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PQ is tangent to a circle… (Class 10 Maths 2024, Q28a)

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MathsCBSE Class 10 · Standard · Board exam 2024, set 30/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2024: question 28(a)

Set in the CBSE Class 10 board exam of 2024, paper 30/2/1, section C, question 28(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28(a) · 3 marks · Short answerCircles

In the given figure, is tangent to a circle centred at and ; show that .

Figure in the paper: Circle with centre O. AB is a diameter drawn horizontally, with A on the left and B on the right, and O at the midpoint. Point P lies on the extension of AB beyond B. PQ is a tangent touching the circle at Q (upper part). Chord AQ and chord BQ are drawn. The angle at A between AB and AQ is marked 30 degrees. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/2/1.

Hints

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

Hint 1

Join OQ; triangle OAQ is isosceles, and AB is a diameter.

Hint 2

Use tangent ⟂ radius at Q to find .

Hint 3

Find as an exterior angle, then in triangle OQP.

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. Join . [radii].

    ½ mark

  2. [angles opposite equal sides].

    ½ mark

  3. [angle in a semicircle], so .

    ½ mark

  4. [tangent ⟂ radius], so .

    ½ mark

  5. [exterior angle of ].

    ½ mark

  6. In : . So in , , hence [sides opposite equal angles].

    ½ mark

Answer: , so .

Where the marks are

  • Joins OQ, OQ = OA (½ mark)
  • (½ mark)
  • (½ mark)
  • (½ mark)
  • (½ mark)
  • , hence BP = BQ (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 paper2024 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.