Question

If ∆ = $\left| \begin{matrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{matrix} \right| = \left| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right|$ then

• A. ∆ = (a-b) (b-c) (c-a)
• B. a, b, c are in G.P.
• C. b, c, a are in G.P.
• D. a, c, b are in G.P.

Answer 1

Answer: (A)

∆ = (a-b) (b-c) (c-a)

Answer 2

Here ∆ = $\left| ⥄ ⥄ \begin{matrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{matrix} ⥄ ⥄ \right| = \frac{1}{abc}\left| \begin{matrix} a & a^{2} & abc \\ b & b^{2} & abc \\ c & c^{2} & abc \end{matrix} \right|$= $\left| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right|$

= (a - b) (b – c) (c – a).