if `f (x) = |{:(x,,a,,a),(a,,x,,a),(a,,a,,x):}|=0` then

Options

A. `f(x) =0" and " f(x) =0` has one common root
B. `f(x) =0" and " f(x) =0` has one common root
C. sum of roots of f(x) =0 is -3a
D. none of these

Answer

B

Explanation

Applying `C_(1) to C_(1)+C_(2) +C_(3)` we get
`Delta = |{:(x+2a,,a,,a),(x+2a,,x,,a),(x+2a,,a,,x):}|=(x +2a) |{:(1,,a,,a),(1,,x,,a),(1,,a,,x):}|`
Applying `R_(1) to R_(2) -R_(2) " and "R_(2) to R_(2) - R_(3)` we get
`Delta =(x+2a) |{:(0,,a-x,,0),(0,,x-a,,a-x),(1,,a,,x):}|=(x-a)^(2) (x+2a)`