If a computer beeps( in the normal way), does it, necessarily, imply that its hardware (especially the cpu) is okay?

Ermmm....

Can you be a little more specific about this "beep" you're referring to?

From my A+ book.

When the computer is turned on or reset, it initiates a special program, also stored on the ROM chip, called the Power On Self Test(POST). The POST program checks out the system every time the computer boots. To perform this check, the POST sends out a standard command thats says to all the devices "check yourselfs out!" All the devices in the computerthen run their own internal diagnostic-the POST doesnt specify what to check. The quality of the diagnostic is up to the people who made that particular device.

All PCs beep on startup to let the user know the speaker is working. Now you know why every computer always beeps when it first starts!

The single beep that the BIOS generates to indicate successful completion of the Power-On Self-Test [POST] indicates that there were no "Fatal Errors" encountered when the BIOS tested all the hardware.
I would take that to mean that all the hardware [including the CPU] passed the tests [and is therefore deemed OK].

Having said that, I don't know the details of the tests.

To further this conversation (if anyone cares to) the “Single” beep is the industry standard for a successful POST. Some BIOS have a short 2 beep code for the same thing. Other combinations of beeps will indicate some problem with the POST and are somewhat proprietary to the BIOS chip.
A visit HERE (http://www.bioscentral.com/) will give you a bit more info them most care for. You can look up different combinations for your BIOS chip here and determine what problem you are having, if in fact you are ;)

Good luck
Frank