1. What is the radian measure of the angle swept by the tip of a clock’s minute hand in 1 hour? (a) \(2\pi\) radians. (b) \(\cfrac{\pi}{2}\) radians. (c) \(\pi\) radians. (d) \(4\pi\) radians.

2. The value of \(1^c\) — (a) \(1^c>60^o\) (b) \(1^c=60^o\) (c) \(1^c<60^o\) (d) None of the above

3. The central angle of each interior angle of a regular hexagon: (a) \(\cfrac{π}{3}\) (b) \(\cfrac{2π}{3}\) (c) \(\cfrac{π}{6}\) (d) \(\cfrac{π}{4}\)

4. The value of 1 radian is: (a) In between 40° and 50° (b) less than 40° (c) In between 50° and 60° (d) greater than 60°

5. If sinθ + cosθ = √2 (where 0° < θ < 90°), then the value of θ is (a) 30° (b) 45° (c) 60° (d) 90°

6. The circular measure of each interior angle of a regular hexagon is – (a) \(\cfrac{π}{3}\) (b) \(\cfrac{2π}{3}\) (c) \(\cfrac{π}{6}\) (d) \(\cfrac{π}{4}\)

7. 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) \(\cfrac{180}{π}\)

8. If the circular (radian) measure of an angle is \( \frac{7\pi}{12} \), what is its value in the sexagesimal (degree) system? (a) 90° (b) 105° (c) 135° (d) 160°

9. If two angles of a triangle are 75° and \( \frac{\pi^c}{6} \), then what is the measure of the third angle? (a) 75° (b) 60° (c) 65° (d) 70°

10. The circular measure of each interior angle of a regular hexagon is _____. (a) \(\cfrac{\pi^c}{4}\) (b) \(\cfrac{\pi^c}{6}\) (c) \(\cfrac{\pi^c}{3}\) (d) \(\cfrac{2\pi^c}{3}\)

11. "If the two acute angles of a right-angled triangle are in the ratio 2:3, what are the radian measures of those two angles? (a) \(\cfrac{π}{5},\cfrac{3π}{10}\) (b) \(\cfrac{π}{10},\cfrac{3π}{5}\) (c) \(\cfrac{π}{5},\cfrac{3π}{20}\) (d) \(\cfrac{π}{5},\cfrac{π}{15}\)

12. The circular measure of each exterior angle of a regular hexagon is – (a) \(\cfrac{π}{3}\) (b) \(\cfrac{2π}{3}\) (c) \(\cfrac{π}{6}\) (d) \(\cfrac{π}{4}\)

13. The sum of two angles is \(\cfrac{13π}{12}\). If one of the angles is \(58°\), find the other angle. (a) 117° (b) 127° (c) 137° (d) 147°

14. In triangle ABC, AB = AC. A line drawn from point C intersects the extended line BA at point D. If AC = AD, what is the measure of \(\angle\)BCD in radians? (a) \(\cfrac{π}{2}\) (b) \(π\) (c) \(\cfrac{π}{4}\) (d) \(\cfrac{π}{3}\)

15. In parallelogram ABCD, the two angles adjacent to side BC are in the ratio 4 : 5. The measures of those two angles are — (a) 40°, 50° (b) 80°, 100° (c) 45°, 135° (d) None of the above

16. What is the ratio of the speeds of the hour hand, minute hand, and second hand of a clock? (a) 1:12:60 (b) 1:60:720 (c) 1:12:720 (d) None of the above

17. The value of \(\cfrac{7π}{12}\) in the sexagesimal system is—? (a) 115° (b) 150° (c) 135° (d) 105°

18. If a ray is rotated counterclockwise around its endpoint, the angle formed will be positive. True / False

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

20. The hour hand of a clock rotates through an angle of \(\cfrac{\pi}{6}\) radians in 2 hours.

21. Express 18° in radians.

22. In a right-angled triangle, the difference between the two acute angles is \(\frac{2\pi}{5}\). Express the measures of these two angles in both radians and degrees.

23. If the radius of a circle is 7 cm, then find the radian measure of the central angle subtended by an arc of length 5.5 cm.

24. In a right-angled triangle, if the difference between the two acute angles is 72°, find their measures in radians.

25. If three angles of a quadrilateral are \(\frac{π}{3}\), \(\frac{5π}{6}\), and \(90^\circ\), then write the measure of the fourth angle in both sexagesimal (degree) and circular (radian) units.

26. Find the radian measure of 52°52'30".

27. In a right-angled triangle, the difference between the two acute angles is 30°. Express the measures of those two angles in both radians and degrees.

28. In a circle, two arcs of unequal lengths are in the ratio 5:2. If the central angle corresponding to the second arc is 30°, what is the radian measure of the central angle corresponding to the first arc?

29. If the three angles of a triangle are in the ratio 2:3:4, then the measure of the largest angle in degrees is ________.

30. Prove that: \(1^\circ < 1^c\)

31. In an equilateral triangle ABC, the base BC is extended to a point E such that CE = BC. A is joined to E to form triangle ACE. Find the circular (radian) measures of the angles of triangle ACE.

32. If three angles of a quadrilateral are \(\frac{π}{5}\), \(\frac{5π}{6}\), and \(90^\circ\), then write the sexagesimal (degree) and circular (radian) measure of the fourth angle.

33. Give the definition of a radian. The angles of a triangle are in the ratio 2:5:3; what is the radian measure of the smallest angle of the triangle?

34. Prove that \(1^c < 60^\circ\)

35. Express 1 radian in degrees.

36. 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 \(64^\circ 3' 16''\)?

37. If the difference between the measures of the two acute angles in a right-angled triangle is 10°, determine their circular measures.

38. In right-angled triangle ∆ABC, AC² = AB² + BC² = 3² + 4² = 9 + 16 = 25 So, AC = √25 = 5 In right-angled ∆ABC, if the base is BC, then the height is AB Again, if the base is AC, then the height is BD ∴ (½) × BC × AB = (½) × AC × BD ⇒ (½) × 4 × 3 = (½) × 5 × BD ⇒ BD = 12⁄5 = 2.4 cm (Answer)

39. If the sum of two angles is 135° and their difference is \(\frac{π}{12}\), write the angles in sexagesimal and circular measure.

40. In a right-angled triangle, if one of the acute angles is 30°, determine the measure of the other acute angle in sexagesimal system.

41. 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?

42. **Give the definition of a radian. Prove that \(1^\circ < 1^c\)** (where \(1^c\) denotes 1 radian)

43. The difference between two given angles is 40°, and their sum is \(\frac{\pi}{2}\) radians. Find the radian measures of the two angles.

44. In a triangle, one angle is 60° and another angle is \(\frac{\pi}{6}\) radians. What is the measure of the third angle in degrees?