The TOSHIBA Satellite 1135-S125 contains the Intel 852GM chipset soldered on your motherboard (non removable) which is capable for speeds of 400mhz system bus. Actually what this means is that it capable of running your Pentium-M P4 2.4. so I would doubt a motherboard incompatibility.
The processor runs at the speed you state (1.2 GHz) in battery-optimized mode (1.2V core)
(24/2multiplier) instead of its full potential when unplugged from the AC adaptor. I believe the speed remains like that even when plugged at the cable due to dynamic switching either at the BIOS or Windows. This means that the system is slowing down deliberately the CPU until a certain % and then it kicks in full power. (24x100Mhz=2400Mhz multiplier) (For heat purposes mainly,also increases the CPU life..so this is normal operation and you should not be worried)
Please download the utility cpu-z at the following link http://www.cpuid.com/cpuz.php
and check what I am saying. It will be displaying the Ghz of the CPU realtime so you can see if when you start any benchmark if it kicks in with a higher multiplier and hence the 2400Mhz speed due to increased stress to the CPU.
Otherwise you may check your BIOS for power management options and switch them to Disable just for testing. (you should enable them back after testing in order to cut down the increased watt consumption)
Lastly if that does not help a Bios Update might be the only resolution. I have found this Motherboad BIOS update for you
ACPI Flash BIOS version 1.70 for Satellite
This bios seems to enable support for CPU's at steps A2 which may be your chip. Unfortunately you did not state the BIOS revision so if it the V1.30 this bios will also add support for CPU's at stepping D1 as well.
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