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PC is a tangent to the circle… (Class 10 Maths 2025, Q31)

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MathsCBSE Class 10 · Standard · Board exam 2025, set 30/3/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 31

Set in the CBSE Class 10 board exam of 2025, paper 30/3/1, 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, PC is a tangent to the circle at C. AOB is the diameter which when extended meets the tangent at P. Find CBA and BCO, if PCA .

Figure in the paper: A circle with centre O. Diameter AOB lies horizontally, with B on the left and A on the right; O between them. The line AB extended beyond B meets the tangent at P on the far left. PC is the tangent touching the circle at C (top). Chords/segments CA, CB and radius CO are drawn. PCA is marked at C. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/3/1.

Hints

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

Hint 1

AB is a diameter. What is the angle ACB?

Hint 2

Split to find . Then use the fact that OC ⟂ PC, and that OB = OC.

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. [angle in a semicircle, since AB is a diameter]

    ½ mark

  2. , so .

    Hence .

    1 mark

  3. [tangent ⟂ radius at the point of contact].

    So .

    1 mark

  4. OB = OC [radii], so [angles opposite equal sides].

    Hence .

    ½ mark

Answer: ,

Where the marks are

  • (angle in semicircle) (½ mark)
  • (1 mark)
  • using tangent ⟂ radius (1 mark)
  • using OB = OC (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 paper2025 board paper, set 30/3/1 (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.