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. 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?

3. The tip of a clock's minute hand rotates in one hour—

(a) \(-\cfrac{\pi}{4}\) radians (b) \(-\cfrac{\pi}{2}\) radians (c) \(-\pi\) radians (d) \(-2\pi\) radians

4. The tip of a clock’s minute hand rotates in 1 hour.

(a) \(\cfrac{π}{4}\) radians (b) \(\cfrac{π}{2}\)radians (c) \(π\) radians (d) \(2π\) radians

5. Write the circular (radian) measure of the angle produced by the tip of a clock’s hour hand after rotating for 1 hour.

6. The tip of the minute hand of a clock — how many radians does it rotate in one hour?

7. 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

8. AC = CE ABC is an equilateral triangle BC = AC = CE ∠BCA = 60° ∠BCE = 180° − 60° = 120° BC = CE ∠CBE = ∠CEB = \(\frac{180° − 120°}{2} = 30°\) 180° = \(\pi\) radians 120° = \(\frac{\pi × 120}{180} = \frac{2\pi}{3}\) radians 30° = \(\frac{\pi × 30}{180} = \frac{\pi}{6}\) radians Angles of triangle ACE in radians: \(\frac{2\pi}{3}\), \(\frac{\pi}{6}\), \(\frac{\pi}{6}\)

9. 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.

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

11. The speed of a boat in still water is 4 km/h. It travels 15 km downstream and 22 km upstream in a total time of 5 hours. Find the speed of the stream.

12. A person travelled 64 km downstream in 4 hours and returned upstream in 32 hours. If the speed of the boat is 5 km/h, what is the speed of the stream in km/h?

(a) 1.5 (b) 2 (c) 2.5 (d) 3

13. The ratio of the speeds of the hour hand and the minute hand of a clock is –

(a) 1:12 (b) 12:1 (c) 1:24 (d) 24:1

14. Standing in the middle of a field, Mohit first saw a flying bird at an angle of elevation of 30° towards the north, and 2 minutes later, he saw it at an angle of elevation of 60° towards the south. If the bird was flying at a constant height of 50\(\sqrt{3}\) meters along a straight path, determine its speed in kilometers per hour.

15. Steady. A boat has a speed of 4 km per hour in still water. The boat travels 15 km downstream in 5 hours and 22 km upstream in 5 hours. Form an equation to determine the speed of the current.

16. Given that the speed of a boat in still water is 4 km/h, the boat travels 15 km downstream in 5 hours and 22 km upstream in 5 hours. Form an equation and determine the speed of the current using Sridharacharya's formula.

17. Standing in the middle of a field, Mohit saw a flying bird—first to the north at an elevation angle of 30°, and then 2 minutes later to the south at an elevation angle of 60°. If the bird is flying along the same straight path at a height of 50√3 meters, calculate its speed in kilometers per hour.

18. If the sum of two angles is 135° and their difference is \(\cfrac{\pi}{12}\), find the values of the two angles in degrees and radians.

19. The speed of a boat in still water is 8 km/h. The boat travels 15 km downstream and 22 km upstream in 5 hours. Calculate and write the speed of the current.

20. A circle is drawn with center at point A of quadrilateral ABCD, which passes through points B, C, and D. Prove that \(\angle\)CBD + \(\angle\)CDB = \(\cfrac{1}{2}\angle\)BAD.

21. A bird was flying from north to south along a path parallel to the ground at a height of 200 meters. Standing at the center of a field, Sushobhan first saw the bird at an angle of elevation of 30° to the north. Three minutes later, he again saw it at an angle of elevation of 45° to the south. Calculate and write the bird’s speed in kilometers per hour, rounded to the nearest whole number. [Take \(\sqrt{3} \approx 1.732\)]

22. Starting from the endpoint of a ray and rotating it clockwise twice around its center produces an angle of 720 degrees.

23. A bird was flying from north to south along a straight path parallel to the ground, at a height of 200 meters. Standing in the middle of a field, Sushovan first saw the bird at an angle of elevation of 30° towards the north. After 3 minutes, he saw it again at an angle of elevation of 45° towards the south. Find the bird’s speed in kilometers per hour, rounded to the nearest whole number. [Given: √3 ≈ 1.732]

24. If the ratio of three consecutive angles of a cyclic quadrilateral is 1:2:3, find the first and third angles in radians.

25. A person first observes a bird flying at an angle of elevation of 60° towards the north, and after 5 minutes, sees it at an angle of elevation of 30° towards the south. If the bird is flying along the same straight line at a height of \(120\sqrt{3}\) meters, determine its speed in kilometers per hour.

26. Standing in the middle of the field, Rohit first saw a flying bird at an elevation angle of 30° towards the north, and then 2 minutes later, saw it at an angle of 60° towards the south. If the bird is flying along a straight path at a constant height of 50√3 meters, what is the bird’s speed in kilometers per hour?