গণিত প্রশ্ন

AI-তৈরি নমুনা উত্তর

L.H.S.
$= \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix}$

Applying column operations $C_1' = C_1 - C_2$ and $C_2' = C_2 - C_3$:
$= \begin{vmatrix} 0 & 0 & 1 \\ a-b & b-c & c \\ a^3-b^3 & b^3-c^3 & c^3 \end{vmatrix}$

Taking $(a-b)$ common from the first column and $(b-c)$ common from the second column:
$= (a-b)(b-c) \begin{vmatrix} 0 & 0 & 1 \\ 1 & 1 & c \\ a^2+ab+b^2 & b^2+bc+c^2 & c^3 \end{vmatrix}$

Expanding along the first row:
$= (a-b)(b-c) \cdot 1 \cdot [1 \cdot (b^2+bc+c^2) - 1 \cdot (a^2+ab+b^2)]$
$= (a-b)(b-c) [b^2 + bc + c^2 - a^2 - ab - b^2]$
$= (a-b)(b-c) [(c^2 - a^2) + b(c - a)]$
$= (a-b)(b-c) [(c-a)(c+a) + b(c-a)]$
$= (a-b)(b-c)(c-a)(a+b+c) = \text{R.H.S. (Proved)}$