MathsCBSE Class 10 · Basic · Board exam 2026, set 430/5/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2026: question 37

Set in the CBSE Class 10 board exam of 2026, paper 430/5/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedSurface Areas and Volumes

There are many varieties of mushrooms available in the world. One such mushroom 'Amanita muscaria' has a upper part which is like red cap (hemispherical) and lower part is like white stem (cylinderical).

The hemispherical cap's radius = 3 cm and cylindrical stem is 2 cm high with diameter 1.4 cm. Considering mushroom a solid object, answer the following questions:

(i)
What is the total height of a mushroom?
1 mark
(ii)
Find the volume of the stem.
1 mark
(iii)(a)
Determine the volume of 7 such mushrooms.
2 marks
OR
(iii)(b)
Find the total surface area of 7 such mushrooms.
2 marks

Figure in the paper: Left: photograph of an Amanita muscaria mushroom with a red-looking hemispherical cap spotted white and a white cylindrical stem. Right: a line diagram of the same mushroom, a semicircular dome on top of a rectangular stem. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 430/5/1.

Hints

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

Hint 1

The height of a hemisphere equals its radius, and the radius of the stem is half its diameter.

Hint 2

Volume of the solid = hemisphere $\frac{2}{3} π R^{3}$ + cylinder $π r^{2} h$ . Find one mushroom first, then multiply by 7.

Hint 3

For the surface area, list every exposed face: the dome, the ring left on the flat underside of the cap, the curved side of the stem and the stem's bottom circle.

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.

(i) Total height = height of stem + radius of hemispherical cap $= 2 + 3 = 5$ cm.
1 mark
(ii) Radius of stem $r = \frac{1.4}{2} = 0.7$ cm and $h = 2$ cm.
Volume of stem $= π r^{2} h = \frac{22}{7} \times 0.7 \times 0.7 \times 2 = 3.08 cm^{3}$ .
1 mark
(iii)(a) Volume of one mushroom = volume of hemisphere + volume of cylinder $= \frac{2}{3} π R^{3} + π r^{2} h$ , with $R = 3$ cm, $r = 0.7$ cm and $h = 2$ cm.
Volume of 7 mushrooms $= 7 (\frac{2}{3} \times \frac{22}{7} \times 3^{3} + \frac{22}{7} \times (0.7)^{2} \times 2)$ .
1 mark
$= 7 (\frac{396}{7} + 3.08) = 396 + 21.56 = 417.56 cm^{3}$ .
1 mark
(iii)(b) Surface of one solid mushroom = curved surface of hemisphere ( $2 π R^{2}$ ) + exposed ring of the flat base of the cap ( $π R^{2} - π r^{2}$ ) + curved surface of stem ( $2 π r h$ ) + bottom of stem ( $π r^{2}$ ) $= 3 π R^{2} + 2 π r h$ .
TSA of 7 mushrooms $= 7 (3 \times \frac{22}{7} \times 3 \times 3 + 2 \times \frac{22}{7} \times 0.7 \times 2)$ .
1 mark
$= 7 (\frac{594}{7} + 8.8) = 594 + 61.6 = 655.6 cm^{2}$ .
1 mark

Answer: (i) 5 cm; (ii) 3.08 cm³; (iii)(a) 417.56 cm³; (iii)(b) 655.6 cm²

Where the marks are

Total height $2 + 3 = 5$ cm (1 mark)
Volume of stem $π r^{2} h = 3.08$ cm³ (1 mark)
(iii)(a) Correct expression $7 (\frac{2}{3} π R^{3} + π r^{2} h)$ with values substituted (1 mark)
(iii)(a) Volume of 7 mushrooms $= 417.56$ cm³ (1 mark)
(iii)(b) Correct surface area expression $7 (3 π R^{2} + 2 π r h)$ with values substituted (1 mark)
(iii)(b) TSA $= 655.6$ cm² (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)
Topic: Volume 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 paper2026 board paper, set 430/5/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2026: question 37

Hints

Step-by-step solution

Revise the chapter

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