1. Khan Academy:
Khan Academy is a widely popular online platform that offers a comprehensive range of video lectures and tutorials for various subjects, including Mathematics. Their Mathematics section covers a wide range of topics from basic arithmetic to advanced calculus, making it suitable for students preparing for the Mathematics optional subject in UPSC exams. The lectures are presented in a clear and concise manner, making it easy for students to understand and revise the concepts. Additionally, Khan Academy provides practice exercises and quizzes to test your understanding and reinforce the learning.

2. MIT OpenCourseWare:
MIT OpenCourseWare is a free and open educational resource that provides access to video lectures and course materials from Massachusetts Institute of Technology (MIT). They offer a variety of courses in Mathematics, including topics like linear algebra, differential equations, and probability theory. The lectures are delivered by renowned professors and experts in the field, ensuring high-quality content. The video lectures are accompanied by lecture notes, assignments, and exams, allowing students to practice and assess their knowledge.

3. NPTEL:
NPTEL (National Programme on Technology Enhanced Learning) is an initiative by the Indian Institutes of Technology (IITs) and the Indian Institute of Science (IISc) to provide quality education through online courses. They offer a wide range of video lectures on Mathematics, covering topics like calculus, linear algebra, and complex analysis. The lectures are delivered by professors from prestigious institutions, ensuring the accuracy and depth of the content. NPTEL also provides course materials, quizzes, and assignments to enhance the learning experience.

4. EduRev:
EduRev is a comprehensive online learning platform that offers a vast collection of video lectures and tutorials for various subjects, including Mathematics. Their Mathematics section covers a wide range of topics, including algebra, geometry, calculus, and statistics. The lectures are presented by experienced teachers and experts in the field, ensuring a clear and detailed explanation of the concepts. EduRev also provides practice questions, quizzes, and previous year question papers to help students prepare effectively for the Mathematics optional subject in UPSC exams.

In conclusion, Khan Academy, MIT OpenCourseWare, NPTEL, and EduRev are highly recommended online platforms for revision of the Mathematics optional subject. Each platform offers a wide range of video lectures and tutorials delivered by experienced teachers and experts in the field. Additionally, they provide supplementary materials such as lecture notes, assignments, and quizzes to enhance the learning experience and assess the understanding of the concepts.