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If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)
- a)positive
- b)negative
- c)non-positive
- d)non -negative
- e)none of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If a, b and c are distinct positive numbers, then the expression (b + ...
We can simplify the given expression as follows:
(b + c - a)(c + a - b)(a + b - c)
Using the identity (x + y - z)^2 = x^2 + y^2 + z^2 + 2xy - 2xz - 2yz, we can expand each of the three factors in the expression:
(b + c - a)^2 = b^2 + c^2 + a^2 + 2bc - 2ab - 2ac
(c + a - b)^2 = c^2 + a^2 + b^2 + 2ca - 2cb - 2ab
(a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2ac - 2bc
Adding these three expressions and simplifying, we get:
2(a^2 + b^2 + c^2) + 2(ab + ac + bc) - (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)
= a^2 + b^2 + c^2 - 2ab - 2ac - 2bc
= (a - b)^2 + (a - c)^2 + (b - c)^2
This is always non-negative, and is equal to zero if and only if a = b = c. Therefore, the given expression is positive for all distinct positive numbers a, b, and c.
(b + c - a)(c + a - b)(a + b - c)
Using the identity (x + y - z)^2 = x^2 + y^2 + z^2 + 2xy - 2xz - 2yz, we can expand each of the three factors in the expression:
(b + c - a)^2 = b^2 + c^2 + a^2 + 2bc - 2ab - 2ac
(c + a - b)^2 = c^2 + a^2 + b^2 + 2ca - 2cb - 2ab
(a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2ac - 2bc
Adding these three expressions and simplifying, we get:
2(a^2 + b^2 + c^2) + 2(ab + ac + bc) - (a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)
= a^2 + b^2 + c^2 - 2ab - 2ac - 2bc
= (a - b)^2 + (a - c)^2 + (b - c)^2
This is always non-negative, and is equal to zero if and only if a = b = c. Therefore, the given expression is positive for all distinct positive numbers a, b, and c.
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Community Answer
If a, b and c are distinct positive numbers, then the expression (b + ...
First term is less than ABC so result will be negative
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If a, b and c are distinct positive numbers, then the expression (b + c – a)(c + a – b)(a + b – c) – abc is (1986 - 2 Marks)a)positiveb)negativec)non-positived)non -negativee)none of theseCorrect answer is option 'B'. Can you explain this answer?
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