1. If the fourth of 5 consecutive geometric numbers is 54 and the fifth is 162, then find the first number.

2. If the average of the first ten natural numbers is A and the median is M, then the relationship between A and M is —

(a) \(A\gt M\) (b) \(A\lt M\) (c) \(A=\cfrac{1}{M}\) (d) \(A=M\)

3. If the product of three consecutive positive geometric numbers is 64, then the mean proportional between the first and the third is ______.

4. If the average of the numbers 6, 7, x, 8, y, and 14 is 9, then

(a) x+y=21 (b) x+y=19 (c) x-y=21 (d) x-y=19

5. If the average of the numbers 6, 7, x, 8, y, and 16 is 9, then—?

(a) x + y = 21 (b) x + y = 17 (c) x - y = 21 (d) x - y = 19

6. If the first of five consecutive numbers in geometric progression is 2 and the second is 6, what is the fifth number?

7. The difference between two positive integers is 3, and the sum of their squares is 117. Find the two numbers.

(a) 6,9 (b) 3,6 (c) 4,9 (d) 5,6

8. If the second and third numbers of four consecutive proportional numbers are 15 and 45, respectively, then what are the first and fourth numbers?

9. If the combined average of the numbers 7, \(x - 3\), 10, \(x + 3\), and \(x - 5\) is 15, then the median is _____.

(a) 16 (b) 10 (c) 18 (d) 24

10. In a group of \(n\) numbers with an average of \(\bar{x}\), if the sum of the first \((n - 1)\) numbers is \(K\), then the \(n\)th number will be \((n - 1)\bar{x} + K\).

11. Translate: If the average of \(p\) numbers is \(b\), and the average of \((p + q)\) numbers is \(a\), then the average of the \(q\) numbers is _____

12. If the average of \( (p + q) \) numbers is \( x \), and the average of \( p \) of those numbers is \( y \), then the average of the remaining \( q \) numbers will be _____ .

13. If the combined average of 5, 15, 22, x, y, 25, and z is 14, then which of the following is correct?

(a) x+y+z=42 (b) x+y+z=31 (c) x+y=z (d) x+y=2z

14. If the middle number of the first \((2n + 1)\) consecutive natural numbers is \(\frac{n + 103}{3}\), then find the value of \(n\).

15. From a ghat on one side of a 600-meter wide river, two boats set off toward the opposite bank in two different directions. If the first boat moves at an angle of 30° with the riverbank, and the second boat moves at a 90° angle to the path of the first boat, then what will be the distance between the two boats after they reach the opposite bank?

16. The heights of two pillars are 180 meters and 60 meters respectively. If the angle of elevation to the top of the second pillar from the base of the first pillar is 30°, then find the angle of elevation to the top of the first pillar from the base of the second pillar.

17. If the difference between the compound interest and the simple interest on a certain sum of money for 2 years at an annual interest rate of 9% is ₹129.60, then find the principal amount.

18. If the difference between the compound interest and simple interest on a certain principal for 2 years at an annual rate of 5% is ₹25, then what is the amount of the principal?

19. 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?

20. If ₹780 is invested for the first 5 years at an interest rate of 4%, for the next 4 years at 5%, and for the last 5 years at 7%, what will be the total interest amount?

(a) ₹ 855 (b) ₹ 685 (c) ₹ 709 (d) ₹ 585

21. The difference between two positive integers is 3, and the sum of their squares is 117. Determine the two numbers.

22. If the average of \(n\) natural numbers is \(\cfrac{n+8}{4}\), then find the value of \(n\).

(a) 4 (b) 6 (c) 8 (d) 10

23. 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.

24. "Four cylindrical concrete pillars each have a diameter of 17.5 cm and are to be plastered all around with a 3.5 cm thick layer of sand-cement mixture. If the height of each pillar is 3 metres, calculate and write how many cubic decimetres of mortar will be required."

25. "Four cylindrical concrete pillars each have a diameter of 17.5 cm and are to be plastered all around with a 3.5 cm thick layer of sand-cement mixture. If the mortar is prepared by mixing sand and cement in a 4:1 ratio, calculate and write how many cubic decimetres of cement will be required."

26. The difference between the outer and inner curved surface areas of a hollow cylindrical pipe is 44 square centimeters, and the length of the pipe is 14 cm. The volume of the material of the pipe is 99 cubic centimeters. Find the outer and inner radii of the pipe.

27. From a point on the roof of a five-storey building, the angle of elevation to the top of a monument and the angle of depression to its base are complementary angles. If the height of the monument is four times the height of the building, find the measures of the elevation and depression angles.

28. 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?

29. The perimeters of two similar triangles are 27 cm and 16 cm respectively. If one side of the first triangle is 9 cm, then find the length of the corresponding side of the second triangle.

30. Standing in the middle of the field, Habu sees a flying bird first at an elevation angle of 30° to the north, and after 2 minutes, at an elevation angle of 60° to the south. If the bird is flying along a straight path at a constant height of \(50\sqrt{3}\) meters, then what is its speed?