Question

If the value of the determinant $\left| \begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{matrix} \right|$ is positive, then

• A. abc> 1
• B. abc > -8
• C. abc < -8
• D. abc > -2

Answer 1

Answer: (B)

abc > -8

Answer 2

Let ∆ = $\left| \begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{matrix} \right|$ $= abc + 2 - a - b - c > 0$

or $abc + 2 > a + b + c$ …… (1)

∴A.M.> G.M.

⇒ $\frac{a + b + c}{3}(abc)^{1/3}$

∴a+b+c > 3 $(abc)^{1/3}$ …… (II)

From (1) and (2) then abc + 2>3 $(abc)^{1/3}$

let $(abc)^{1/3}$=x then $x^{3} + 2 > 3x$

⇒ $(x - 1)^{2}(x + 2) > 0$

∴x+2 > 0

⇒ x> -2 ; abc > - 8