Relation between A.M. and G.M. - One Line Questions
1.
What is the A.M. of 'a' and 'b' if their G.M. is √ab? —
(a+b)/2
2.
For any positive real numbers a, b, c, the inequality A.M. ≥ G.M. states that: —
(a+b+c)/3 ≥ ³√(abc)
3.
For any positive real number 'a', the minimum value of a + 1/a is: —
2
4.
What is the minimum value of x + 1/x for x > 0? —
2
5.
The A.M. of two numbers is 5. If their G.M. is 4, find the numbers. —
1 and 9
6.
If x, y > 0 and A.M.(x, y) = 3 * G.M.(x, y), find the ratio x/y. —
9+4√5 : 1
7.
If x, y > 0 and A.M.(x, y) = 2 * G.M.(x, y), find the ratio x/y. —
3+2√2 : 1
8.
If x, y > 0 and x/y + y/x = 2.5, find the ratio x/y. —
1:2 or 2:1
9.
If x, y > 0 and x/y + y/x = 5/2, find the ratio x/y. —
1:2 or 2:1
10.
If A.M. = 25 and G.M. = 20 for two positive numbers, find the numbers. —
10 and 40
11.
Let 'a' and 'b' be two positive numbers. If A.M. = 10 and G.M. = 8, find the value of a³ + b³. —
1728
12.
Let 'a' and 'b' be two positive numbers. If A.M. = 13 and G.M. = 5, find the value of |a - b|. —
24
13.
Let 'a' and 'b' be positive real numbers. If A.M. = 15 and G.M. = 9, find the value of |a - b|. —
24
14.
If x and y are positive real numbers, and A.M.(x, y) = 10, G.M.(x, y) = 6, what is x² + y²? —
260
15.
If A.M. = 10 and G.M. = 8 for two positive numbers, what is the value of the sum of squares of the numbers? —
260
16.
The A.M. of two positive numbers is 10. If their G.M. is 8, find the sum of their squares. —
260
17.
If the G.M. of two positive numbers is 9 and their A.M. is 10, find the difference between the numbers. —
4
18.
If the G.M. of two numbers is 6 and their A.M. is 10, find the numbers. —
2 and 18
19.
If the A.M. of two positive numbers is 18 and their G.M. is 6, find the numbers. —
6 and 30
20.
If the G.M. of two numbers is 7 and their A.M. is 12.5, find the numbers. —
5 and 10
21.
If the G.M. of two positive numbers is 6 and their A.M. is 7.5, find the numbers. —
3 and 12
22.
The A.M. of two numbers is 15. If their G.M. is 9, find the numbers. —
3 and 27
23.
The A.M. of two positive numbers is 6. If their G.M. is 3, find the numbers. —
3 and 9
24.
Let 'a' and 'b' be two positive real numbers. If A.M. = 5 and G.M. = 3, find the value of a² + b². —
61
25.
The A.M. of two positive numbers is 10. If their G.M. is 6, find the difference between the numbers. —
16
26.
If the Arithmetic Mean of two positive numbers is 10 and their Geometric Mean is 8, what are the two numbers? —
4 and 16
27.
The A.M. of two numbers is 20. If their G.M. is 16, find the numbers. —
8 and 32
28.
If the A.M. of two positive numbers is 13 and their G.M. is 12, find the numbers. —
4 and 9
29.
If the G.M. of two numbers is 5 and their A.M. is 7.5, find the numbers. —
5 and 10
30.
Let 'a' and 'b' be two positive numbers. If A.M. = 15 and G.M. = 12, find the value of |a - b|. —
12
31.
If A.M. = 15 and G.M. = 9 for two positive numbers, find the value of |a - b|. —
12
32.
The A.M. of two numbers is 18. If their G.M. is 12, find the numbers. —
6 and 30
33.
The A.M. of two numbers is 18 and their G.M. is 12. What are the numbers? —
6 and 30
34.
If A.M. = 20 and G.M. = 16 for two positive numbers, find the value of |a - b|. —
8
35.
If A is the A.M. and G is the G.M. of two positive numbers, then the numbers are given by: —
A ± √(A² - G²)
36.
What is the condition for the equality of Arithmetic Mean and Geometric Mean for two positive real numbers 'a' and 'b'? —
a = b
37.
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 ≥ G
38.
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 ≥ G
39.
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.M. = 5, G.M. = √21
40.
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.M. = 6, G.M. = √35
41.
Find the A.M. and G.M. of the numbers 4 and 9. —
A.M. = 6.5, G.M. = 6
42.
If x and y are positive real numbers and x + y = 14, and xy = 48, find their A.M. and G.M. —
A.M. = 7, G.M. = √48
43.
If x and y are positive real numbers such that x + y = 16 and xy = 60, find their A.M. and G.M. —
A.M. = 8, G.M. = √60
44.
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.M. ≥ G.M.
45.
If A.M. of a and b is A and G.M. is G, then (a-b)² is equal to: —
4(A² - G²)
46.
For any set of 'n' positive real numbers, the equality A.M. = G.M. holds if and only if: —
All numbers are equal
47.
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: —
AM-GM Inequality
48.
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? —
k > 1
49.
If x and y are positive real numbers and x/y + y/x = 2, what is the relation between x and y? —
x = y
50.
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? —
x² - 2Ax + G² = 0