Question

If a, b, c are unequal then what is the condition that the value of the determinant – $\Delta = \left| \begin{matrix} a & a^{2} & a^{3} + 1 \\ b & b^{2} & b^{3} + 1 \\ c & c^{2} & c^{3} + 1 \end{matrix} \right| = 0$

• A. (a – b) (b – c)(c – a) = 0
• B. a + b + c + 1 = 0
• C. abc = –1
• D. None of these

Answer 1

Answer: (C)

abc = –1

Answer 2

a & a^{2} & a^{3} \\ b & b^{2} & b^{3} \\ c & c^{2} & c^{3} \end{matrix} \right| + \left| \begin{matrix} a & a^{2} & 1 \\ b & b^{2} & 1 \\ c & c^{2} & 1 \end{matrix} \right| = 0$$ = $\Delta = \left| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} \right|$ (abc + 1) = 0 = (a – b) (b – c) (c – a)(abc + 1) = 0, \\ abc = – 1 (\\ a ≠ b ≠ c)