MCQ
$\left| {\,\begin{array}{*{20}{c}}1&1&1\\a&b&c\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}\,} \right| = $
- A${a^3} + {b^3} + {c^3} - 3abc$
- B${a^3} + {b^3} + {c^3} + 3abc$
- ✓$(a + b + c)(a - b)(b - c)(c - a)$
- DNone of these
Hence $(a - b),\,(b - c),\,(c - a)$ are factors of $\Delta $. Since $\Delta $ is symmetric in $a,b,c $ and of $4th$ degree, $(a + b + c)$ is also a factor, so that we can write
$\Delta$=$k(a-b)(b-c)(c-a)(a+b+c) $ ......................$(i)$
Where by comparing the coefficients of the leading term $b{c^3}$ on both the sides of identity $(i).$ We get $1 = k( - 1)\,( - 1) \Rightarrow k = 1$
$\Delta$=$(a-b)(b-c)(c-a)(a+b+c) $
Trick : Put $a = 1,\,b = 2,\,c = 3$, so that determinant $\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&2&3\\1&8&{27}\end{array}\,} \right| = 1(30) - 1(24) + 1(8 - 2) = 12$
which is given by $(c)$ . i.e. $(1 + 2 + 3)\,(1 - 2)\,(2 - 3)(3 - 1) = 12$.
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