Relation between A.M. and G.M. - Question Bank

1. If A.M. = 20 and G.M. = 16 for two positive numbers, find the value of |a - b|.
A) 8
B) 12
C) 16
D) 24
2. If x, y > 0 and x/y + y/x = 5/2, find the ratio x/y.
A) 1:2
B) 2:1
C) 1:2 or 2:1
D) 1:5
3. The A.M. of two numbers is 5. If their G.M. is 4, find the numbers.
A) 1 and 9
B) 2 and 8
C) 3 and 7
D) 4 and 6
4. If x and y are positive real numbers, and A.M.(x, y) = 10, G.M.(x, y) = 6, what is x² + y²?
A) 136
B) 172
C) 200
D) 260
5. If the G.M. of two numbers is 6 and their A.M. is 10, find the numbers.
A) 2 and 18
B) 3 and 12
C) 4 and 9
D) 6 and 6
6. For a set of 'n' positive real numbers, the A.M. is always greater than or equal to the G.M. This is known as:
A) Cauchy-Schwarz Inequality
B) AM-GM Inequality
C) Jensen's Inequality
D) Bernoulli's Inequality
7. If x and y are positive real numbers such that x + y = 16 and xy = 60, find their A.M. and G.M.
A) A.M. = 8, G.M. = √60
B) A.M. = 16, G.M. = 60
C) A.M. = 8, G.M. = 60
D) A.M. = 16, G.M. = √60
8. The A.M. of two numbers is 18. If their G.M. is 12, find the numbers.
A) 6 and 30
B) 8 and 22
C) 10 and 26
D) 12 and 24
9. If A.M. = 15 and G.M. = 9 for two positive numbers, find the value of |a - b|.
A) 6
B) 12
C) 18
D) 24
10. If x, y > 0 and x/y + y/x = 2.5, find the ratio x/y.
A) 1:2
B) 2:1
C) 1:2 or 2:1
D) 1:1
11. The A.M. of two positive numbers is 10. If their G.M. is 8, find the sum of their squares.
A) 164
B) 200
C) 260
D) 324
12. If A is the A.M. and G is the G.M. of two positive numbers, then the numbers are given by:
A) A ± √(A² - G²)
B) A ± √(A² + G²)
C) G ± √(A² - G²)
D) G ± √(A² + G²)
13. If the G.M. of two numbers is 7 and their A.M. is 12.5, find the numbers.
A) 3 and 10
B) 4 and 9
C) 5 and 10
D) 3 and 12
14. For any positive real numbers a, b, c, the inequality A.M. ≥ G.M. states that:
A) (a+b+c)/3 < ³√(abc)
B) (a+b+c)/3 > ³√(abc)
C) (a+b+c)/3 = ³√(abc)
D) (a+b+c)/3 ≥ ³√(abc)
15. If x and y are positive real numbers and x + y = 14, and xy = 48, find their A.M. and G.M.
A) A.M. = 7, G.M. = √48
B) A.M. = 14, G.M. = 48
C) A.M. = 7, G.M. = 48
D) A.M. = 14, G.M. = √48
16. The A.M. of two numbers is 20. If their G.M. is 16, find the numbers.
A) 4 and 36
B) 8 and 32
C) 10 and 30
D) 16 and 20
17. Let 'a' and 'b' be two positive numbers. If A.M. = 15 and G.M. = 12, find the value of |a - b|.
A) 6
B) 12
C) 18
D) 24
18. If x, y > 0 and A.M.(x, y) = k * G.M.(x, y), for what value of k does a unique solution for x/y exist?
A) k < 1
B) k = 1
C) k > 1
D) k = 0
19. If A.M. = 10 and G.M. = 8 for two positive numbers, what is the value of the sum of squares of the numbers?
A) 164
B) 200
C) 260
D) 324
20. If the A.M. of two positive numbers is 18 and their G.M. is 6, find the numbers.
A) 2 and 34
B) 3 and 33
C) 6 and 30
D) 9 and 27
21. For any positive real number 'a', the minimum value of a + 1/a is:
A) 1
B) 2
C) 0.5
D) 4
22. If x, y > 0 and A.M.(x, y) = 3 * G.M.(x, y), find the ratio x/y.
A) 1:1
B) 3:1
C) 9+4√5 : 1
D) 9-4√5 : 1
23. The A.M. of two numbers is 15. If their G.M. is 9, find the numbers.
A) 3 and 27
B) 5 and 25
C) 6 and 24
D) 9 and 15
24. Let 'a' and 'b' be two positive numbers. If A.M. = 13 and G.M. = 5, find the value of |a - b|.
A) 12
B) 18
C) 24
D) 30
25. If the G.M. of two numbers is 5 and their A.M. is 7.5, find the numbers.
A) 5 and 10
B) 2.5 and 12.5
C) 3 and 12
D) 4 and 11
26. If x and y are positive real numbers and x/y + y/x = 2, what is the relation between x and y?
A) x > y
B) x < y
C) x = y
D) x = 2y
27. The A.M. of two positive numbers is 10. If their G.M. is 6, find the difference between the numbers.
A) 4
B) 8
C) 12
D) 16
28. If A.M. of a and b is A and G.M. is G, then (a-b)² is equal to:
A) A² - G²
B) 4(A² - G²)
C) A² + G²
D) 4(A² + G²)
29. If A.M. = 25 and G.M. = 20 for two positive numbers, find the numbers.
A) 10 and 40
B) 15 and 35
C) 20 and 30
D) 25 and 25
30. What is the minimum value of x + 1/x for x > 0?
A) 1
B) 2
C) 0.5
D) 4
31. If x and y are positive real numbers such that x + y = 12 and xy = 35, find the A.M. and G.M. respectively.
A) A.M. = 6, G.M. = √35
B) A.M. = 12, G.M. = 35
C) A.M. = 6, G.M. = 35
D) A.M. = 12, G.M. = √35
32. The A.M. of two positive numbers is 6. If their G.M. is 3, find the numbers.
A) 3 and 9
B) 2 and 18
C) 1 and 36
D) 3 and 3
33. Let 'a' and 'b' be two positive numbers. If A.M. = 10 and G.M. = 8, find the value of a³ + b³.
A) 1024
B) 1312
C) 1728
D) 2048
34. If x, y > 0 and A.M.(x, y) = 2 * G.M.(x, y), find the ratio x/y.
A) 1:1
B) 2:1
C) 3+2√2 : 1
D) 3-2√2 : 1
35. Find the A.M. and G.M. of the numbers 4 and 9.
A) A.M. = 6.5, G.M. = 6
B) A.M. = 5, G.M. = 36
C) A.M. = 6.5, G.M. = 36
D) A.M. = 5, G.M. = 6
36. If A is the A.M. and G is the G.M. of two positive numbers 'a' and 'b', then the numbers are the roots of which quadratic equation?
A) x² - 2Ax + G² = 0
B) x² + 2Ax + G² = 0
C) x² - Ax + G = 0
D) x² - 2Ax + G = 0
37. The A.M. of two numbers is 18 and their G.M. is 12. What are the numbers?
A) 6 and 30
B) 8 and 22
C) 10 and 26
D) 12 and 24
38. Let 'a' and 'b' be positive real numbers. If A.M. = 15 and G.M. = 9, find the value of |a - b|.
A) 12
B) 16
C) 20
D) 24
39. If the G.M. of two positive numbers is 9 and their A.M. is 10, find the difference between the numbers.
A) 2
B) 4
C) 6
D) 8
40. For any set of 'n' positive real numbers, the equality A.M. = G.M. holds if and only if:
A) All numbers are distinct
B) All numbers are equal
C) The sum of the numbers is maximum
D) The product of the numbers is maximum
41. If the A.M. of two positive numbers is 13 and their G.M. is 12, find the numbers.
A) 4 and 9
B) 3 and 10
C) 2 and 11
D) 1 and 12
42. What is the A.M. of 'a' and 'b' if their G.M. is √ab?
A) (a+b)/2
B) a+b
C) ab
D) √(a²+b²)
43. If x and y are positive real numbers such that x + y = 10 and xy = 21, find the A.M. and G.M. respectively.
A) A.M. = 5, G.M. = √21
B) A.M. = 10, G.M. = 21
C) A.M. = 5, G.M. = 21
D) A.M. = 10, G.M. = √21
44. For a set of 'n' positive real numbers x₁, x₂, ..., xn, their Arithmetic Mean is denoted by A and their Geometric Mean is denoted by G. The relationship between A and G is:
A) A > G
B) A < G
C) A = G
D) A ≥ G
45. If the G.M. of two positive numbers is 6 and their A.M. is 7.5, find the numbers.
A) 3 and 12
B) 4 and 9
C) 5 and 10
D) 6 and 9
46. Consider three positive real numbers x, y, and z. If their Arithmetic Mean is A and their Geometric Mean is G, what is the generalized inequality between A and G?
A) A > G
B) A < G
C) A = G
D) A ≥ G
47. Let 'a' and 'b' be two positive real numbers. If A.M. = 5 and G.M. = 3, find the value of a² + b².
A) 34
B) 45
C) 50
D) 61
48. If the Arithmetic Mean of two positive numbers is 10 and their Geometric Mean is 8, what are the two numbers?
A) 4 and 16
B) 2 and 32
C) 5 and 20
D) 8 and 12
49. What is the condition for the equality of Arithmetic Mean and Geometric Mean for two positive real numbers 'a' and 'b'?
A) a > b
B) a < b
C) a = b
D) a ≠ b
50. For two positive real numbers 'a' and 'b', which inequality correctly represents the relationship between their Arithmetic Mean (A.M.) and Geometric Mean (G.M.)?
A) A.M. > G.M.
B) A.M. < G.M.
C) A.M. = G.M.
D) A.M. ≥ G.M.
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