MathsCBSE Class 10 · Basic · Sample paper, session 2026-27 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic sample paper 2026-27: question 34

From the official CBSE sample question paper for 2026-27, section D, question 34, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34 · 5 marks · Long answerSurface Areas and Volumes

The largest possible hemisphere is drilled out of a wooden cubical block of side 21 cm such that the base of the hemisphere is on one of the faces of the cube. Then, find

i.
the volume of the wood left in the block.
0 marks
ii.
the total surface area of the remaining solid.
0 marks

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

The largest hemisphere has diameter equal to the side of the cube.

Hint 2

For the surface area, remove the circular base area from the cube's surface and add the hemisphere's curved surface $2 π r^{2}$ .

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.

Side of cube $a = 21$ cm. The largest hemisphere has diameter equal to the side, so $r = \frac{21}{2}$ cm.
½ mark
(i) Volume of wood left = Volume of cube − Volume of hemisphere $= a^{3} - \frac{2}{3} π r^{3}$
½ mark
$= 2 1^{3} - \frac{2}{3} \times \frac{22}{7} \times \frac{21 \times 21 \times 21}{2 \times 2 \times 2} = 9261 - 2425.5$
1 mark
$= 6835.5 cm^{3}$
½ mark
(ii) TSA of remaining solid = TSA of cube − area of circular top removed + curved surface of hemisphere
$= 6 a^{2} - π r^{2} + 2 π r^{2} = 6 a^{2} + π r^{2}$
½ mark
$= 6 (21)^{2} + \frac{22}{7} \times (\frac{21}{2})^{2} = 2646 + 346.5$
1 mark
$= 2992.5 cm^{2}$
1 mark

Answer: (i) 6835.5 cm³; (ii) 2992.5 cm²

Where the marks are

r = 21/2 cm (½ mark)
Volume formula: cube − hemisphere (½ mark)
Correct substitution (1 mark)
Volume = 6835.5 cm³ (½ mark)
TSA formula $6 a^{2} - π r^{2} + 2 π r^{2}$ (½ mark)
Correct substitution (1 mark)
TSA = 2992.5 cm² (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Surface Area and Volume Formulae Revision

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 2026-27 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2026-27: question 34

Hints

Step-by-step solution

Revise the chapter

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