Question:
If $ a, b $ and $ c$ are distinct positive numbers, then the expression $(b + c - a ) (c+ a - 6) (a + b - c) - abc$ is
If $ a, b $ and $ c$ are distinct positive numbers, then the expression $(b + c - a ) (c+ a - 6) (a + b - c) - abc$ is
Updated On: June 02, 2025
- positive
- negative
- non-positive
- non-negative
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The Correct Option is B
Solution and Explanation
The correct answer is B:Negative
Given that;
\(a,b,c\) are distinct positive numbers
By using \(AM≥GM\)
\(x+y≥2\sqrt{xy}\)
\(y+z≥2\sqrt{yz}\)
\(z+x≥2\sqrt{zx}\)
So, \((x+y)(y+z)(z+x)≥2\sqrt{xy}.2\sqrt{yz}.2\sqrt{zx}\)
⇒\((x+y)((y+z)(z+x)≥8(xyz)\)
Now let \(x=b+c-a,\space y=c+a-b,\space z=a+b-c\)
or \(x+y=2c,\space y+z=2a,\space z+x=2b\)
\(\therefore (2a)(2b)(2c)≥8(b+c-a)(c+a-b)(a+b-c)\)
⇒\((b+c-a)(c+a-b)(a+b-c)-abc≤0\space (negative)\)
Given that;
\(a,b,c\) are distinct positive numbers
By using \(AM≥GM\)
\(x+y≥2\sqrt{xy}\)
\(y+z≥2\sqrt{yz}\)
\(z+x≥2\sqrt{zx}\)
So, \((x+y)(y+z)(z+x)≥2\sqrt{xy}.2\sqrt{yz}.2\sqrt{zx}\)
⇒\((x+y)((y+z)(z+x)≥8(xyz)\)
Now let \(x=b+c-a,\space y=c+a-b,\space z=a+b-c\)
or \(x+y=2c,\space y+z=2a,\space z+x=2b\)
\(\therefore (2a)(2b)(2c)≥8(b+c-a)(c+a-b)(a+b-c)\)
⇒\((b+c-a)(c+a-b)(a+b-c)-abc≤0\space (negative)\)
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Concepts Used:
Geometric Progression
What is Geometric Sequence?
A geometric progression is the sequence, in which each term is varied by another by a common ratio. The next term of the sequence is produced when we multiply a constant to the previous term. It is represented by: a, ar1, ar2, ar3, ar4, and so on.
Properties of Geometric Progression (GP)
Important properties of GP are as follows:
- Three non-zero terms a, b, c are in GP if b2 = ac
- In a GP,
Three consecutive terms are as a/r, a, ar
Four consecutive terms are as a/r3, a/r, ar, ar3 - In a finite GP, the product of the terms equidistant from the beginning and the end term is the same that means, t1.tn = t2.tn-1 = t3.tn-2 = …..
- If each term of a GP is multiplied or divided by a non-zero constant, then the resulting sequence is also a GP with a common ratio
- The product and quotient of two GP’s is again a GP
- If each term of a GP is raised to power by the same non-zero quantity, the resultant sequence is also a GP.
If a1, a2, a3,… is a GP of positive terms then log a1, log a2, log a3,… is an AP (arithmetic progression) and vice versa