I have an Exide invamaster 1500imst short tubeler normal battery of 12 volt, 150AH, C10. Now I want to connect this battery with a solar battery of 12 volt, 150AH, C20. Can I connect this normal 12 Volt 150AH C10 battery with a solar 12 volt 150AH C20 battery in series for my 24Volt solar system. Will it damage the battery or what will be happened if I use them in series for a solar inverter of 24 volt battery system with 1200 watt solar pannel.
Can you - yes. Is it a good idea - no.
A 150AH @ C10 and 150AH @ C20 battery are not the same "size" because one is rated at the 10-hour discharge rate and the other is rated at the 20 hour discharge rate.
Since you have not specified some other chemistry, I'm assuming both are lead-acid 12V batteries, as the majority of 12V batteries are.
If by "normal" you mean "automobile starting" battery, that one may have short life due to construction optimized for a very different service than a deep cycle (or "solar") battery. On the other hand, by pairing them you are limited to the solar battery's discharge rate if you want to get rated capacity (150AH @ 20hr - 7.5 A - at 24V about 180 Watts before inverter losses, perhaps 150 Watts after inverter losses.)
If that seems like a very low power output, welcome to the ugly realities of solar power, where almost everyone starting out undersizes their battery bank until they have the expensive lesson that it costs a lot to be cheap.
You're not only mixing different models. You're also mixing new and used batteries. This means, as a pack, you will get the performance of the worst battery. When it gives up the ghost, you will be left with 1 battery that still has some useful life, and you will be tempted to pair it with a new battery. Don't. At that point replace with all new.
Keep in mind...
The problem with lead-acid as a solar battery is that they give very short life. They give a lot shorter life when they are deep cycled.
Now you think "Full use of the battery's range from 100% to 0% is the entire point of a battery, that is its most basic function. Surely Harper, you can't be saying a lead-acid battery can't do that!"
Yeah, that's exactly what I'm saying.
And that's not entirely unheard of in the battery business; even with lithium (though it's not nearly as bad). Tesla locks out (in software) the bottom 20% of the battery's range, which greatly increase the service life of the battery.
With lead-acid, it is much worse. You should research this yourself, but many wise solar designers say the bottom 70% of nameplate capacity is unusuable. Do not draw more than 30% of capacity from a lead-acid on a daily basis - that is, don't plan to dip it below 70% charged. If you have 150AH @ 24V, plan to use 45AH @ 24V.
It's OK to dip it to say 50% or 40% SOC on an occasional basis, but don't make a habit of that, or it'll significantly shorten the life of your pack. Dipping it all the way down to 10% SOC, your battery will be toast in 20-100 deep cycles. And that's assuming it's built as a deep-cycle battery; a car starting battery will fare worse.