MathsCBSE Class 10 · Basic · Sample paper, session 2025-26 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic sample paper 2025-26: question 35

From the official CBSE sample question paper for 2025-26, section D, question 35, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35 · 5 marks · Long answerSurface Areas and Volumes

A solid toy is in the form of a hemisphere surmounted by a right circular cone of height 2cm and diameter of base 4cm. If a right circular cylinder circumscribes the toy, find the difference of the volumes of the cylinder and the toy. [Use $π = 3.14$ ]

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

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

Hint 1

The radius of the hemisphere equals the radius of the cone's base, which is half the diameter.

Hint 2

The cylinder's height is the radius of the hemisphere plus the height of the cone.

Hint 3

Find the toy's volume as $\frac{2}{3} π r^{3} + \frac{1}{3} π r^{2} h$ , then subtract it from $π r^{2} H$ .

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.

Figure: A hemisphere of radius $r = 2$ cm, surmounted by a cone of base radius 2 cm and height $h = 2$ cm, all enclosed in a cylinder. The cylinder has radius $r = 2$ cm and height $H = r + h = 2 + 2 = 4$ cm.
1 mark
Volume of toy $=$ volume of hemisphere $+$ volume of cone
$= \frac{2}{3} π r^{3} + \frac{1}{3} π r^{2} h = \frac{1}{3} π r^{2} (2 r + h) = \frac{1}{3} \times 3.14 \times 4 \times (4 + 2) = 8 \times 3.14 = 25.12 cm^{3}$
2 marks
Volume of circumscribing cylinder $= π r^{2} H = 3.14 \times 4 \times 4 = 50.24 cm^{3}$
1 mark
Required difference $= 50.24 - 25.12 = 25.12 cm^{3}$
1 mark

Answer: 25.12 cm³

Where the marks are

Correct figure with $r = 2$ cm, cone height 2 cm and cylinder height 4 cm (1 mark)
Volume of toy (hemisphere + cone) = 25.12 cm³ (2 marks)
Volume of cylinder = 50.24 cm³ (1 mark)
Difference = 25.12 cm³ (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Cone on a Hemisphere and Similar Shapes

Practise this chapterBoard-style questions from Surface Areas and Volumes, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2025-26 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2025-26: question 35

Hints

Step-by-step solution

Revise the chapter

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