As you said, you are a maths student, it is always welcome to pursue whatever UG you can. It is good to choose management studies. After aspiring what you want to achieve in UPSC with managerial skills. It is also known that MBA is in boom and easily to get recruited and gives you confidence to stand for UPSC many times.

There is a huge difference between the two. Sarat Chandra IAS Academy is run by Sarat sir who got 195 marks in UPSC interview and cracked several competitive exams. He is a graduate from IIT Madras and he produced several IAS, IPS Rankers. He is very strong in almost every subject of UPSC Civil services exam. All the faculty of Sarat Chandra IAS Academy (Mathews, Kalyan, Kiran, Mukesh etc) are experts and he frequently invites several IAS and IPS officers (Ketan Garg IAS for Economy, Saveesh Varma IRS for polity etc).Students of Sarat Chandra IAS Academy won almost all the inter college competitions. Sarat Chandra IAS Academy is undoubtedly the Best IAS academy for Degree + IAS, and these students are able to crack UPSC Civil services, RBI and many central government exams. That is why students from several states across India come to this institute vto join Degree + IAS and Inter + IAS. They finish syllabus in first 2 years and revise it several times. They take students for extra curricular activities also for competitions. Sarat sir's wife Vijaya Keerthi is an IRSE officer and she too helps the students. Takshasila Academy is not even comparable to Sarat Chandra IAS Academy.

Sarat Chandra IAS Academy is there in Vajayawada. They have some kind of integrated course for 3 years where you can pursue your graduation and at the same time prepare for UPSC examination. But if you want to take only the IAS course it is for 1 year along with all the levels of the examination. But in the integrated course they will prepare for all other similar exams. 

Following are a few good colleges for BA in Hyderabad and you can also have ias coaching in hyderabad. 

Osmania University College for women 

St. Francis college for women 

Lakhotia institute of design 

St. Joseph degree and PG college

Moreover Osmania University would be recommended to you for its reputation and course structure that might be helpful for your ias preparation 

No institute give coaching on the basis of board exams. If you start your preparation it advisable that you start it from your home or do a group study with like-minded friends as simply investing in coaching is not good, rather than that you purchased books and there is a lot of stuff on the internet where you can study literally all subjects free of cost. Unacademy is the site who give coaching free of cost online. 

• Sharma Academy
• PS Academy
• Anand Super 100
• Chanakya IAS Academy
• Mathematica- The complete Education
• Arjun IAS Academy
• Tathagat IAS

No matters at all whatever you choose either BA or BSc. The only eligibility criteria to the UPSC exam is to be a graduate only. What matters the most is your preparation because all subjects and all papers are equally important in UPSC. Whether it is History, geography, Physics, Chemistry, Economics. You have to focus on syllabus and have to stick to that. 

You start your preparation for UPSC from initial college days so that you have 3 years in your hand.