Sequences And Series Ques 10
- If $a, b$ and $c$ are distinct positive numbers, then the expression $(b+c-a)(c+a-b)(a+b-c)-a b c$ is
(1991, 2M)
(a) positive
(b) negative
(c) non-positive
(d) non-negative
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Answer:
Correct Answer: 10.(b)
Solution: (b) Since, $\mathrm{AM}>\mathrm{GM}$
$\therefore \frac{(b+c-a)+(c+a-b)}{2}>(b+c-a)(c+a-b)^{1 / 2}$
$\Rightarrow \quad c>[(b+c-a)(c+a-b)]^{1 / 2}$ $\quad$ ……..(i)
Similarly $ \quad b>[(a+b-c)(b+c-a)]^{1 / 2} $ $\quad$ ……..(ii)
and $ \quad a>[(a+b-c)(c+a-b)]^{1 / 2} $ $\quad$ ……..(iii)
On multiplying Eqs. (i), (ii) and (iii), we get
$ a b c>(a+b-c)(b+c-a)(c+a-b) $
Hence, $(a+b-c)(b+c-a)(c+a-b)-a b c<0$
BETA
