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Hello,The National Eligibility Test is conducted by National Testing Agency to select the candidates for the post of Assistant Professor and for awarding Junior Research Fellowship. If you qualify this exam for the post of Assistant Professor (Lecturer), there is no need to appear in any entrance exam for teaching. Here is the process that can help you to know how to become lecturer after qualifying NET Exam. https://mahawarravi036.wixsite.com/studentexamalert/single-post/2018/10/16/Job-Process-After-Qualifying-UGC-NET
If you want to be a government teacher in the state or central government then you can appear in Teaching Eligibility Test (TET). So, my advice is, go for only one exam preparation out of these two. Hope it helps.

Him,
Eligibility Criteria:
Candidates seeking admission to the B.Sc B.Ed.
The programme should have passed CBSE Senior Secondary examination/ Pre-University examination of Karnataka or an equivalent examination of Kerala, Andhra Pradesh, Tamil Nadu, Pondicherry or the UT of Lakshadweep with 45% marks in the aggregate. Relaxation up to 5% of marks shall be given to the SC/ST candidates. Candidates should have passed the qualifying examination with the following combinations of subjects. For admission to the PCM stream: Physics, Chemistry, Mathematics/ and any other subject approved the university of Mysore; For admission to CBZ stream: Physics, Chemistry, Mathematics, Biology / Physics, Chemistry, Biology / Chemistry, Botany, Zoology. Admission Process
Admission shall be regulated through selection on the basis of marks in the qualifying examination or performance in a specially designed selection test or both as per the admission policies of NCERT from time to time. Yes, you are eligible.