If a, b, c are distinct and aa2a3−1bb2b3−1cc2c3−1=0 ,then abc equals

# If  are distinct and then $abc$ equals

1. A
2. B
3. C
4. D

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### Solution:

Write the determinant as  $=abc{\mathrm{\Delta }}_{1}-{\mathrm{\Delta }}_{2}$

where ${\mathrm{\Delta }}_{1}=\left|\begin{array}{ccc}1& a& {a}^{2}\\ 1& b& {b}^{2}\\ 1& c& {c}^{2}\end{array}\right|=\left(a-b\right)\left(b-c\right)\left(c-a\right)$

and ${\mathrm{\Delta }}_{2}=\left|\begin{array}{ccc}a& {a}^{2}& 1\\ b& {b}^{2}& 1\\ c& {c}^{2}& 1\end{array}\right|={\mathrm{\Delta }}_{1}$

$\therefore \left(a-b\right)\left(b-c\right)\left(c-a\right)\left(abc-1\right)=0$

Since, are distinct, we get

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