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O is the centre of the circle… (Class 10 Maths 2023, Q31)

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MathsCBSE Class 10 · Standard · Board exam 2023, set 30/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2023: question 31

Set in the CBSE Class 10 board exam of 2023, paper 30/2/1, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerCircles

In the given figure, O is the centre of the circle and QPR is a tangent to it at P. Prove that .

Figure in the paper: A circle with centre O. Points B and A lie on the circle, with chord BA going up-right to A. Q is on the left and R on the right of a tangent line QPR touching the circle at P at the bottom. OP is a dashed radius. Angle QAP is marked at A and angle APR is marked at P. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/2/1.

Hints

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

Hint 1

Look at triangle OAP: what is special about OA and OP?

Hint 2

What is the angle between the tangent QR and the radius OP?

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.

  1. [radii of the same circle], so in , …(i) [angles opposite equal sides]

    1 mark

  2. [tangent ⟂ radius at the point of contact], so , i.e. .

    1 mark

  3. Using (i):

    ½ mark

  4. Since O lies on the line AQ, . Hence . Proved.

    ½ mark

Answer: (proved).

Where the marks are

  • Uses to get (1 mark)
  • Uses tangent ⟂ radius: (1 mark)
  • Substitutes from (i) (½ mark)
  • Replaces by and concludes (½ mark)

Revise the chapter

Practise this chapterBoard-style questions from Circles, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 30/2/1 (Standard): 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.