MathsCBSE Class 10 · Basic · Board exam 2023, set 430/5/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 33(b)

Set in the CBSE Class 10 board exam of 2023, paper 430/5/1, section D, question 33(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(b) · 5 marks · Long answerAreas Related to Circles

AB and CD are arcs of two concentric circles of radii 3.5 cm and 10.5 cm respectively and centred at O. Find the area of the shaded region if $∠ A O B = 6 0^{\circ}$ . Also, find the length of arc CD.

Figure in the paper: Two concentric circles centred at O. Points A and B lie on the inner arc and C and D lie on the outer arc; OA, OB, OC and OD are joined. $∠ A O B = 6 0^{\circ}$ is marked at O. The region between the two arcs bounded by OC, OD is shaded. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/5/1.

Hints

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

Hint 1

The shaded part lies between two sectors with the same angle 60°.

Hint 2

Subtract the smaller sector from the larger: $\frac{θ}{360} π (R^{2} - r^{2})$ .

Hint 3

Arc CD is on the outer circle, so use r = 10.5 cm in the arc-length formula.

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.

OA = 3.5 cm, OC = 10.5 cm, $θ = 6 0^{\circ}$ . Shaded area = sector OCD − sector OAB $= \frac{60}{360} \times π (10. 5^{2} - 3. 5^{2})$
2 marks
$10. 5^{2} - 3. 5^{2} = (10.5 + 3.5) (10.5 - 3.5) = 14 \times 7 = 98$ , so area $= \frac{22}{7} \times \frac{1}{6} \times 98$
1 mark
$= \frac{154}{3}$ cm² $\approx 51.33$ cm²
½ mark
Length of arc CD $= \frac{60}{360} \times 2 \times \frac{22}{7} \times 10.5$
1 mark
$= 11$ cm
½ mark

Answer: Shaded area = 154/3 cm² ≈ 51.33 cm²; arc CD = 11 cm

Where the marks are

Expresses shaded area as difference of two sectors $\frac{60}{360} π (R^{2} - r^{2})$ (2 marks)
Correct simplification with 98 (1 mark)
Area = 154/3 cm² (½ mark)
Arc length formula with substitution (1 mark)
Arc CD = 11 cm (½ mark)

Revise the chapter

Chapter: Areas Related to Circles (notes, formulas, where marks are lost and every past question)
Topic: Area of a Sector 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 paper2023 board paper, set 430/5/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 33(b)

Hints

Step-by-step solution

Revise the chapter

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