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The cracker scans for open ports on the victim's pc and attempts to connect to the victims pc. Secondly, the intruder will start making use of the open port and install backdoor/trojan to gain control of the user's computer?
an open port is a possibility, not a vulnerability. Attacker needs a open port AND a vulnerability to attack
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Thirdly, using the Task Manager, can i be able to view all applications running in the background. Any possibility that the backdoor is not shown in the task manager as a process?
Unfortunatelly, there are still ways to avoid a program to be showed on task manager. However, you can see those on "hijackthis" utility, except if the malware is installed as a system driver or something like that (i dont know anyone that do that)