Windows 7 Does core i5 CPU with H67 chipset support monitors with resolution 2560x1600?

According to the Core i5-2400 Specs: Link Removed

Your processor has the following:

Graphics Specifications
Processor GraphicsLink Removed
Graphics Base Frequency850 MHz
Graphics Max Dynamic Frequency1.10 GHz
Intel® Quick Sync VideoLink Removed
Intel® InTru™ 3D TechnologyLink Removed
Intel® Wireless DisplayNo
Intel® Flexible Display Interface (Intel® FDI)Link Removed
Intel® Clear Video HD TechnologyLink Removed
Dual Display Capable ‡Link Removed

As you can see it has the HD 2000 graphics built in.

• Intel® HD Graphics 2000 and Intel® HD Graphics 3000 are the second generation of processor graphics and perform much higher than the original version. The Intel® HD Graphics 2000 and Intel® HD Graphics 3000 were introduced with the Intel® Core™ i7-2xxx, Intel® Core™ i5-2xxx, and Intel® Core™ i3-2xxx Processors.

The problem now lies in the fact that it seems the HD 2000 resolution can depend on what exact generation etc.. so I'm not sure. I see some posts elsewhere mentioning running at 1680x1050, however they also mention that the HD Graphics 2000 does not do very well in gaming if you push it too far.

Seeing that resolution mentioned makes me think no it will not do the resolution that you wish to accomplish, however it will do 1080 HD video.