Question

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

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

Answer 1

The correct option is A 1+abc = 0

Splitting the determinant into two determinants, we get $\Delta =\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|+abc\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=0=(1+abc)\left[\right(a-b\left)\right(b-c\left)\right(c-a\left)\right]=0$
Because a, b, c are different, the second factor cannot be zero. Hence, option (a), 1+abc = 0, is correct.