1. 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
2. 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
3. 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
4. If \(\cfrac{1}{3x}+\cfrac{1}{4x}+\cfrac{1}{2x}+\cfrac{1}{6x}=5\), then what is the value of \(x\)?
(a) \(\cfrac{1}{7}\) (b) \(\cfrac{1}{5}\) (c) \(\cfrac{1}{4}\) (d) None of the above
5. If the measure of an angle in degrees is \(x\) and in radians is \(y\), then what is the value of \( \frac{x}{y} \)?
(a) \(\cfrac{π}{180}\) (b) \(\cfrac{π}{90}\) (c) \(\cfrac{90}{π}\) (d) \(\cfrac{180}{π}\)
6. 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) \(\cfrac{3}{5}\)
7. If \(x\sin45^\circ \cos45^\circ \tan60^\circ = \tan^2 45^\circ - \cos^2 60^\circ\), then what is the value of \(x\)?
(a) 1 (b) \(\cfrac{2}{\sqrt3}\) (c) \(\cfrac{1}{\sqrt3}\) (d) \(\cfrac{\sqrt3}{2}\)
8. In a right-angled triangle, the two acute angles are \(\theta\) and \(\phi\). If \( \tan\theta = \cfrac{5}{12} \), then what is the value of \( \sin\phi \)?
(a) \(\cfrac{12}{13}\) (b) \(\cfrac{5}{13}\) (c) \(\cfrac{1}{4}\) (d) \(\cfrac{10}{13}\)
9. \(\theta\) is a positive acute angle, and if \( \tan\theta = \cot\theta \), then what is the value of \(\theta\)?
(a) 40° (b) 45° (c) 60° (d) 20°
10. In a right-angled quadrilateral, if the number of vertices is denoted by \(x\), the number of sides by \(y\), and the number of diagonals by \(z\), then what is the value of \(x + 3y - 5z\)?
(a) 14 (b) 44 (c) 20 (d) 24
11. 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
12. \(y\) varies directly with the square of \(x\), and \(y = 9\) when \(x = 9\); if \(y = 4\), then what is the value of \(x\)?
13. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?
14. If \(\theta\) is a positive acute angle and \(\sin \theta = \frac{\sqrt{3}}{2}\), then what is the value of \(\tan(\theta - 15^\circ)\)?
15. If the compound ratio of \(2 : 3\), \(3 : 5\), and \(x : 8\) is \(1 : 4\), then what is the value of \(x\)?
(a) 3 (b) 4 (c) 5 (d) 6
16. If 8, 14, \(7x + 2\), and 28 are in continued proportion, then what is the value of \(x\)?
(a) 1 (b) 3 (c) 2 (d) 4
17. If \(4\), \(5 + x\), \(2x + 1\), and \(14\) are in continued proportion and \(x\) is a positive number, then what is the value of \(x\)?
(a) 1 (b) 2 (c) 4 (d) 3
18. If the numbers \((2 + x)\), \((8 + x)\), and \((26 + x)\) are in continued proportion, then what is the value of \(x\)?
(a) 4 (b) 3 (c) 2 (d) 1
19. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then what is the value of \(x\)?
20. If \(\theta\) is a positive acute angle and \( \sin\theta = \cos(2\theta + 15^\circ) \), then what is the value of \(\theta\)?
(a) 30° (b) 25° (c) 60° (d) 90°
21. In the figure, triangle ABC is inscribed in a circle and touches the circle at points P, Q, and R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, and BC = x cm, then what is the value of x?
22. In the cyclic quadrilateral ABCD, if \(\angle\)DBA = 50° and \(\angle\)ADB = 33°, then what is the value of \(\angle\)BCD?
23. 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?
24. 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?
25. If sinA+sinB=2, then what is the value of cosA-cosB
(a) 0 (b) 1 (c) 2 (d) 3
26. 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) -\(\cfrac{1}{3}\)
27. In triangle ABC , what is the value of sin\(\cfrac{(B+C)}{2} \) ?
(a) sin\(\cfrac{A}{2}\) (b) sinA (c) cosA (d) cos \(\cfrac{A}{2}\)
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. If PQRS is a cyclic parellelogram, what is the value of \( \angle P \) ?
(a) 45° (b) 60° (c) 90° (d) 75°
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°
