I have installed a modem but it keeps on being undetected after every reboot. I always have to search for plug and play devices after every reboot to return it into the Device Manager hardware list.

Why does this happen?

Try physically removing the modem from the computer. If you have a network card installed try removing that also. Turn off the computer, reboot. Now install just the modem. Power up. and let windows try to recognize the installation and find/install drivers for it. If it does not find the drivers you will have to download the right ones or off CD that came with the modem. After reinstall the network card NIC if needed.

Also, in some older PC Bios configs you had to enable modem IRQ ports. You might want to check your BIOS also just to make sure.

It's an external modem and I have installed the driver from the cd that came with it.

If it's on when Windows boots it gets detected but if it's off it can't be. Shouldn't Windows detect it automatically after being turned on?