Find the following:-

From given information:

f(x)-2=k(x-1)  Hence f(1)=2

and f(x)-1==k'(x-2), Hence f(2)=1

Let remainder be ax+b when divided by  (x - 1)(x - 2).

We have:

f(x)=l(x - 1)(x - 2).+ax+b

Putting x=1 and 2 respectively we get,

f(1)=a+b=2

and f(2)=2a+b=1

Solving,a=-1 and b=3

Hence remainder is -x+3.