Question

The least value of the product xyz for which the determinant |x111y1x1z| is non-negative, is
(1) –8 (2) –1 (3) -2√2 (4) -16√2

Answer 1

Your answer is  -8 ...

Answer 2

The required "option 2) – 1" is correct.

Step-by-step explanation:

We have,

$\left[\begin{array}{ccc}x&1&1\\1&y&1\\x&1&z\end{array}\right]$  non-negative

To find, the least value of the product xyz = ?

∴ $\left[\begin{array}{ccc}x&1&1\\1&y&1\\x&1&z\end{array}\right] >=0$

⇒ x(yz - 1) - 1(z-x) + 1(1 - xy) ≥ 0

⇒ xyz - x - z + x + 1 - xy ≥ 0

⇒ xyz  - z  + 1 - xy ≥ 0

⇒ xy(z - 1) -(z - 1) ≥ 0

∴ The least value of the product xyz = - 1

Hence, the required "option 2) – 1" is correct.