Evaluate: `|(1,a ,a^2), (1,b,b^2) ,(1,c,c^2)|` .

Options

A. `(a -b) (b -c) (c -a)`
B. `(a^(2) -b^(2)) (b^(2) -c^(2)) (c^(2) -a^(2))`
C. `(a -b+ c) (b -c +a) (c -a +b)`
D. none of these

Answer

A

Explanation

We have,
`Delta = |(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2))|`
`rArr Delta = |(1,a,a^(2)),(0,b -a,b^(2) -a^(2)),(0,c -a,c^(2) -a^(2))|` Applying `R_(2) rarr R_(2) - R_(1) and R_(3) rarr R_(3) - R_(1)`
`rArr Delta = (b -a) (c -a) |(1,a,a^(2)),(0,1,b +a),(0,1,c +a)|` Taking `(b -c) and (c -a)` common from `R_(2) and R_(3)` respectively
`rArr Delta = -(b -a) (c -a) (c-b) = (a -b) (b -c) (c -a)`