MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2026: question 26

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section C, question 26, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26 · 3 marks · Short answerCompetency-basedDifferential Equations

A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Translate 'rate of decrease of volume is proportional to surface area' into an equation with a constant $k$ .

Hint 2

Differentiate $V = \frac{4}{3} π r^{3}$ with respect to time and compare the two expressions for $\frac{d V}{d t}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Let $r$ be the radius and $V = \frac{4}{3} π r^{3}$ the volume at time $t$ . The volume decreases at a rate proportional to the surface area, so $\frac{d V}{d t} = - k (4 π r^{2})$ , where $k > 0$ is a constant.
1 mark
Differentiating $V = \frac{4}{3} π r^{3}$ with respect to $t$ : $\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t}$ . Hence $4 π r^{2} \frac{d r}{d t} = - 4 k π r^{2}$ .
1 mark
Therefore $\frac{d r}{d t} = - k$ , which is a negative constant. So the radius decreases at a constant rate.
1 mark

Answer: $\frac{d r}{d t} = - k$ (a constant), so the radius decreases at a constant rate

Where the marks are

Forms $\frac{d V}{d t} = - k (4 π r^{2})$ with $k > 0$ (1 mark)
Uses $\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t}$ and equates the two (1 mark)
Gets $\frac{d r}{d t} = - k$ and concludes (1 mark)

Revise the chapter

Chapter: Differential Equations (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Differential Equations, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 26

Hints

Step-by-step solution

Revise the chapter

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