Pre-calculus | Topic: Complex Roots | Level One: Complex Numbers

The expression z-z* represents the difference between a complex number z and its complex conjugate z*. The complex conjugate of a complex number has the same real part but the imaginary part with opposite sign. Therefore, when we subtract z* from z, the real parts cancel out, leaving only the imaginary part. This means that z-z* is always purely imaginary or zero.

The given equation is a complex number equation in the form (a+i)(3+bi)=40-74i. By multiplying the terms using the FOIL method, we get 3a + 3bi + ai - b = 40 - 74i. Equating the real and imaginary parts of the equation separately, we get 3a - b = 40 and 3b + a = -74. Solving these two equations simultaneously, we find that a = -11 and b = 7, which matches the given answer.

The correct answer is Option 2 because to find z+w, we need to add the real parts and the imaginary parts separately. The real part of z is 1 and the real part of w is 3, so the real part of z+w is 1+3=4. The imaginary part of z is 2i and the imaginary part of w is -i, so the imaginary part of z+w is 2i+(-i)=i. Therefore, z+w is equal to 4+i, which is represented by Option 2.

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Port 20 is used for the "Data" transfer in FTP (File Transfer Protocol) and it works in conjunction with port "21" which is used for the control commands.

SSH (Secure Shell) is a network protocol that allows secure remote access to a computer or server. It is commonly used for secure remote administration and file transfers. The correct answer is 22 because SSH is typically assigned to port number 22.

TFTP stands for Trivial File Transfer Protocol. It is a simple and lightweight file transfer protocol that is commonly used for transferring files between network devices. Unlike other file transfer protocols, TFTP does not have features like authentication or directory listing, making it less secure but faster and easier to implement. TFTP is often used in scenarios where speed and simplicity are more important than security, such as booting diskless workstations or transferring firmware updates to network devices.

TFTP (Trivial File Transfer Protocol) is assigned to port number 69. This port is used for transferring files between a client and a server in a simple and lightweight manner. TFTP is commonly used for network booting, firmware updates, and other similar operations where a small and fast file transfer protocol is needed.

DHCP (Dynamic Host Configuration Protocol) uses port numbers 67 and 68. Port 67 is used by DHCP servers to receive client requests and provide IP configuration information, while port 68 is used by DHCP clients to send requests and receive IP configuration from DHCP servers. These port numbers are standardized and widely used for DHCP communication.

The correct answer is "IMAP: 143, POP3: 110". This choice includes both of the Mail Receiving Protocols (IMAP and POP3) with their proper assigned port numbers (IMAP: 143, POP3: 110).

The complex conjugate of a real number is itself, so if z is real, then z* will also be a real number.