Assumethespeedoflightinairtobe3\times 10}$^8 m/s. The frequency/frequencies of TM waves which can propagate in this waveguide is/are _______.

The value of the integral

_D 3(x^2+y^2)\ dxdy \ \

where D is the shaded triangular region shown in the diagram, is _____ (rounded off to the nearest integer).

A linear 2-port network is shown in Fig. (a). An ideal DC voltage source of 10 V is connected across Port 1. A variable resistance R is connected across Port 2. As R is varied, the measured voltage and current at Port 2 is shown in Fig. (b) as a V₂ versus -I₂ plot. Note that for V₂ = 5 V, I₂ = 0 mA, and for V₂ = 4 V, I₂= −4 mA.

When the variable resistance R at Port 2 is replaced by the load shown in Fig. (c), the current I₂ is _______ mA (rounded off to one decimal place).

For a vector x̅ = [x[0], x[1],......x[7]] the 8-point discrete Fourier transform (DFT) is denoted by X̅ = DFT (x̅) = X[0], X[1], ...., X[7]], where

$\mathrm{X}[\mathrm{k}]={\sum }_{\mathrm{n}=0}^7\mathrm{x}[\mathrm{n}]\exp \left(-\mathrm{j}\frac{2\pi }{8}\mathrm{nk}\right)$

Here j = √-1, if X̅ = [1, 0, 0, 0, 2, 0, 0, 0] and y̅ = (DFT (x̅)), then the value of y[0] is ________ (rounded off to one decimal place).