CBSE 2026 · Region 5 · Set 2 · Q26 · 3 marks
A thin metallic wire in the shape of a circular ring has its enclosed area increasing at a uniform rate when heated. Show that the rate of change of circumference varies inversely as the radius.
Marking-scheme solution
Let r be the radius of circular ring. Area $\displaystyle A=\pi r^{2}$
$\displaystyle \dfrac{dA}{dt}=2 \pi r \dfrac{dr}{dt}=K$ ($\displaystyle K>0$ is a constant)
Circumference $\displaystyle C=2 \pi r$
$\displaystyle \dfrac{dC}{dt}=2 \pi \dfrac{dr}{dt}=2 \pi \dfrac{K}{2 \pi r}$ using (i)
$\displaystyle =\dfrac{K}{r}$
Thus, rate of change of circumference varies inversely with radius.
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CBSE Class 12 Mathematics past-paper question from the 2026board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.