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If f(x)=|(a, -1, 0), (ax, a,-1 ) ,(ax^(2...

If `f(x)`=`|(a, -1, 0), (ax, a,-1 ) ,(ax^(2), ax, a)|`, then `f(2x)-f(x)` is divisible by

A

`x`

B

`a`

C

`2a+3x`

D

`x^(2)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
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)`
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