GATE EC 2018 Question Paper (10-Feb-2018) (Shift 1)

In the circuit shown below, a positive edge-triggered D Flip-Flop is used for sampling input data 𝐷${}_{\text{in}}$ using clock 𝐶𝐾. The XOR gate outputs 3.3 volts for logic HIGH and 0 volts for logic LOW levels. The data bit and clock periods are equal and the value of Δ𝑇/𝑇${}_{\text{CK}}$ = 0.15, where the parameters Δ𝑇 and 𝑇${}_{\text{CK}}$ are shown in the figure. Assume that the Flip-Flop and the XOR gate are ideal.



If the probability of input data bit (𝐷${}_{\text{in}}$) transition in each clock period is 0.3, the average

value (in volts, accurate to two decimal places) of the voltage at node 𝑋, is _______.

The logic gates shown in the digital circuit below use strong pull-down nMOS transistors for LOW logic level at the outputs. When the pull-downs are off, high-value resistors set the output logic levels to HIGH (i.e. the pull-ups are weak). Note that some nodes are intentionally shorted to implement “wired logic”. Such shorted nodes will be HIGH only if the outputs of all the gates whose outputs are shorted are HIGH.



The number of distinct values of 𝑋₃𝑋₂𝑋₁𝑋₀ (out of the 16 possible values) that give 𝑌 = 1 is _______.

The cut-off frequency of TE₀₁ mode of an air-filled rectangular waveguide having inner dimensions a cm × b cm (a > b) is twice that of the dominant TE₁₀ mode. When the waveguide is operated at a frequency which is 25% higher than the cut-off frequency of the dominant mode, the guide wavelength is found to be 4 cm. The value of b (in cm, correct to two decimal places) is _______.

A uniform plane wave traveling in free space and having the electric field.

$\vec{\mathrm{E}}=\left(\sqrt{2}{\hat{\mathrm{a}}}_{\mathrm{x}}-{\hat{\mathrm{a}}}_{\mathrm{z}}\right)\cos \left[6\sqrt{3}\pi \times {10}^8\mathrm{t}-2\pi \left(\mathrm{x}+\sqrt{2}\mathrm{z}\right)\right]\frac{\mathrm{V}}{\mathrm{m}}$

is incident on a dielectric medium (relative permittivity > 1, relative permeability = 1) as shown in the figure and there is no reflected wave.



The relative permittivity (correct to two decimal places) of the dielectric medium is

___________.

The position of a particle (𝑡) is described by the differential equation:

$\frac{{\mathrm{d}}^2\mathrm{y}}{\mathrm{d}{\mathrm{t}}^2}=-\frac{\mathrm{dy}}{\mathrm{dt}}-\frac{5\mathrm{y}}{4}$

The initial conditions are y(0) = 1 and ${\frac{\mathrm{dy}}{\mathrm{dt}}|}_{\mathrm{t}=0}=0$. The position (accurate to two decimal places) of the particle at t = π is _______.