1. PQRS is a cyclic quadrilateral. PQ is the diameter of the circle, and if \(\angle\)PRS = 56°, then what is the value of \(\angle\)QPS?

2. ABCD is a cyclic quadrilateral. If BC is the diameter and ∠ADC = 130°, then what is the value of ∠ACB?

3. In the cyclic quadrilateral ABCD, AB is the diameter and AB || DC. If \(\angle\)ACD = 25°, determine the value of \(\angle\)CAD.

4. O is the center of a circle, and AB is its diameter. ABCD is a cyclic quadrilateral where AB\(\parallel\)DC, and \(\angle\)BAC = 75°. The value of \(\angle\)BCD will be-

(a) 60° (b) 45° (c) 75° (d) 50°

5. In the given figure, \(O\) is the center of the circle, and \(AB\) is the diameter. If \(\angle ADC = 120^\circ\), then determine the value of \(\angle BAC\).

(a) 50° (b) 60° (c) 40° (d) 30°

6. PQRS is a cyclic quadrilateral, and the sides PQ and SR are extended to meet at point T. The center of the circle is O; if ∠POQ = 110°, ∠QOR = 60°, and ∠ROS = 80°, then calculate and write the values of ∠RQS and ∠QTR.

7. In a circle with center O, where BOC is the diameter and ABCD is a cyclic quadrilateral, if \(\angle ADC = 110^\circ\), then find the value of \(\angle ACB\).

8. 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

9. The center of a circle is O, and AB is its diameter. ABCD is a cyclic quadrilateral. If \(\angle\)ABC = 65° and \(\angle\)DAC = 40°, then the measure of \(\angle\)BCD is—?

(a) 75° (b) 105° (c) 115° (d) 80°

10. 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.

11. 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 extended line BA at point T. If \(\angle PBO = 30^\circ\), then what is the measure of \(\angle PTA\)?

12. 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.

13. In the cyclic quadrilateral ABCD, AB is the diameter and O is the center of the circle. If \(\angle\)ADC = 120°, then \(\angle\)BAC = _____

14. In the cyclic quadrilateral ABCD, AB is the diameter and \(\angle\)ACD = 50°, then what is the measure of \(\angle\)BAD?

(a) 30° (b) 40° (c) 50° (d) 60°

15. In the cyclic quadrilateral ABCD, if \(\angle\)DBA = 50° and \(\angle\)ADB = 33°, then what is the value of \(\angle\)BCD?

16. O is the center of the circle, and AB is its diameter. ABCD is a cyclic quadrilateral. Given that \(\angle\)ABC = 65° and \(\angle\)DAC = 60°, find the measure of \(\angle\)BCD.

17. In a circle centered at O, AB is a diameter. ABCD is a cyclic quadrilateral. Given \(\angle\)ABC = 65° and \(\angle\)DAC = 40°, find the value of \(\angle\)BCD.

18. ABCD is a cyclic quadrilateral. If \(\angle\)ABC = 65° and \(\angle\)CAD = 40°, then find the value of \(\angle\)BCD.

(a) 70° (b) 25° (c) 115° (d) 90°

19. ABCD is a cyclic quadrilateral. If \(\angle\) BAD = 65°, \(\angle\) ABD = 70°, and \(\angle\) BDC = 45°, then find the value of \(\angle\) ACB.

20. In a circle centered at O, PQ is a diameter; R is a point on the circle, and PR = RQ. Then, the measure of \(\angle\)RPQ is: ____?

(a) 30° (b) 90° (c) 60° (d) 45°

21. In the adjacent figure, O is the center of the circle, AC is the diameter, and chord DE is parallel to diameter AC. If \(\angle\)CBD = 60°, find the value of \(\angle\)CDE.

22. ABCD is a cyclic quadrilateral and O is the center of the circle. If \(\angle COD = 120^\circ\) and \(\angle BAC = 30^\circ\), then calculate the values of \(\angle BOC\) and \(\angle BCD\).

23. 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.

24. In the given figure, O is the center of the circle and AB is the diameter. ABCD is a cyclic quadrilateral. If ∠ADC = 120°, then find the measure of ∠BAC.

(a) 50° (b) 60° (c) 30° (d) 40°

25. In the given figure, O is the center of the circle and AB is the diameter. ABCD is a cyclic quadrilateral. If ∠ABC = 65° and ∠DAC = 40°, then find the measure of ∠BCD.

(a) 75° (b) 105° (c) 115° (d) 80°

26. In the given figure, O is the center of the circle and AB is the diameter. ABCD is a cyclic quadrilateral where AB || DC and ∠BAC = 25°. Find the measure of ∠DAC.

27. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

(a) 3 (b) -3 (c) \(\cfrac{1}{3}\) (d) -\(\cfrac{1}{3}\)

28. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

(a) -\(\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

29. In a circle with center O, a tangent PT is drawn from an external point P to the circle, with T being the point of tangency. If PT = 12 cm and OP = 13 cm, the diameter of the circle will be:

(a) 5 cm (b) 8 cm (c) 6 cm (d) 10 cm

30. 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°