MathsCBSE Class 10 · Basic · Board exam 2024, set 430/1/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic 2024: question 29(A)

Set in the CBSE Class 10 board exam of 2024, paper 430/1/1, section C, question 29(A), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q29(A) · 3 marks · Short answerCircles

In two concentric circles, the radii are $O A = r$ cm and $O Q = 6$ cm, as shown in the figure. Chord CD of larger circle is a tangent to smaller circle at Q. PA is tangent to larger circle. If $P A = 16$ cm and $O P = 20$ cm, find the length CD.

Figure in the paper: Two concentric circles with centre O. A is a point on the outer circle, with tangent PA drawn from external point P, where OP = 20 cm; OA is joined. Chord CD of the outer circle is horizontal, touching the inner circle at Q, with OQ perpendicular to CD. D lies on the outer circle to the right of Q and OD is joined. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/1/1.

Hints

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

Hint 1

The tangent PA is perpendicular to radius OA, so $△ O A P$ is right-angled.

Hint 2

OD is also a radius of the larger circle and OQ ⟂ CD; use Pythagoras in $△ O Q D$ .

Hint 3

The perpendicular from the centre bisects the chord CD.

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.

$O A ⊥ P A$ [tangent ⟂ radius]. In right $△ O A P$ : $O A^{2} = O P^{2} - P A^{2} = 2 0^{2} - 1 6^{2} = 144 \Rightarrow r = O A = 12$ cm. Also $O D = 12$ cm (radius of larger circle).
1 mark
$O Q ⊥ C D$ [tangent CD ⟂ radius OQ of smaller circle]. In right $△ O Q D$ : $Q D^{2} = O D^{2} - O Q^{2} = 1 2^{2} - 6^{2} = 108 \Rightarrow Q D = 63$ cm
1 mark
The perpendicular from the centre to a chord bisects it, so $C D = 2 Q D = 123$ cm $(\approx 20.78$ cm $)$ .
1 mark

Answer: $C D = 123$ cm

Where the marks are

Uses tangent ⟂ radius and Pythagoras to get $r = 12$ cm (1 mark)
Finds $Q D = 63$ cm in $△ O Q D$ (1 mark)
Uses perpendicular bisects chord to get $C D = 123$ cm (1 mark)

Revise the chapter

Chapter: Circles (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Circles, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 430/1/1 (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 2024: question 29(A)

Hints

Step-by-step solution

Revise the chapter

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