Question

Assertion (A) : If $a,b,c$ are distinct positive real numbers such that $a^2+b^2+c^2=1$, then$\Sigma ab=2$
Reason (R) : $A.M.\ge G.M$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true

Answer 1

Answer: (D)

$A.M.\ge G.M$ as $a,b,c$ are distinct real numbers
$\therefore \frac{a^2+b^2}{2}>\sqrt{a^2{b}^2}ie{a}^2+b^2>2ab...\left(i\right)$
Similarly $b^2+c^2>2bc.....\left(ii\right)$
$\&c^2+a^2>2ac....\left(iii\right)$
Now by adding (i), (ii) (iii), we get
$2\left(a^2+b^2+c^2\right)>2\left(ab+bc+ca\right)$
$\Rightarrow ab+bc+ca<1$ (using $a^2+b^2+c^2=1$)
$\therefore$ Assertion (A) is false & Reason (R) is true.