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If A be the A.M. of two positive unequal quantities x and y and G be their G. M, then
- a)A < G
- b)A>G
- c)A ≥ G
- d)A ≤ G
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If A be the A.M. of two positive unequal quantities x and y and G be t...
Given:
A = Arithmetic mean of x and y
G = Geometric mean of x and y
To find:
The correct expression for A and G
Solution:
Arithmetic mean (A) of x and y is given by:
A = (x + y)/2
Geometric mean (G) of x and y is given by:
G = √(x × y)
Substituting the values of A and G, we get:
A × G = [(x + y)/2] × √(x × y)
Multiplying both sides by 2, we get:
2AG = (x + y)√(xy)
Squaring both sides, we get:
(2AG)² = (x + y)² × xy
Simplifying the above expression, we get:
4A²G² = (x² + 2xy + y²) × xy
Rearranging the terms, we get:
4A²G² = (x²y + xy²) + (x²y + xy²) + 2xy²
Simplifying further, we get:
4A²G² = 2xy(x + y) + 2xy²
Dividing both sides by 4xy, we get:
A²G² = [(x + y)/2]² × [(x + y)/2 + y]/2
Simplifying the above expression, we get:
A²G² = [(x + y)/2]³/2
Taking the square root on both sides, we get:
AG = [(x + y)/2]√[(x + y)/2]
Therefore, the correct expression for A and G is:
AG.
A = Arithmetic mean of x and y
G = Geometric mean of x and y
To find:
The correct expression for A and G
Solution:
Arithmetic mean (A) of x and y is given by:
A = (x + y)/2
Geometric mean (G) of x and y is given by:
G = √(x × y)
Substituting the values of A and G, we get:
A × G = [(x + y)/2] × √(x × y)
Multiplying both sides by 2, we get:
2AG = (x + y)√(xy)
Squaring both sides, we get:
(2AG)² = (x + y)² × xy
Simplifying the above expression, we get:
4A²G² = (x² + 2xy + y²) × xy
Rearranging the terms, we get:
4A²G² = (x²y + xy²) + (x²y + xy²) + 2xy²
Simplifying further, we get:
4A²G² = 2xy(x + y) + 2xy²
Dividing both sides by 4xy, we get:
A²G² = [(x + y)/2]² × [(x + y)/2 + y]/2
Simplifying the above expression, we get:
A²G² = [(x + y)/2]³/2
Taking the square root on both sides, we get:
AG = [(x + y)/2]√[(x + y)/2]
Therefore, the correct expression for A and G is:
AG.
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Community Answer
If A be the A.M. of two positive unequal quantities x and y and G be t...
Arithmetic mean ≥ geometric mean ( generally)
arithmetic mean= geometric mean (equal numbers)
arithmetic mean>geometric mean (unequal or distinct numbers)
arithmetic mean= geometric mean (equal numbers)
arithmetic mean>geometric mean (unequal or distinct numbers)
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If A be the A.M. of two positive unequal quantities x and y and G be their G. M, thena)A < Gb)A>Gc)A ≥ Gd)A ≤ GCorrect answer is option 'B'. Can you explain this answer?
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If A be the A.M. of two positive unequal quantities x and y and G be their G. M, thena)A < Gb)A>Gc)A ≥ Gd)A ≤ GCorrect answer is option 'B'. Can you explain this answer? for CA Foundation 2025 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about If A be the A.M. of two positive unequal quantities x and y and G be their G. M, thena)A < Gb)A>Gc)A ≥ Gd)A ≤ GCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for CA Foundation 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A be the A.M. of two positive unequal quantities x and y and G be their G. M, thena)A < Gb)A>Gc)A ≥ Gd)A ≤ GCorrect answer is option 'B'. Can you explain this answer?.
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