Consider the function f(x) = `|{:(a^(2)+x,,ab,,ac),(ab,,b^(2)+x,,bc),(ac,,bc,,c^(2)+x):}|`
Which of the following is true ?

Options

A. f(x) =0 and f(x) =0 have one positive common root
B. f(x)=0 and f(x)=0 have one negative common root
C. f(x) =0 and f(x) =0 have no common root
D. None of these

Answer

D

Explanation

`Delta =(1)/(a) |{:(a^(3)+ax,,ab,,ac),(a^(2)b,,b^(2)+x,,bc),(a^(2)c,,bc,,c^(2)+x):}|`
Applying `C_(1) to C_(1)+bC_(2) +cC_(2)` and taking `a^(2) +b^(2)+c^(2)+x` common we get
`Delta =(1)/(a)(a^(2)+b^(2)+c^(2)+x) |{:(a,,ab,,ac),(b,,b^(2)+x,,bc),(c,,bc,,c^(2)+x):}|`
Applying `C_(2) to C_(2)-bC_(1) " and " C_(3) to C_(3)-cC_(1)` we get
`Delta =(1)/(a)(a^(2)+b^(2)+c^(2)+x) |{:(a,,0,,0),(b,,x,,0),(c,,0,,x):}|`
`=(1)/(a) (a^(2) +b^(2)+c^(2)+x) (ax^(2))`
`=x^(2) (a^(2)+b^(2)+c^(2)+x)`
Thus `Delta ` is divisib le by x and `x^(2)`. also graph of f(x) is