MathsCBSE Class 12 · Sample paper, session 2026-27 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2026-27: question 27

From the official CBSE sample question paper for 2026-27, section C, question 27, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27 · 3 marks · Short answerApplication of Derivatives

The circular area of vision on the ground by a surveillance drone fitted with camera increases at the rate of $44 m^{2}$ per metre as it gains vertical height. The radius of the circle on the ground increases at the rate of 6 m/s. Find the rate at which the height of the drone increases with time, at the instant when the radius of the circular area on the ground is 14 m. (Use $π = \frac{22}{7}$ )

Figure in the paper: Picture of a drone with camera hovering above the ground, a downward arrow indicating its vertical height, and a dashed ellipse/circle on the ground showing the area of vision. See it in the official paper.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2026-27.

Hints

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

Hint 1

Write $A = π r^{2}$ and differentiate with respect to time.

Hint 2

Use the chain rule $\frac{d A}{d t} = \frac{d A}{d h} \cdot \frac{d h}{d t}$ to bring in the given 44 m² per metre.

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 $A$ be the area of the circle of vision, $r$ its radius and $h$ the height of the drone at time $t$ . Given $\frac{d A}{d h} = 44$ m²/m and $\frac{d r}{d t} = 6$ m/s. We need $\frac{d h}{d t}$ when $r = 14$ m.
½ mark
$A = π r^{2} \Rightarrow \frac{d A}{d t} = 2 π r \frac{d r}{d t}$ . By the chain rule, $\frac{d A}{d t} = \frac{d A}{d h} \cdot \frac{d h}{d t}$ , so $\frac{d A}{d h} \cdot \frac{d h}{d t} = 2 π r \frac{d r}{d t}$ .
1½ marks
$44 \frac{d h}{d t} = 2 \times \frac{22}{7} \times 14 \times 6 = 528 \Rightarrow \frac{d h}{d t} = 12$ m/s. So the height of the drone increases at 12 m/s at that instant.
1 mark

Answer: 12 m/s

Where the marks are

Defines the variables and writes the given rates (½ mark)
Differentiates $A = π r^{2}$ and links the rates by the chain rule (1½ marks)
Substitutes $r = 14$ to get $\frac{d h}{d t} = 12$ m/s (1 mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Application of Derivatives, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2026-27: every question, by section.

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CBSE Class 12 Mathematics sample paper 2026-27: question 27

Hints

Step-by-step solution

Revise the chapter

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