GATE CS 2022 Q30 — Combinational Circuits

To determine the functionality of components $\text{P}$, $\text{Q}$, and $\text{T}$, we analyze their roles in the digital display system:

1. Functionality of $\text{P}$ (Addressing Memory $\text{S}$):
   • Memory segment $\text{S}$ contains $1024$ words ($1024 = 2^{10}$).
   • To select one specific word out of $1024$ words, a $10$-bit address is required.
   • A decoder takes $n$ input lines and activates one of the $2^n$ output lines. Therefore, a $10 : 2^{10}$ decoder is needed to decode the $10$-bit address and select the corresponding word in $\text{S}$.
   • Thus, $\text{P}$ is a $10 : 2^{10}$ decoder.
2. Functionality of $\text{Q}$ (Addressing Register File $\text{R}$):
   • Register file $\text{R}$ contains $32$ words ($32 = 2^5$).
   • To select one specific register out of $32$, a $5$-bit address is required.
   • Similarly, a $5 : 2^5$ decoder is needed to decode the $5$-bit address and select the corresponding register in $\text{R}$.
   • Thus, $\text{Q}$ is a $5 : 2^5$ decoder.
3. Functionality of $\text{T}$ (Selecting the Input to Register $\text{X}$):
   • $\text{T}$ takes two data inputs (one from memory $\text{S}$ and one from register file $\text{R}$) and routes one of them to register $\text{X}$.
   • The selection is controlled by a single mode bit $\text{M}$.
   • A combinational circuit that selects one of $2^n$ inputs based on $n$ select lines is a multiplexer (MUX). Since there are $2$ inputs and $1$ select line, $\text{T}$ is a $2:1$ multiplexer.

Combining these results, we have:

• $\text{P}$: $10:2^{10}$ decoder
• $\text{Q}$: $5:2^5$ decoder
• $\text{T}$: $2:1$ multiplexer

This corresponds to Option C.