Assertion (A): When a hemisphere of same radius (r) is carved out from one side of a solid wooden cylinder, the total surface area of remaining solid is increased by 2π r². Reason (R): Curved surface area of hemisphere is 2π r². (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/5/1 · 1 mark · Assertion-reason

CBSE Class 10 Mathematics Basic 2025: question 19

Set in the CBSE Class 10 board exam of 2025, paper 430/5/1, section A, question 19, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q19 · 1 mark · Assertion-reasonCompetency-basedSurface Areas and Volumes

Assertion (A): When a hemisphere of same radius (r) is carved out from one side of a solid wooden cylinder, the total surface area of remaining solid is increased by $2 π r^{2}$ .

Reason (R): Curved surface area of hemisphere is $2 π r^{2}$ .

(A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C)Assertion (A) is true, but Reason (R) is false.
(D)Assertion (A) is false, but Reason (R) is true.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/5/1.

Hints

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

Hint 1

Which surface is lost when you carve out the hemisphere?

Hint 2

Net change = area added − area removed.

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.

Carving removes a flat circle of area $π r^{2}$ and adds the hemisphere's curved surface $2 π r^{2}$ : net increase $= π r^{2}$ , not $2 π r^{2}$ . So A is false.
CSA of hemisphere $= 2 π r^{2}$ is true. Option (D).
1 mark

Answer: Option (D)

Where the marks are

Shows increase is $π r^{2}$ , A false, R true (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)

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 430/5/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2025: question 19

Hints

Step-by-step solution

Revise the chapter

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