JEE Main - Relation between A.M. and G.M. Online Test (MCQ Quiz)

1. 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.

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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)

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. What is the minimum value of x + 1/x for x > 0?

A) 1 B) 2 C) 0.5 D) 4

22. 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

23. 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)

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. For any positive real number 'a', the minimum value of a + 1/a is:

A) 1 B) 2 C) 0.5 D) 4

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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)

38. 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

39. 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)

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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

49. 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

50. 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

Relation between A.M. and G.M. - Online Test

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