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Acceleration or mass ratio for blocks hung over a pulley (Atwood machine)

This type of question, asked 2 times: 2 May 2023 S1, 18 Apr 2026 S2. All 2 questions below, oldest first.

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  1. Question 1

    2 May 2023 S1

    12
    Two identical blocks each of mass MM attached to the ends of a massless inextensible string which passes over a pulley with a fixed axis as shown below. A small mass mm is now placed on the block B. The acceleration with which the two blocks move together is [g=g = gravitational acceleration]

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  2. Question 2

    18 Apr 2026 S2

    104
    An inextensible string passing over a smooth pulley connects two blocks of masses m1m_1 and m2m_2 (m2>m1)(m_2 > m_1) vertically. If the acceleration of the system is (gn)\left(\dfrac{g}{n}\right), then the ratio of masses (m2m1)\left(\dfrac{m_2}{m_1}\right) is

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