Roots of the equations `|{:(x,,m,,n,,1),(a,,x,,n,,1),(a,,b,,x,,1),(a,,b,,c,,1):}|=0` are

Options

A. independent of m and n
B. independent of a,b and c
C. depend on m,n and a,b,c
D. inedependent of m,n and a,b,c

Answer

A

Explanation

`|{:(x,,m,,n,,1),(a,,x,,n,,1),(a,,b,,x,,1),(a,,b,,c,,1):}| " "[R_(1) to R_(1) -R_(2) ,R_(2) to R_(2)-R_(3),R_(3) to R_(3) -R_(4)]`
`" or " |{:(x-a,,m-x,,0,,0),(0,,x-b,,n-x,,0),(0,,0,,x-c,,0),(a,,b,,c,,1):}|=0`
`" or "|{:(x-a,,m-x,,0),(0,,x-b,,n-x),(0,,0,,x-c):}|=0`
`" or " (x-a) |{:((x-b),,n-x),(0,,(x-c)):}|=0`
`" or " (x-a) (x-b) (x-c) =0`
Thus roots are independent of m and n.