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No. You do not require to have biology as a subject to become a psychologist. After 12th, you can pursue either BSc or BA, then MSc or MA, and PhD. To become a clinical psychologist, you can do MPhil in clinical psychology. However, biology helps you study clinical psychology better. There is one subject in masters course, psycho-biology or biological basis of bahavior.
If you are looking forward to bece psychiatrist, you need biology and complete MBBS. After MBBS, you can opt for MD in psychiatry to become psychiatrist.
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