1. 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

2. 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.

3. The ratio of the capital invested by three people is 4:7:9, and the profit of the first person is ₹100 less than the profit of the third person. What is the amount of profit earned by the second person?

4. Two unequal arcs of a circle subtend two angles at the center in the ratio 5:3, and the sexagesimal (degree) measure of the second angle is 45°. What is the radian measure of the first angle?

5. Two friends start a partnership business by investing ₹40,000 and ₹50,000 respectively. They agree that 50% of the profit will be shared equally, and the remaining 50% will be divided in the ratio of their investments. If the first friend’s share of the profit is ₹800 less than the second friend’s share, then what is the first friend’s profit?

6. \[ \frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6} \] This is a mathematical equation involving rational expressions. Here's the English translation of the statement: **"The sum of the following three fractions equals one-sixth:"** - The first fraction is: one divided by \((x - 1)(x - 2)\) - The second fraction is: one divided by \((x - 2)(x - 3)\) - The third fraction is: one divided by \((x - 3)(x - 4)\) And their total is equal to \(\frac{1}{6}\).

7. A person rows a boat to a point 8 km away and returns in 4 hours 16 minutes. If the speed of the stream is 1 km/h, what is the speed of the boat in still water?

8. In a joint business, the capital ratio of three individuals is 5:8:3, and the second person's profit is ₹100 more than the third person's. What is the total profit in the business?

9. The capital ratio of three people is 4:8:9, and the difference in profit between the first and third person is ₹200. What is the profit share of the second person?

10. In a business, three friends have their capital investment in the ratio 2:3:4. If in the following year they increase their capital by 10%, 15%, and 20%, respectively, in what ratio will the profit be distributed among them at the end of the second year?

(a) 44 : 69 : 96 (b) 69 : 44 : 96 (c) 96 : 69 : 44 (d) 45 : 96 : 63

11. 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

12. 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.

13. Standing in the middle of a field, Mohit first sees a flying bird at an angle of elevation of 30° to the north. Two minutes later, he sees the same bird at an angle of elevation of 60° to the south. If the bird is flying along a straight line at a height of \(50\sqrt{3}\) meters, determine its speed in kilometers per hour.

14. Under the condition that no interest will be charged if the money is repaid within two years, three friends borrowed ₹6000, ₹8000, and ₹5000 respectively from a cooperative bank to jointly purchase four cycle rickshaws. After two years, after deducting all expenses, their total income amounted to ₹30,400. They divided this income in proportion to their capital contributions and then repaid their individual bank loans. Now, calculate how much money each person will have left in hand and what the ratio of the remaining amounts will be.

15. 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.

16. In a circle, two arcs of unequal length subtend angles at the center in the ratio 5:2. If the sexagesimal (degree) measure of the second angle is 30°, then calculate and write both the sexagesimal and radian measures of the first angle.

17. 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

18. If the rate of simple interest and compound interest is 10% per annum, find the ratio of simple interest to compound interest for the second year on any principal.

(a) 20:21 (b) 10:11 (c) 5:6 (d) 1:1

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

20. 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.

21. 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?

22. If the ratio of the radii of two perpendicular solid cones is 2:3 and the ratio of their heights is 5:3, what is the ratio of their volumes ?

(a) 27:20 (b) 20:27 (c) 4:9 (d) 9:4

23. The volume ratio of a sphere and the circumscribed cube of the sphere is

(a) π:2 (b) π:3 (c) π:4 (d) π:6

24. If the ratio of the capital investments of A, B, and C is \(\cfrac{1}{2} : \cfrac{1}{3} : \cfrac{1}{4}\) and the total profit is ₹520, then C's share of the profit will be

(a) ₹120 (b) ₹160 (c) ₹180 (d) ₹140

25. Pallavi invests ₹500 for 9 months, and Rajiya invests ₹600 for 5 months in a business. The profit will be distributed between them in the ratio –

(a) 3:2 (b) 5:6 (c) 6:5 (d) 9:5

26. If the ratio of the radii of two vertical solid circular cylinders is 2:3, and the ratio of their heights is 5:3, then the ratio of their curved surface areas will be –

(a) 2:5 (b) 8:7 (c) 10:9 (d) 16:9

27. If the volumes of two cones are in the ratio 1:4 and the ratio of the diameters of their bases is 4:5, find the ratio of their heights

(a) 1:5 (b) 5:4 (c) 5:16 (d) 25:64

28. If the ratio of the principal and the total amount is 10:11, what is the annual interest rate?

(a) 10% (b) 11% (c) \(10\cfrac{1}{11}\)% (d) 12%

29. What is the combined ratio of \(a:bc\), \(b:ca\), and \(c:ab\)?

(a) \(1:1\) (b) \(1:abc\) (c) \(abc:1\) (d) None of these

30. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC?

(a) 1:2 (b) 2:1 (c) 1:1 (d) None of these