Assertion (A): If the radius of sector of a circle is reduced to its half and angle is doubled then the perimeter of the sector remains the same. Reason (R): The length of the arc subtending angle θ at the centre of a circle of radius r = (π rθ)/(180). (A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) (B) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of 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 · Standard · Sample paper, session 2024-25 · 1 mark · Assertion-reason

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 20

From the official CBSE sample question paper for 2024-25, section A, question 20, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q20 · 1 mark · Assertion-reasonCompetency-basedAreas Related to Circles

Assertion (A): If the radius of sector of a circle is reduced to its half and angle is doubled then the perimeter of the sector remains the same.

Reason (R): The length of the arc subtending angle $θ$ at the centre of a circle of radius $r = \frac{π r θ}{180}$ .

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

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

The perimeter of a sector is two radii plus the arc length.

Hint 2

The arc length stays the same, but what happens to the two radii?

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.

Original perimeter of the sector $= 2 r + \frac{π r θ}{180}$ .
With radius $\frac{r}{2}$ and angle $2 θ$ : perimeter $= r + \frac{π ( r /2 ) ( 2 θ )}{180} = r + \frac{π r θ}{180}$ . This is less than before, so (A) is false.
(R) is the correct formula for arc length, so it is true. The answer is option (D).
1 mark

Answer: Option (D)

Where the marks are

Shows the new perimeter is $r + \frac{π r θ}{180}$ , so A is false while R is true (1 mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)
Topic: Length of an Arc of a Circle

Practise this chapterBoard-style questions from Areas Related to Circles, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2024-25 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2024-25: question 20

Hints

Step-by-step solution

Revise the chapter

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