Question

If f(x) = \(\begin{vmatrix}1 & a & bc \\1 & b & ca \\1 & c & ab \end {vmatrix}\) = \(\begin{vmatrix}1 & a & a^2 \\1 & b & b^2 \\1 & c & c^2 \end {vmatrix}\) then which one among the following is correct?

(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

I got this question by my college professor while I was bunking the class.

This question is from Application of Determinants in portion Determinants of Mathematics – Class 12

Answer 1

Correct choice is (b) a, b, c are in G.P

Easiest explanation: Here, f(x) = \(\begin{vmatrix}1 & a & bc \\1 & b & ca \\1 & c & ab \end {vmatrix}\)

Multiplying and diving by abc,

= (1/abc) \(\begin{vmatrix}a & a^2 & abc \\b & b^2 & abc \\c & c^2& abc \end {vmatrix}\)  

= \(\begin{vmatrix}1 & a & a^2 \\1 & b & b^2 \\1 & c & c^2 \end {vmatrix}\)

= (a – b)(b – c)(c – a)