Will the GTX1050 Ti Mini DDR5 4GB graphics card support the Intel i5 3rd Gen PC ?

You can actually use any Graphics card in any PC, regardless of it being outdated or the latest one. GPUs that we get for last 6–8 years are all based on PCI express slot and they need the same slot to support, in most machines it is there yet to may reconfirm before you actually buy the card.

There are more options than MINI in GTX1050Ti Range - you may consider opting for a better stronger card than MINI - check out various models with prices over here ..

http://bwindia.net/catalog/hardwares/graphics-card-latest-models-suggested/nvidia-geforce-10-series/graphics-c...

However, you shall check your smps wattage and compatibility with the card beforehand.