GATE CSE 2024 Paper 1 — page 17

Questions 49–51 of 65. Imported answers require review unless explicitly marked reviewed.

Let

be any

matrix, where

. Which of the following statements is/are TRUE about the system of linear equations

Your answerSelect all correct options. You can choose more than one.

There exist at least

linearly independent solutions to this system

There exist

linearly independent vectors such that every solution is a linear combination of these vectors

There exists a non-zero solution in which at least

variables are 0

There exists a solution in which at least

variables are non-zero

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Consider the 5-state DFA

accepting the language

shown below. For any string

let

be the number of 0's in

and

be the number of 1's in

Which of the following statements is/are FALSE?

Your answerSelect all correct options. You can choose more than one.

States 2 and 4 are distinguishable in

States 3 and 4 are distinguishable in

States 2 and 5 are distinguishable in

Any string

with

is in

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The chromatic number of a graph is the minimum number of colours used in a proper coloring of the graph. Let

be any graph with

vertices and chromatic number

. Which of the following statements is/are always TRUE?

Your answerSelect all correct options. You can choose more than one.

contains a complete subgraph with

vertices

contains an independent set of size at least

contains at least

edges

contains a vertex of degree at least

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2024 · Paper 1