Question

Consider the system of equations
$(a-1)x-y-z=0,$ $x-(b-1)y+z=0,$ $x+y-(c-1)z=0$ Where $a,b$ and $c$ are non-zero real numbers.
Statement-1: if $x,y,z$ are not all zero, then $ab+bc+ca=abc$
Statement-2: $abc\ge 27$

• A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
• B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
• C. Statement-1 is True, Statement-2 is False.
• D. Statement-1 is False, Statement-2 is True.

Answer 1

Answer: (B)

Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1

Given system of equation can be written as
$AX=O$
where$A=\left[\begin{array}{ccc}a-1& -1& -1\\ 1& -(b-1)& 1\\ 1& 1& -(c-1)\end{array}\right]$
Now, since, x,y,z are not all zero,so for the consistency of system of equations
$|A|=0$
$\left|\begin{array}{ccc}a-1& -1& -1\\ 1& -(b-1)& 1\\ 1& 1& -(c-1)\end{array}\right|=0$
$\Rightarrow (a-1)bc-ab-bc-ac=0$
$\Rightarrow abc=ab+bc+ac$
Hence, statement 1 is true
Now, since, $\Delta >0$
$\Rightarrow abc>ab+bc+ac$ ....(1)
Since, $AM\ge GM$
$\Rightarrow \frac{ab+bc+ac}{3}\ge (abc{)}^{2/3}$
$\Rightarrow ab+bc+ca\ge 3(abc{)}^{2/3}$ ....(2)
From (1) and (2),
$abc>3(abc{)}^{2/3}$
$(abc{)}^{1/3}\ge 3$
$\Rightarrow abc\ge 27$
Hence, statement 2 is true.
But it is not the correct explanation of statement 1