Question

If a² + b² + c² + ab + bc + ca ≤ 0 ∀ a, b, c ∈ R, then value of the determinant $\left| \begin{matrix} (a + b + 2)^{2} & a^{2} + b^{2} & 1 \\ 1 & (b + c + 2)^{2} & b^{2} + c^{2} \\ c^{2} + a^{2} & 1 & (c + a + 2)^{2} \end{matrix} \right|$

Equals

• A. 65
• B. a² + b² + c² + 31
• C. 4(a² + b² + c²)
• D. 0

Answer 1

Answer: (A)

65

Answer 2

a² +b² + c² + ab + bc + ca ≤ 0

2a² + 2b² + 2c² + 2ab + 2bc + 2ca ≤ 0

(a + b)² + (b + c)² + (c + a)² ≤ 0 ⇒ a + b = b + c = c + a = 0 ⇒ a = b = c = 0

∆ = $\left| \begin{matrix} 4 & 0 & 1 \\ 1 & 4 & 0 \\ 0 & 1 & 4 \end{matrix} \right|$= 65