Let `p(x)=x^(2)-5x" and "q(x)-3x+b`, where a and b are positive integers. Suppose hef `(p(x),q((x))=x-1" and "k(x)=1cm(p(x),q(x))`. If the coefficient of the highest degree term of k(x) is 1, the sum of the roots of (x - 1)+ k(x) is-

Options

A. 4
B. 5
C. 6
D. 7

Answer

D

Explanation

`p(x)=x^(2)-5x+a,q(x)=x^(2)-3x+b" "{"where, "a, b in N}`
hef `(p(x),q(x))=x-1`
so x - 1 is root of both `p(x) & q(x)`
`p(1)=1-5+a=0rArra=4`
`q(1)=1-3+b=0rArrb=2`
`p(x)=x^(2)-5x+4=(x-1)(x-4)`
`q(x)=x^(2)-3x+2=(x-1)(x-2)`
`k(x)=1cm(p(c),q(x))=(x-1)(x-2)(x-4)`
`(x-1)+k(x)=(x-1)+(x-1)(x-2)(x-4)`
`=(x-1)[1+x^(2)-6x+8]`
`=(x-1)(x^(2)-6x+9)`
`=(x-1)(x-3)^(2)`
roots 1, 3, 3
Sum of roots = 1 + 3 + = 7