MathsCBSE Class 10 · Basic · Board exam 2025, set 430/1/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2025: question 34(b)

Set in the CBSE Class 10 board exam of 2025, paper 430/1/1, section D, question 34(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(b) · 5 marks · Long answerSurface Areas and Volumes

A cubical block is surmounted by a hemisphere of radius 3·5 cm. What is the smallest possible length of the edge of the cube so that the hemisphere can totally lie on the cube ? Find the total surface area of the solid so formed.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/1/1.

Hints

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

Hint 1

The flat circular face of the hemisphere must fit on the top face of the cube, so compare the edge with the diameter.

Hint 2

The circle covered on the cube is lost from the surface, but the curved surface $2 π r^{2}$ is added.

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.

For the hemisphere to lie totally on the cube, the edge must be at least the diameter of the hemisphere. Smallest edge $a = 2 \times 3.5 = 7$ cm.
1 mark
TSA of solid = TSA of cube − base area of hemisphere + CSA of hemisphere $= 6 a^{2} - π r^{2} + 2 π r^{2} = 6 a^{2} + π r^{2}$ .
1 mark
$6 a^{2} = 6 \times 7 \times 7 = 294$ cm²
1½ marks
$π r^{2} = \frac{22}{7} \times 3.5 \times 3.5 = 38.5$ cm²
1½ marks
TSA $= 294 + 38.5 = 332.5$ cm² $= \frac{665}{2}$ cm²
1 mark

Answer: Edge = 7 cm; TSA = 332.5 cm²

Where the marks are

Smallest edge $= 7$ cm (1 mark)
Correct expression $6 a^{2} + 2 π r^{2} - π r^{2}$ (1 mark)
Computes $6 a^{2} = 294$ cm² (1½ marks)
Computes $π r^{2} = 38.5$ cm² (1½ marks)
TSA $= 332.5$ cm² (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Surface Area of a Combination of Solids

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 paper2025 board paper, set 430/1/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2025: question 34(b)

Hints

Step-by-step solution

Revise the chapter

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