1. 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) \(\cfrac{10}{13}\)

2. Two angles of a triangle are 35°57′4″ and 39°2′56″. What is the radian measure of the third angle?

3. What will be the measure of the third angle of a triangle in radians if the other two angles are \(65^\circ 56' 44''\) and \(64^\circ 3' 16''\)?

4. The base of a triangle is \(16\sqrt{3}\) cm, and the two angles adjacent to the base are 30° and 60°. What is the height of the triangle?

5. Draw a triangle where one side is 6.7 cm and the two adjacent angles are 75° and 55°. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]

6. The length of one side is 6.2 cm, and the measures of the two angles adjacent to that side are 50° and 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

7. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?

8. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

9. In triangle \(\triangle ABC\), D and E are the midpoints of sides AB and AC respectively. If the area of triangle ABC is 36 square centimeters, then what is the area of triangle ADE?

(a) 6 square centimeters (b) 9 square centimeters (c) 12 square centimeters (d) 16 square centimeters

10. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

11. 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

12. In triangle ABC, the circumcenter is O. If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then what is the measure of \(\angle\)OAC?

(a) 70° (b) 40° (c) 110° (d) 140°

13. ABC and POR are two similar triangles. If BC = 5 cm, QR = 4 cm, and the height AD = 3 cm, then what is the length of the height PE?

(a) 4.2 cm (b) 1.25 cm (c) 5.4 cm (d) 2.4 cm

14. A rotating ray, starting from a certain position, rotates two full turns in the counterclockwise direction (opposite to the clock hands) and then produces an additional angle of 30°. What are the sexagesimal (degree) and circular (radian) measures of this angle?

15. 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?

16. If the sides of a triangle are in the ratio \(3x : 4x : 5x\), what is the measure of the largest angle of the triangle?

(a) 105\(^o\) (b) 75\(^o\) (c) 60\(^o\) (d) None of the above

17. If the angles of a triangle are in the ratio \(1 : 1 : 2\), then what will be the ratio of the sides of the triangle?

(a) \(1:\sqrt{2}:1 \) (b) \(1:1:\sqrt{2}\) (c) \(1:1:2\) (d) \(1:2:1\)

18. 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°

19. 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?

20. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, then what are the measures of the first and third angles?

21. AB and CD are two chords of a circle with center O, and both have equal lengths. If \(\angle\)AOB = 60°, then what is the measure of \(\angle\)COD?

22. In triangle ABC, the circumcenter is \(O\). If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then what is the measure of \(\angle\)AOC?

23. The perimeters of two similar triangles are 20 cm and 16 cm respectively. If one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle?

24. The lengths of two sides are 7.6 cm and 6 cm, and the included angle between them is 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

25. If the sum of two angles is 135° and their difference is \(\cfrac{\pi}{12}\), then calculate and write the measures of the two angles in both sexagesimal (degree-minute-second) and radian systems.

26. If two angles of a triangle measure \(65^\circ 56' 55''\) and \(64^\circ 3' 5''\), find the circular (radian) measure of the third angle.

27. The three sides of a triangle are 6 cm, 8 cm, and 10 cm respectively. What is the circumradius of the triangle? This triangle is special—it’s a right triangle (since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). And for right triangles, the circumradius is half the hypotenuse. So the circumradius = \(\frac{10}{2} = 5\) cm.

28. 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) -\(\cfrac{1}{3}\)

29. 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

30. If the lengths of the perpendicular sides of a right-angled triangle are \(a\) and \(b\), and the length of the perpendicular drawn from the right-angled vertex to the hypotenuse is \(p\), then –

(a) \( \cfrac{1}{p^2} =\cfrac{1}{a^2} +\cfrac{1}{b^2} \) (b) \( \cfrac{1}{p^2} =\cfrac{1}{a^2} -\cfrac{1}{b^2} \) (c) \(p^2=a^2+b^2\) (d) \(p^2=a^2-b^2\)