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CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Q263 Marksrelated_rates

CBSE Class 12 Mathematics 2026 Main Paper 65/5/1 · Question 26 (Applications Of Derivatives)

A spherical balloon loses volume at a rate proportional to its surface area. Prove that its radius decreases at a constant rate.

✅ Step-by-Step Solution

Final Answer: dr/dt=−k, where k>0 is the proportionality constant.
  1. Write dV/dt=−k·4πr².
  2. As V=4πr³/3, the chain rule gives dV/dt=4πr² dr/dt.
  3. For r>0 cancel the surface area to obtain dr/dt=−k.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Write dV/dt=−k·4πr².
  • [2 marks] As V=4πr³/3, the chain rule gives dV/dt=4πr² dr/dt.
  • [0.5 marks] For r>0 cancel the surface area to obtain dr/dt=−k.

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All Questions in CBSE Class 12 Mathematics 2026 Main Paper 65/5/1

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