MathsCBSE Class 10 · Standard · Board exam 2025, set 30/6/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2025: question 31

Set in the CBSE Class 10 board exam of 2025, paper 30/6/1, 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 adjoining figure, TP and TQ are tangents drawn to a circle with centre O. If $∠ O P Q = 1 5^{\circ}$ and $∠ P T Q = θ$ , then find the value of $sin 2 θ$ .

Figure in the paper: A circle with centre O. T is an external point with tangents TP (upper) and TQ (lower) touching the circle at P and Q. A chord PQ is drawn, and OP is a radius. The angle between OP and PQ at P is marked $1 5^{\circ}$ ; the angle at T between TP and TQ is marked $θ$ . See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/6/1.

Hints

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

Hint 1

The radius OP is perpendicular to the tangent TP.

Hint 2

Triangle TPQ is isosceles because TP = TQ; use the angle sum.

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.

$∠ O P T = 9 0^{\circ}$ [tangent ⟂ radius], so $∠ QP T = 9 0^{\circ} - 1 5^{\circ} = 7 5^{\circ}$ .
½ mark
$T P = T Q$ [tangents from an external point are equal], so $∠ P QT = ∠ QP T = 7 5^{\circ}$ .
½ mark
In $△ P T Q$ : $θ = 18 0^{\circ} - 7 5^{\circ} - 7 5^{\circ} = 3 0^{\circ}$
1 mark
$sin 2 θ = sin 6 0^{\circ}$
½ mark
$= \frac{3}{2}$
½ mark

Answer: $sin 2 θ = \frac{3}{2}$ ( $θ = 3 0^{\circ}$ )

Where the marks are

$∠ QP T = 7 5^{\circ}$ using tangent ⟂ radius (½ mark)
$∠ P QT = 7 5^{\circ}$ using equal tangents (½ mark)
$θ = 3 0^{\circ}$ (1 mark)
$sin 2 θ = sin 6 0^{\circ}$ (½ mark)
Value $3 /2$ (½ mark)

Revise the chapter

Chapter: Circles (notes, formulas, where marks are lost and every past question)
Topic: Solving Problems Using Tangent Properties

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 30/6/1 (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.

CBSE Class 10 Mathematics Standard 2025: question 31

Hints

Step-by-step solution

Revise the chapter

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