If a, b, c are mutually unequal real numbers, then the value of \(\frac{\begin{vmatrix} 1 & a & a^3\\ 1 & b & b^3\\ 1 & c & c^3 \end{vmatrix}}{\begin{vmatrix} 1 & a & a^2\\ 1 & b & b^2\\ 1 & c & c^2 \end{vmatrix}}  = \)

a + b + c

\(\frac{\begin{vmatrix} 1 & a & a^3\\ 0 & b-a & b^3-a^3\\ 0 & c-a & c^3-a^3 \end{vmatrix}}{\begin{vmatrix} 1 & a & a^2\\ 0 & b-a & b^2-a^2\\ 0 & c-a & c^2-a^2 \end{vmatrix}}\)

\(\frac{(b-a)(c-a)\begin{vmatrix} 1 & a & a^3\\ 0 & 1 & b^2-a^2+ab\\ 0 & 1 & c^2-a^2+ac \end{vmatrix}}{(b-a)(c-a)\begin{vmatrix} 1 & a & a^2\\ 0 & 1 & b+a\\ 0 & 1 & c+a \end{vmatrix}}  =\frac{c^2+ac-b^2-ab}{(c-b)} \)

$=(c+b)+a$

$=a+b+c$

Option 2 is correct.