Question

If a + b + c > 0 and ∆ = $\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right|$ , then

• A. ∆< 0
• B. ∆≤ 0
• C. ∆> 0
• D. ∆ = 0

Answer 1

Answer: (B)

∆≤ 0

Answer 2

∆ = – (a + b + c) (a² + b² + c² – ab – bc – ac)

we know $\frac{a^{2} + b^{2}}{2}$≥ ab (by AM ≥ GM)

$\frac{b^{2} + c^{2}}{2}$ ≥ bc ⇒ $\frac{a^{2} + c^{2}}{2}$ ≥ ac ⇒ a² + b² + c² ≥ ab + bc + ac

a² + b² + c² – ab – bc – ac ≥ 0

Now, ∆ = – ( +) ( + ) ∆ ≤ 0