CUET · MATHS · PYQ PAPER 2023

A cone, whose height is always equal to its diameter, is increasing in volume at the rate of \(40 cm^3 / sec\). The rate at which radius is increasing when circular base area is \(1 m^2\) is :

A 1 mm/sec
B 2 mm/sec
C 0.01 mm/sec
D 0.002 mm/sec

Verified Solution

Answer & Solution

Correct Answer

(D) 0.002 mm/sec

Step-by-step Solution

Detailed explanation

\(h = 2r\) \(V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi r^2 (2r) = \frac{2}{3}\pi r^3\) \(\frac{dV}{dt} = 2\pi r^2 \frac{dr}{dt}\) \(A = \pi r^2 = 1 m^2 = 10000 cm^2\) \(40 cm^3/sec = 2 (10000 cm^2) \frac{dr}{dt}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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