MathsCBSE Class 10 · Standard · Board exam 2025, set 30/6/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2025: question 34

Set in the CBSE Class 10 board exam of 2025, paper 30/6/1, 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

From one face of a solid cube of side 14 cm, the largest possible cone is carved out. Find the volume and surface area of the remaining solid. (Use $π = \frac{22}{7}$ , $5 = 2.2$ )

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/6/1.

Hints

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

Hint 1

The largest cone has base diameter and height both equal to the side of the cube.

Hint 2

For surface area: remove the circular base from the cube's surface but add the cone's curved surface (the hollow inside).

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.

Largest cone: diameter of base = side of cube = 14 cm, so radius $r = 7$ cm.
½ mark
Height of cone $h = 14$ cm.
½ mark
Slant height $l = h^{2} + r^{2} = 196 + 49 = 75 = 7 \times 2.2 = 15.4$ cm
1 mark
Volume of remaining solid = volume of cube − volume of cone $= 1 4^{3} - \frac{1}{3} \times \frac{22}{7} \times 7^{2} \times 14 = 2744 - \frac{2156}{3}$
1 mark
$= \frac{6076}{3}$ cm³ $\approx 2025.33$ cm³
½ mark
Surface area = TSA of cube − area of base circle + CSA of cone $= 6 \times 1 4^{2} - \frac{22}{7} \times 7^{2} + \frac{22}{7} \times 7 \times 15.4 = 1176 - 154 + 338.8$
1 mark
$= 1360.8$ cm²
½ mark

Answer: Volume $= \frac{6076}{3}$ cm³ ( $\approx 2025.33$ cm³); surface area $= 1360.8$ cm²

Where the marks are

Radius 7 cm (½ mark)
Height 14 cm (½ mark)
Slant height 15.4 cm (1 mark)
Volume expression cube − cone (1 mark)
Volume $6076/3$ cm³ (½ mark)
Surface area expression (1 mark)
Surface area 1360.8 cm² (½ 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 paper2025 board paper, set 30/6/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 34

Hints

Step-by-step solution

Revise the chapter

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