1. In the cyclic quadrilateral ABCD, if \(\angle\)DBA = 50° and \(\angle\)ADB = 33°, then what is the value of \(\angle\)BCD?
2. 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?
3. If ABCD is a cyclic quadrilateral and \(\angle\)A=120°, what is the measure of \(\angle\)C ?
(a) \(\cfrac{π}{3}\) (b) \(\cfrac{π}{6}\) (c) \(\cfrac{π}{2}\) (d) \(\cfrac{2π}{3}\)
4. In the cyclic quadrilateral ABCD, if AB = AD, \(\angle\)DAC = 70° and \(\angle\)BDC = 50°, then what is the measure of \(\angle\)ACD?
(a) 30\(^o\) (b) 40\(^o\) (c) 50\(^o\) (d) 70\(^o\)
5. If PQ is a tangent to a circle centered at O and touches the circle at point C, then what is the value of \(\angle\)BCP?
(a) 90\(^o\) (b) 80\(^o\) (c) 70\(^o\) (d) None of the above
6. ABCD is a cyclic quadrilateral, and CD is extended up to point E. If \(\angle\)ADE = 92°, then what is the measure of \(\angle\)ABC?
(a) 88\(^o\) (b) 29\(^o\) (c) 92\(^o\) (d) 60\(^o\)
7. 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°
8. In triangle ABC, \(\angle C = 90^\circ\) and AC : BC = 3 : 4, then what is the value of cosec A?
(a) \(\cfrac{3}{4}\) (b) \(\cfrac{5}{3}\) (c) \(\cfrac{5}{4}\) (d) \(\cfrac{3}{5}\)
9. 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
10. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)?
(a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt2}\) (c) \(\frac{\sqrt3}{2}\) (d) 1
11. ABCD is a cyclic trapezium in which sides AD and BC are parallel to each other. If \(\angle\)ABC = 75°, then what is the measure of \(\angle\)BCD?
(a) 105° (b) 90° (c) 150° (d) 75°
12. AB is extended to point X in the cyclic quadrilateral ABCD. If \(\angle\)XBC = 98° and \(\angle\)ADB = 45°, then what is the measure of \(\angle\)BAC?
13. 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°
14. AB is a chord of a circle centered at O. The tangent drawn at point B intersects the extended line AO at point T. If \(\angle\)BAT = 25°, then what is the value of \(\angle\)BTA?
15. ABCD is a cyclic quadrilateral and O is the center of the circle. If \(\angle\)COD = 120° and \(\angle\)BAC = 30°, then what is the measure of \(\angle\)BOC?
16. In the cyclic quadrilateral ABCD, side AB is extended up to point X. If \(\angle\)XBC = 82° and \(\angle\)ADB = 47°, then what is the measure of \(\angle\)BAC?
(a) 45° (b) 45° (c) 35° (d) 60°
17. 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) -\(\cfrac{1}{3}\)
18. In triangle ABC , what is the value of sin\(\cfrac{(B+C)}{2} \) ?
(a) sin\(\cfrac{A}{2}\) (b) sinA (c) cosA (d) cos \(\cfrac{A}{2}\)
19. 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
20. If \( 2 \cos \theta = 1 \), what is the value of \( \theta \) ?
(a) 10° (b) 15° (c) 60° (d) 30°
21. If \( 2\sqrt{6} \) is a rationalizing factor of \( \sqrt{2x} \), what is the value of \( x \) ?
(a) 2 (b) 3 (c) 6 (d) √6
22. If the number of vertices, faces, and edges of a cuboid are \( p \), \( q \), and \( r \) respectively, what is the value of \( \frac{3(p + r)}{2q} \) ?
(a) 10 (b) 12 (c) 5 (d) 6
23. PQRS is a cyclic trapezium. PQ is a diameter of the circle, and PO || SR. If \(\angle\)QRS = 110°, then the value of \(\angle\)QSR is -
(a) 20° (b) 25° (c) 30° (d) 40°
24. At an annual simple interest rate of 12%, if the ratio of principal to interest after \(x\) years is 25:24, what is the value of \(x\)?
(a) 8 (b) 10 (c) 12 (d) 5
25. 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°
26. ABCD is a cyclic trapezium, where AD \(\parallel\) BC. If \(\angle\)ABC = 70°, then the measure of \(\angle\)BCD will be—?
(a) 110° (b) 80° (c) 70° (d) 120°
27. If a right-angled quadrilateral has \(x\) number of vertices, \(y\) number of edges, and \(z\) number of faces, then what is the value of \(x - y + z\)?
(a) 8 (b) 6 (c) 2 (d) 12
28. If a right-angled quadrilateral has \(a\) number of vertices, \(b\) number of edges, and \(c\) number of faces, then what is the value of \(2a - b + 3c\)?
(a) 16 (b) 18 (c) 20 (d) 22
29. If sin 51° = \(\cfrac{a}{\sqrt{a^2 + b^2}}\), then what is the value of tan 51° + tan 39°?
(a) \(\cfrac{a^2-b^2}{ab}\) (b) \(\cfrac{a^2+b^2}{2ab}\) (c) \(\cfrac{a^2+b^2}{ab}\) (d) \(\cfrac{a^2-b^2}{2ab}\)
30. If \( \tan^4\theta + \tan^2\theta = 1 \), then what is the value of \( \cos^4\theta + \cos^2\theta - 1 \)?
(a) 1 (b) 1 (c) 0 (d) None of the above
