There are various universities and colleges in India that conduct their own entrance tests and exams for admission into undergraduate courses in arts.
Some of them are explained below:

TISS BAT : Tata institute of social sciences annually conducts Bachelor Admission test and it is scheduled for April 28,2019. It is conducted to offer Admission to their different Bachelor of Arts programs. The applications for the Entrance test has started and it will be closed on march 20,2019. You can apply for this Application in online mode.

BHU UET : Benaras Hindu University conducts Undergraduate Entrance Test for admission  to undergraduate courses in arts.It is a national level entrance test. This year, BHU UET 2019 is scheduled from May 12 to 16, 2019 in offline mode.
Candidates must pass 10+2 with minimum 35% marks from the recognized board/university to appear for the exam. The courses offered are

B.A.(H)
B.Com. (H)
B.A. LL.B. (Hons)
LL.B
B.P.Ed
B.P.A.
B.F.A.
B.Voc.
B.Ed.

Patna Women’s College Entrance Exam :

The college offers an entrance exam for admission to undergraduate studies.
Candidates who will be appearing for the qualifying examination for the current academic year are also eligible to apply subject to the condition that they produce of having passed the qualifying examination with at least 45% marks at the time of admission.
Students who want to applying for English Honours need to appear in the proficiency test.Only, the students who  qualify the entrance test will be able to appear for Proficiency Test. The final result will be made on the basis of entrance and PET exam.Compartment / failed candidates or candidates with less than 45% marks secured in the qualifying examination, will not be considered for admission even if they have qualified in the Entrance Test 2019.
Minimum Qualifying Marks-  The candidates need to score 50% marks (45% for SC/ST/EBT) as per the last year information.

JMIEE by Jamia Millia Islamia:

Jamia Millia Islamia (JMI) conducts an entrance exam for the admission to various Undergraduate programs. Last date of application is march 14,2019.
A valid score in JMI Entrance Test followed by an Interview is the primary requirement to secure admission in
BA
B.Com
BBA
B.Sc
BA-LL.B
BHM
BPT
B.Li.I.Sc
BFA
BTTM

AUAT by Alia University :

Aliah University Admission Test (AUAT) is a national level entrance exam conducted by Aliah University for offering admission to various UG courses.
AUAT 2019 will comprise of objective type questions except for few language coursed.Examination will be conducted in offline mode in English language only.

Jawaharlal Nehru University Entrance exam:
It is conducted for admission into undergraduate courses as well as pg courses.The exam is scheduled for December 2019 in the offline mode.
Candidates must have passed their 10+2 examination with a minimum of 45% marks.Candidates who will be appearing in 10+2 examination can also apply.
Selected candidates of JNUEE 2019 will be called for viva-voce. After the viva-voce, a merit list will be published by JNU.

I hope the above details are helpful to you.