If `f(x)=|a-1 0a x a-1a x^2a x a|,t h e nf(2x)-f(x)` is divisible by `a` b. `b` c.`c ,d ,e` d. none of these

Options

A. x
B. a
C. `2a+3x`
D. `x^(2)`

Answer

A, B, C

Explanation

Applying `R_(3) to R_(3) -xR_(2) " and " R_(2) to R_(2)-xR_(1)` we get
`f(x)= |{:(a,,-1,,0),(0,,a+x,,-1),(0,,0,,a+x):}|=a(a+x)^(2)`
hence
`f(2x)- f(x) =a [(a+2x)^(2) -(a-x)^(2)]`
`=a(a+2x-a-x) (a+2x+a+x)`
`=ax (2a+3x)`