1. AOB is a diameter of the circle with center O, and C is a point on the circle. If \(\angle\)OBC = 60°, then find the measure of \(\angle\)OCA.

2. AOB is a diameter of a circle. Point C lies on the circle. If ∠OBC = 60°, find the value of ∠OCA.

3. In a circle with center \(O\), \(AB\) is the diameter, and \(P\) is a point on the circle. If \(\angle AOP = 104°\), find the value of \(\angle BPO\).

(a) 54° (b) 72° (c) 36° (d) 27°

4. AOB is a diameter of a circle. C is a point on the circle. If \(\angle\)OBC = 55°, find the value of \(\angle\)OCA.

5. AB is the diameter of a circle centered at O. C is any point on the circle. Given ∠BAC = 50° and CD is perpendicular to AB, Find the value of ∠BCD.

6. AOB is a diameter of the circle. C is a point on the circle. If \(\angle\)OBC = 60°, determine the value of \(\angle\)OCA.

7. In the adjacent figure, AB is the diameter of a circle centered at O. Point C is any point on the circle. Given that ∠BAC = 50° and CD is perpendicular to AB, find the value of ∠BCD.

8. O is the center of a circle. PQ is a diameter, and R is a point on the circumference. If \(\angle\)PQR = 40°, then what is the measure of \(\angle\)POR?

(a) 80° (b) 40° (c) 20° (d) 100°

9. AOB is the diameter of a circle. AC and BD are two chords that intersect at point E. If \(\angle\)COD = 40°, find the value of \(\angle\)CED.

10. AOB is the diameter of a circle. Chords AC and BD, when extended, meet at point E. If ∠COD = 40°, find the value of ∠CED.

(a) 40° (b) 80° (c) 20° (d) 70°

11. If the angle of elevation of the top of a chimney from a point on level ground at its base is 60°, and from another point on the same straight line 24 meters further away from the base, the angle of elevation is 30°, then calculate and write the height of the chimney. [Take the approximate value of \(\sqrt{3} = 1.732\) and find the result correct to three decimal places.]

12. In a circle with center \(O\), \(\bar{AB}\) is a diameter. On the opposite side of the circumference from the diameter \(\bar{AB}\), there are two points \(C\) and \(D\) such that \(\angle AOC = 130°\) and \(\angle BDC = x°\). Find the value of \(x\).

(a) 25° (b) 50° (c) 60° (d) 65°

13. ABCD is a cyclic quadrilateral and O is the center of the circle. If ∠COD = 120° and ∠BAC = 30°, then find the values of ∠BOC and ∠BCD.

14. AB is a chord of a circle centered at point O. OP is perpendicular to AB, where AB = 8 cm and OP = 3 cm. Find the diameter of the circle.

15. AB is a diameter of a circle centered at O. P is any point on the circumference. If \(\angle\)POA = 120°, then find the measure of \(\angle\)PBO.

16. AOB is the diameter of a circle, and point C lies on the circumference. Given: AC = 3 cm and BC = 4 cm Then, find the length of AB.

17. In a circle centered at O, AB is a diameter, and C is any point on the circle. Given that \(\angle\)OAC = 45°, determine the measure of \(\angle\)OCB.

(a) 90° (b) 45° (c) 30° (d) 60°

18. In the circle centered at O, AB is the diameter. M is a point on the circumference. If \(\angle\) MAB = 72°, find the value of \(\angle\) MBA.

(a) 72° (b) 18° (c) 108° (d) none of the above

19. In the adjacent figure, O is the center of the circle and AB is the diameter. The length of chord CD is equal to the radius of the circle. AC and BD are extended to meet at point P. Find the value of ∠APB.

20. AB is a diameter of a circle with center O. P is any point on the circle. Tangents are drawn at points A and B, which are intersected by the tangent drawn from point P at points Q and R respectively. If the radius of the circle is \(r\), prove that \(PQ \cdot PR = r^2\).

21. In the adjacent figure, O is the center of the circle, and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extension of BA at point T. If ∠PBO = 30°, then find the measure of ∠PTA.

22. In the center circle O, AB is a diameter. C is any point on the circumference of the circle where AC = 3 cm and BC = 4 cm. Find the length of AB.

(a) 3 cm (b) 4 cm (c) 5 cm (d) 7 cm

23. QR is a chord of the circle, and POR is a diameter of the circle. OD is perpendicular to the chord QR. If OD = 4 cm, then the length of PQ is –

(a) 4 cm (b) 2 cm (c) 8 cm (d) None of these

24. In a circle with center \(O\), \(AB\) is a diameter. \(P\) is any point on the circumference. If \(\angle POA = 120°\), then the measurement of \(\angle PBO\) is –

(a) 30° (b) 60° (c) 90° (d) 120°

25. "AB is a diameter of a circle with center O. P is any point on the circumference. If ∠ POA = 120°, then what is the measure of ∠ PBO?"

(a) 90\(^o\) (b) 60\(^o\) (c) 75\(^o\) (d) None of the above

26. In the cyclic quadrilateral PQRS, the side PS is a diameter of the circle. If \(\angle\)PQR = 120°, then what is the measure of \(\angle\)SPR?

(a) 90° (b) 30° (c) 60° (d) 120°

27. In the cyclic quadrilateral PORS, the side PS is a diameter of the circle. If \(\angle\)PQR = 128°, then what is the measure of \(\angle\)SPR?

(a) 30° (b) 38° (c) 60° (d) None of the above

28. AB is a chord of a circle with center O. A tangent is drawn at point B, which intersects the extended line AO at point T. If ∠BAT = 21°, then find the value of ∠BTA.

29. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), then find the value of \(k\).

30. 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?