1. In a circle centered at O, the chords AB and CD are equal in length. If \(\angle\)AOB = 60°, then find the measure of \(\angle\)COD.

(a) 40° (b) 30° (c) 60° (d) 90°

2. From an external point P, two tangents PA and PB are drawn to a circle centered at O. If PA = 9 cm and ∠APB = 60°, then what are the measures of ∠PAB and the length of chord AB?

(a) 9 cm (b) 3 cm (c) 6 cm (d) 12 cm

3. In a circle centered at O, the chords AB and CD are of equal length. If \(\angle\)AOB = 60°, then determine the value of \(\angle\)COD.

(a) 40° (b) 30° (c) 60° (d) 90°

4. In a circle centered at O, the chords AB and AC are perpendicular to each other. If AB = 6 cm and AC = 8 cm, then the radius of the circle will be -.

(a) 7 cm (b) 10 cm (c) 5 cm (d) 4 cm

5. In a circle centered at O, the chords AB and CD are equal in length. If \( \angle \)AOB = 60°, then find the value of \( \angle \)COD.

(a) 30° (b) 40° (c) 120° (d) 60°

6. In a circle centered at O, the chords AB and CD have equal lengths. If \(\angle\)AOB = 60°, then the value of \(\angle\)COD is -

(a) 60° (b) 30° (c) 120° (d) 90°

7. If the volumes of two vertical solid cylinders are equal, and their heights are in the ratio 1:2, then the ratio of the lengths of their radii will be –

(a) 1: √2 (b) √2:1 (c) 1:2 (d) 2:1

8. In a circle with center O, PQ and PR are two chords. Tangents drawn at points Q and P intersect at point S. If ∠QSR = 70°, then what is the measure of ∠QPR?

9. If two tangents are drawn from points P and Q on a circle and intersect at point A such that ∠PAQ = 60°, then what is the measure of ∠APQ?

10. In a circle centered at O, chords AB and CD are equidistant from the center. If ∠AOB = 60° and CD = 6 cm, then what is the radius of the circle?

11. In a circle, AB and AC are two chords. Tangents are drawn at points B and C, and they intersect at point P. If \(\angle\)BAC = 54°, then what is the measure of \(\angle\)BPC?

12. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.

13. In a circle centered at O, if the lengths of chords PQ and RS are in the ratio 1:1, then the ratio of \(\angle\)POQ to \(\angle\)ROS is _____ |

14. If the volumes of two vertical circular cylinders are equal, and their heights are in the ratio 4:9, then the ratio of their radii will be –

(a) 3:2 (b) 2:3 (c) 4:9 (d) 8:9

15. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?

(a) 145° (b) 55° (c) 110° (d) None of the above

16. Two tangents are drawn to a circle from points A and B on its circumference, and they intersect at point P. If \(\angle\)APB = 68°, then what is the measure of \(\angle\)PAB?

17. In a circle with center O, AB and CD are two equal chords. If \(\angle\)AOB = 60° and CD = 10 cm, then what is the length of OD?

18. In a joint business, if 6 times A's capital, 8 times B's capital, and 10 times C's capital are all equal, then what will be the ratio of their capitals?

(a) 15 : 20 : 12 (b) 20 : 12 : 15 (c) 20 : 15 : 12 (d) 15 : 12 : 17

19. In a circle centered at \(O\), two chords \(AB\) and \(CD\) have equal lengths. Given that \(\angle AOB = 60°\) and the radius of the circle is \(6\) cm, find the area of \(\triangle COD\).

(a) \(9\sqrt3\) square c.m (b) \(6\sqrt3\) square c.m (c) \(2\sqrt3\) square c.m (d) \(3\sqrt3\) square c.m

20. If the lengths of two equal chords are _____ cm and the distance between them is 4 cm, then the diameter of the circle will be 10 cm.

21. In a circle centered at O, the chords AB and CD are of equal length. If \(\angle\)AOB = 60° and CD = 6 cm, determine the radius of the circle.

22. In a circle centered at \(O\), two equal chords \(AB\) and \(CD\) are given. If \(\angle AOB = 60^\circ\) and the radius of the circle is 6 cm, determine the area of \( \triangle COD \).

(a) \(6\sqrt3\) square cm (b) \(2\sqrt3\) square cm (c) \(2\sqrt3\) square cm (d) \(9\sqrt3\) square cm

23. The chords AB and CD of a central circle are equal in length. If \(\angle\)COD = 60°, then \(\angle\)AOB = ———.

24. In a circle centered at \( O \), the chords \( AB \) and \( AC \) are perpendicular to each other. If \( AB = 8 \) cm and \( AC = 6 \) cm, determine the radius of the circle.

(a) 7 cm (b) 10 cm (c) 5 cm (d) 9 cm

25. If the chords AB and AC of a central circle are located on opposite sides of the radius OA, then \(\angle\)OAB = \(\angle\)OAC.

26. In a circle centered at O, PQ and PR are two chords. The tangents drawn at points Q and R intersect at point S. If \(\angle\)QSR = 70°, determine the value of \(\angle\)QPR.

27. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are real and unequal, then \(b^2 - 4ac\) will be —

(a) >0 (b) =0 (c) <0 (d) none of the above

28. In a circle centered at O, there are two chords of lengths 6 cm and 8 cm. If the distance from the center to the shorter chord is 4 cm, calculate and write the distance from the center to the other chord.

29. In a circle centered at O, the chords AB and CD are of equal length. If \(\angle\)AOB = 60° and CD = 6 cm, find the radius of the circle.

30. Two chords AB and CD of a circle with center O intersect each other at point P. Prove that \(\angle AOD + \angle BOC = 2\angle BPC\) If \(\angle AOD\) and \(\angle BOC\) are supplementary, then prove that the two chords are perpendicular to each other.