GATE IN 2021 Question Paper (06-Feb-2021) (Shift 1)

A Boolean function F of three variables X, Y, and Z is given as

F (X, Y, Z) = (Xʹ + Y + Z)˖(X + Yʹ + Zʹ )˖(Xʹ + Y + Zʹ )˖( Xʹ Yʹ Zʹ + Xʹ Y Zʹ + X Y Zʹ )

Which one of the following is true

A $10\frac{1}{2}$ digit Counter-timer is set in the ‘frequency mode’ of operation (with Tₛ = 1 s). For a specific input, the reading obtained is 1000. Without disconnecting this input, the Counter-timer is changed to operate in the ‘Period mode’ and the range selected is microseconds (μs, with fₛ= 1 MHz). The counter will then display

A J-type thermocouple has an output voltage V${}_{\theta }$ = (13650 + 50 θₓ) μV, where θₓ is the junction temperature in Celsius (°C). The thermocouple is used with reference junction compensation, as shown in the figure. The Instrumentation amplifier used has a gain G = 20. If θ${}_{Ref}$ is 1 °C, for an input θₓ of 100 °C, the output Vₒ of the instrumentation amplifier in millivolt is

A laser pulse is sent from ground level to the bottom of a concrete water tank at normal incidence. The tank is filled with water up to 2 m below the ground level. The reflected pulse from the bottom of the tank travels back and hits the detector. The round-trip time elapsed between sending the laser pulse, the pulse hitting the bottom of the tank, reflecting back and sensed by the detector is 100 ns. The depth of the tank from ground level marked as x in metre is ____.
(Refractive index of water n${}_{water}$ = 1.3 and velocity of light in air cₐᵢᵣ = 3 × 10⁸m/s)

A 4 × 1 multiplexer with two selector lines is used to realize a Boolean function F having four Boolean variables X, Y, Z and W as shown below. S0 and S1 denote the least significant bit (LSB) and most significant bit (MSB) of the selector lines of the multiplexer respectively. I0, I1, I2, I3 are the input lines of the multiplexer.



The canonical sum of product representation of F is